Purpose: To use the ballistic pendulum to determine the initial velocity of a projectile using conservation of momentum and conservation of energy.
Equipment: Ballistic pendulum, carbon paper, meter stick, clamp, box, triple beam balance, plumb.
Introduction: In this experiment a steel ball will be shot into the bob of a pendulum and the height, h, to which the pendulum bob moves, as shown in Figure 1, will determine the initial velocity, V, of the bob after it receives the moving ball. If we equate the kinetic energy of the bob and ball at the bottom to the potential energy of the bob and ball at the height, h, that they are raised to, we get:
( K.E ) bottom = ( P.E)top
½ ( M + m ) V² = ( M + m ) g.h
where M is the mass of the pendulum and m is the mass of the ball. Solving for V we get:
V = √ 2gh ……………. ( 1 )
Using conservation of momentum we know the momentum before impact (collision) should be the same as the momentum after impact. Therefore:
Pi = Pf
or
mv0 = ( M + m) V …………… ( 2 )
where v0 is the initial velocity of the ball before impact. By using equations (1) and (2) we can therefore find the initial velocity, v0, of the ball.
We can also determine the initial velocity of the ball by shooting the ball as above but this time allowing the ball to miss the pendulum bob and travel horizontally under the influence of gravity. In this case we simply have a projectile problem where we can measure the distance traveled horizontally and vertically (see Figure 2 from the ballistic pendulum at http://www.hartnell.edu/physics/labs4a.htm) and then determine the initial velocity, v0, of the ball.
M + m Starting with equations:
∆x = voxt + ½axt² ……………… ( 3 )
∆y = voyt + ½ayt²
……………… ( 4 )
You should be able to derive the initial velocity of the ball in the horizontal direction (assuming that
∆x and ∆y are known). Include this derivation in your lab report.
Procedure:
Part I Determination of Initial Velocity from Conservation of Energy
1. Set the apparatus near one edge of the table as shown in figure 2. Make sure that the base is accurately horizontal, as shown by a level. Clamp the frame to the table. To make the gun ready for shooting, rest the pendulum on the rack, put the ball in position on the end of the rod and, holding the base with one hand, pull back on the ball with the other until the collar on the rod engages the trigger. This compresses the spring a definite amount, and the ball is given the same initial velocity every time the gun is shot.
2. Release the pendulum from the rack and allow it to hang freely. When the pendulum is at rest, pull the trigger, thereby propelling the ball into the pendulum bob with a definite velocity. This causes the pendulum to swing from a vertical position to an inclined position with the pawl engaged in some particular tooth of the rack.
3. Shoot the ball into the cylinder about nine times, recording each point on the rack at which the pendulum comes to rest. This in general will not be exactly the same for all cases but may vary by several teeth of the rack. The mean of these observations gives the mean highest position of the pendulum. Raise the pendulum until its pawl is engaged in the tooth corresponding most closely to the mean value and measure h1, the elevation above the surface of the base of the index point for the center of gravity. Next release the pendulum and allow it to hang in its lower most position and measure h2. The difference between these two values gives h, the vertical distance through which the center of gravity of the system is raised after shooting the ball. Record h:
4. Carefully remove the pendulum from its support. Weigh and record the masses of the pendulum and of the ball. Replace the pendulum and carefully adjust the thumb screw.
5. From these data calculate the initial velocity v using equations (1) and (2). Part II Determination of Initial Velocity from Measurements of Range and Fall
1. To obtain the data for this part of the experiment the pendulum is positioned up on the rack so that it will not interfere with the free flight of the ball. One observer should watch carefully to determine the point at which the ball strikes the floor. The measurements in this part of the experiment are made with reference to this point and the point of departure of the ball. Clamp the frame to the table, as it is important that the apparatus not be moved until the measurements have been completed. A piece of paper taped to the floor at the proper place and covered with carbon paper will help in the exact determination of the spot at which the ball strikes the floor.
Note: Use caution in shooting the gun.
2. Shoot the ball a number of times, noting each time the point at which it strikes the floor. Determine, relative to the edge of the paper, the average position of impact of the ball. Determine the distances ∆x using this average position on the floor. From the the values of ∆x and ∆y calculate v0 by the use of equations (3) and (4). Make careful sketches in your lab report that clearly show all of the distances involved.
3. Find the percentage difference between the values of v0 determined by the two methods in parts I and II. Try to analyze, the probable errors of the two methods and estimate which one should give the more accurate result.
Note: Before leaving the apparatus, put the ball on its peg and be sure that the spring gun is released.
Conclusions:
I learned that the conservation of energy theorem applies accurately when all external and internal forces are taken into consideration such as friction and centripetal force. Considering dK = W and dK = -Ug, W = -Ug. K = .5(mv^2) and Ug = mgh. Since dK = Kf - Ki, we have that W = Kf - Ki = Ug = -mgh.
Sources of error are strictly through human error.
Sunday, December 16, 2012
Centripetal Force
Purpose: To verify Newton’s second law of motion for the case of uniform circular
motion.
Equipment: Centripetal force apparatus, metric scale, vernier caliper, stop watch,
slotted weight set, weight hanger, triple beam balance.
Introduction:
The centripetal force apparatus is designed to rotate a known mass trough
a circular path of known radius. By timing the motion for a definite
number of revolutions and knowing the total distance that the mass has
traveled, the velocity can be calculated. Thus the centripetal force, F,
necessary to cause the mass to follow its circular path can be
determined from Newton’s second law.
r
Where m is the mass, v is the velocity, and r is the radius of the circular
path.
Here we have used the fact that for uniform circular motion, the
acceleration, a, is given by:
Procedure:
1. For each trial the position of the horizontal crossarm and the vertical indicator
post must be such that the mass hangs freely over the post when the spring is
detached. After making this adjustment, connect the spring to the mass and
practice aligning the bottom of the hanging mass with the indicator post while
rotating the assembly.
2. Measure the time for 50 revolutions of the apparatus. Keep the velocity as
constant as possible by keeping the pointer on the bottom of the mass aligned with
the indicator post. A white sheet of paper placed as a background behind the
apparatus can be helpful in getting the alignment as close as possible. Using the
same mass and radius, measure the time for three different trials. Record all data
in a neat excel table (see 6).
3. Using the average time obtained above, calculate the velocity of the mass. From
this calculate the centripetal force exerted on the mass during its motion.
F = (mv^2)/r
a(c) = (v^2)/r
4. Independently determine the centripetal force by attaching a hanging weight to
the mass until it once again is positioned over the indicator post (this time at rest).
Since the spring is being stretched by the same amount as when the apparatus was
rotating, the force stretching the spring should be the same in each case.
a. Calculate this force and compare with the centripetal force obtained
in part 3 by finding the percent difference.
b. Draw a force diagram for the hanging weight and draw a force
diagram for the spring attached to the hanging mass:
5. Add 100 g to the mass and repeat steps 2, 3 and 4 above.
6. The following data should be calculated and recorded in your excel table:
a. Mass and radius for each trial.
b. Average number of revolutions/sec (frequency) for each trial.
c. Linear speed for each trial.
d. Calculated and measured centripetal force for each trial and their percent
difference.
Conclusions:
I learned that in order to calculate the centripetal force of an object undergoing circular motion, the formula F = (mv^2)/r must be used where m = mass, v = linear velocity (tangential), and r = radius. I also learned that when extra mass is added to the system but centripetal force and the radius are constant, then the deciding factor must be that the velocity depreciated, that is, the velocity decreased to account for equilibrium on both sides of the equation.
Possible sources of error are of course resultant from human interference and quite possibly significant figure usage during calculations.
motion.
Equipment: Centripetal force apparatus, metric scale, vernier caliper, stop watch,
slotted weight set, weight hanger, triple beam balance.
Introduction:
The centripetal force apparatus is designed to rotate a known mass trough
a circular path of known radius. By timing the motion for a definite
number of revolutions and knowing the total distance that the mass has
traveled, the velocity can be calculated. Thus the centripetal force, F,
necessary to cause the mass to follow its circular path can be
determined from Newton’s second law.
r
Where m is the mass, v is the velocity, and r is the radius of the circular
path.
Here we have used the fact that for uniform circular motion, the
acceleration, a, is given by:
Procedure:
1. For each trial the position of the horizontal crossarm and the vertical indicator
post must be such that the mass hangs freely over the post when the spring is
detached. After making this adjustment, connect the spring to the mass and
practice aligning the bottom of the hanging mass with the indicator post while
rotating the assembly.
2. Measure the time for 50 revolutions of the apparatus. Keep the velocity as
constant as possible by keeping the pointer on the bottom of the mass aligned with
the indicator post. A white sheet of paper placed as a background behind the
apparatus can be helpful in getting the alignment as close as possible. Using the
same mass and radius, measure the time for three different trials. Record all data
in a neat excel table (see 6).
3. Using the average time obtained above, calculate the velocity of the mass. From
this calculate the centripetal force exerted on the mass during its motion.
F = (mv^2)/r
a(c) = (v^2)/r
4. Independently determine the centripetal force by attaching a hanging weight to
the mass until it once again is positioned over the indicator post (this time at rest).
Since the spring is being stretched by the same amount as when the apparatus was
rotating, the force stretching the spring should be the same in each case.
a. Calculate this force and compare with the centripetal force obtained
in part 3 by finding the percent difference.
b. Draw a force diagram for the hanging weight and draw a force
diagram for the spring attached to the hanging mass:
5. Add 100 g to the mass and repeat steps 2, 3 and 4 above.
6. The following data should be calculated and recorded in your excel table:
a. Mass and radius for each trial.
b. Average number of revolutions/sec (frequency) for each trial.
c. Linear speed for each trial.
d. Calculated and measured centripetal force for each trial and their percent
difference.
Conclusions:
I learned that in order to calculate the centripetal force of an object undergoing circular motion, the formula F = (mv^2)/r must be used where m = mass, v = linear velocity (tangential), and r = radius. I also learned that when extra mass is added to the system but centripetal force and the radius are constant, then the deciding factor must be that the velocity depreciated, that is, the velocity decreased to account for equilibrium on both sides of the equation.
Possible sources of error are of course resultant from human interference and quite possibly significant figure usage during calculations.
Projectile Simulation
Purpose: 1. To gain a better understanding of motion in two dimensions by working a projectile
problem
2. To gain experience using a computer simulation of a physical problem
Equipment Needed: windows based computer with Interactive Physics software
Introduction: The computer will be used to create a simulation of a rubber ball being thrown through a
window on the side of a building. The computer, of course, uses the laws of physics
(previously programmed into the computer) to simulate the motion. In a similar way, we
can use the laws of physics to do the calculations ourselves and predict the correct initial
velocity needed and thus check the accuracy of our simulation. The only other means of
verifying any simulation (and the best!) is doing the actual experiment, but depending on
the situation this may be difficult and/or costly.
By adjusting the initial velocity of the ball we can use the computer to repeat the
experiment over and over again until we achieve the desired result (getting the ball
through the window). The ball's initial velocity is controlled by using the slider bars for
Vx and Vy or by simply typing in values in the boxes below each slider. Be sure that the
simulation has been reset before making any changes. The components, Vx and Vy, of
the ball's velocity are displayed by digital meters. Vectors attached to the ball indicate
the ball's velocity as well as the x and y components of the velocity.
Procedure:
1. Turn on the computer and load the Interactive Physics software located within the Physics Apps
folder. Select File/Open and double click on FLYBALL file.
2. Set the initial velocity of the ball by adjusting the sliders for Vx and Vy and observe the initial
velocity vector attached to the ball change as you adjust the sliders. Run the simulation and observe
the motion of the ball. Watch the velocity vector and its components as they change throughout the
motion.
Repeat the simulation with a different initial velocity until the ball goes through the center of the
window. Record V0x, V0y, and the total elapsed time. Also locate the initial position of the ball by
moving the mouse until the cursor on the center of the ball. Read the values for x and y in the
coordinate boxes in the lower left portion of the screen. In a similar manner, determine the
coordinates of the center of the window. Make a motion diagram showing all relevant variables.
Using your motion equations, calculate what the time of flight should be and compare with the
simulation time.
V0x
V0y
∆t
x0
y0
xf
yf3. Using the player controls, step the motion back to where your projectile is at the very top of its path.
Record the values for Vx, Vy, and t at this point. From your initial velocity, calculate what the time
should be at the top. How does your calculated time compare with the simulation’s time?
4. Run the simulation using a ball speed of V0x = 6.00 m/s and V0y = 8.02 m/s. Record the horizontal
and vertical components of the velocity at various times (using the player controls), completing a
table like the one shown below. Be careful to record the sign (positive or negative) in all cases.
Time (s) Vx (m/s) Vy (m/s)
0.0 6.0 8.02
0.2
0.4
0.6
0.8
1.0
1.2
1.4
5. Using the data from the table, plot the horizontal and vertical components of the velocity, Vx and Vy,
versus time on a single graph in Graphical Analysis. Use different symbols to mark the data points
for the two velocity components. You should observe that one component of velocity changes while
the other is constant. Explain. What should the slope for each curve be?
6. Fit the graphs and find the slopes of the lines. Did they agree with your expectations? Explain.
Print a copy of the graphs for everyone in the group. 7. The horizontal and vertical positions of a projectile in free fall are given by
x = x0 + V0xt + ½axt
2
y = y0 + V0yt + ½ayt
2
Solve each of these equations algebraically for V0x and V0y.
8. Using the results from part 7, calculate V0x and V0y
for a time of flight of 1.6 s and then run the
simulation. Verify that the ball does go through the center of the window for your calculated values
of V0x and V0y . Repeat for a time of flight of 1.33 s and 0.5 s. Sketch the shape of each trajectory
and discuss the differences in the trajectories. What determines the amount of time the projectile
spends in the air?
9. Calculate the initial speed (magnitude of the velocity) and direction (angle above the horizontal) for
each of the three simulations discussed in part 8. Put the results from parts 8 and 9 in a table.
V0 θ
Conclusions:
The purpose of the projectile simulation lab was to estimate the trajectory an object would take to undergo a predetermined behavior and by mechanics of kinematics, we were to ascertain the velcocities of the x and y directions to make the object fulfil the desired behavior given initial and final positions as well as acceleration. Before this lab, I was completely confused as to how a
"split-up" kinematic equations were related and afterward I was able to relate them by solving for certain variables and interpolating them into the other equation, much like one of the rocket problems discussed during the course of the semester.
problem
2. To gain experience using a computer simulation of a physical problem
Equipment Needed: windows based computer with Interactive Physics software
Introduction: The computer will be used to create a simulation of a rubber ball being thrown through a
window on the side of a building. The computer, of course, uses the laws of physics
(previously programmed into the computer) to simulate the motion. In a similar way, we
can use the laws of physics to do the calculations ourselves and predict the correct initial
velocity needed and thus check the accuracy of our simulation. The only other means of
verifying any simulation (and the best!) is doing the actual experiment, but depending on
the situation this may be difficult and/or costly.
By adjusting the initial velocity of the ball we can use the computer to repeat the
experiment over and over again until we achieve the desired result (getting the ball
through the window). The ball's initial velocity is controlled by using the slider bars for
Vx and Vy or by simply typing in values in the boxes below each slider. Be sure that the
simulation has been reset before making any changes. The components, Vx and Vy, of
the ball's velocity are displayed by digital meters. Vectors attached to the ball indicate
the ball's velocity as well as the x and y components of the velocity.
Procedure:
1. Turn on the computer and load the Interactive Physics software located within the Physics Apps
folder. Select File/Open and double click on FLYBALL file.
2. Set the initial velocity of the ball by adjusting the sliders for Vx and Vy and observe the initial
velocity vector attached to the ball change as you adjust the sliders. Run the simulation and observe
the motion of the ball. Watch the velocity vector and its components as they change throughout the
motion.
Repeat the simulation with a different initial velocity until the ball goes through the center of the
window. Record V0x, V0y, and the total elapsed time. Also locate the initial position of the ball by
moving the mouse until the cursor on the center of the ball. Read the values for x and y in the
coordinate boxes in the lower left portion of the screen. In a similar manner, determine the
coordinates of the center of the window. Make a motion diagram showing all relevant variables.
Using your motion equations, calculate what the time of flight should be and compare with the
simulation time.
V0x
V0y
∆t
x0
y0
xf
yf3. Using the player controls, step the motion back to where your projectile is at the very top of its path.
Record the values for Vx, Vy, and t at this point. From your initial velocity, calculate what the time
should be at the top. How does your calculated time compare with the simulation’s time?
4. Run the simulation using a ball speed of V0x = 6.00 m/s and V0y = 8.02 m/s. Record the horizontal
and vertical components of the velocity at various times (using the player controls), completing a
table like the one shown below. Be careful to record the sign (positive or negative) in all cases.
Time (s) Vx (m/s) Vy (m/s)
0.0 6.0 8.02
0.2
0.4
0.6
0.8
1.0
1.2
1.4
5. Using the data from the table, plot the horizontal and vertical components of the velocity, Vx and Vy,
versus time on a single graph in Graphical Analysis. Use different symbols to mark the data points
for the two velocity components. You should observe that one component of velocity changes while
the other is constant. Explain. What should the slope for each curve be?
6. Fit the graphs and find the slopes of the lines. Did they agree with your expectations? Explain.
Print a copy of the graphs for everyone in the group. 7. The horizontal and vertical positions of a projectile in free fall are given by
x = x0 + V0xt + ½axt
2
y = y0 + V0yt + ½ayt
2
Solve each of these equations algebraically for V0x and V0y.
8. Using the results from part 7, calculate V0x and V0y
for a time of flight of 1.6 s and then run the
simulation. Verify that the ball does go through the center of the window for your calculated values
of V0x and V0y . Repeat for a time of flight of 1.33 s and 0.5 s. Sketch the shape of each trajectory
and discuss the differences in the trajectories. What determines the amount of time the projectile
spends in the air?
9. Calculate the initial speed (magnitude of the velocity) and direction (angle above the horizontal) for
each of the three simulations discussed in part 8. Put the results from parts 8 and 9 in a table.
V0 θ
Conclusions:
The purpose of the projectile simulation lab was to estimate the trajectory an object would take to undergo a predetermined behavior and by mechanics of kinematics, we were to ascertain the velcocities of the x and y directions to make the object fulfil the desired behavior given initial and final positions as well as acceleration. Before this lab, I was completely confused as to how a
"split-up" kinematic equations were related and afterward I was able to relate them by solving for certain variables and interpolating them into the other equation, much like one of the rocket problems discussed during the course of the semester.
Hooke’s Law and the Simple Harmonic Motion of a Spring
Hooke’s Law and the Simple Harmonic Motion of a Spring
Purpose: To determine the force constant of a spring and to study the motion of a spring and mass
when vibrating under influence of gravity.
Equipment: Spring, masses, weight hanger, meter stick, support stand with clamps, motion detector,
LabPro interface, wire basket.
Introduction: When a spring is stretched a distance x from its equilibrium position, it will exert a restoring
force directly proportional to this distance. We write this restoring force, F, as:
1) F = -kx
where k is the spring constant and depends on the stiffness of the spring. The minus
sign remind us that the direction of the force is opposite to the displacement.
Equation 1 is valid for most springs and is called Hooke’s Law.
If a mass is attached to a spring that is hung vertically, and the mass is pulled down
and released, the spring and the mass will oscillate about the original point of
equilibrium. Using Newton’s second law and some calculus we can show that the
motion is periodic (repeats itself over and over) and has period, T (in sec), given by
2) T =
4
where m is the mass supported by the spring.
table
table clamp
spring
hanging
mass
support rods
π
2
k
m
Procedure:
1. Hang the spring on the support rod, as shown in the diagram, and measure the position of the lower end
of the spring. Place 350 gm mass on the spring and observe its position. Now attach, in turn, masses of
450, 550, 650, 750, 850, 950, 1050 gm and measure how far the spring is stretched for each of these
masses.
2. Make a plot of the downward force applied to the spring (y-axis) versus the displacement of the spring (xaxis). Remembering Equation 1, determine the force constant, k.
3. Start up the Logger Pro software by clicking on its icon in the Physics Apps folder. From the program
click on Open/Mechanics/Hooke’s Law to open the file for this experiment. A graph of linear position
vs time should appear. Place the motion detector on the floor beneath the hanging mass. Place the wire
basket over the motion detector to protect it from any accidentally dropped masses. Once again place the
350 gm mass on the spring and pull the mass downward until the spring has been stretched 10 cm.
Release the mass and observe the subsequent motion. Start collecting data by clicking on the Collect
button. The time scale on your graph should allow for at least five cycles of the motion to be seen. Press
the “x=” button and determine the time for five cycles. From this number calculate the period of motion.
Record your data. Repeat this for the other masses used in part 1. Create a data table which gives
average T and m values.
4. Using the data for the last trial (with m = 1050 g), fit the data to a sinusoidal function using the
Analyze/Curve Fit option. Determine the period and the amplitude from your function. Compare the
period with the value obtained in part 3.
5. Make a plot of T2 (y-axis) vs m (x-axis).
Conclusion:
The whole point of Hooke's Law in this experiment was to understand the conservation of momentum, potential gravitational energy, potential spring energy, and the conservation of energy with regards to projectile motion. Since Hooke's Law is F=-kx where k is the spring constant and x is the displacement of the spring due to compression or extension, it's understood that the force is in the opposite direction of the displacement because of the signs in the equation.
Inelastic Collision
INELASTIC COLLISIONS
Purpose:
To analyze the motion of two low friction carts during an inelastic collision and verify that the law of
conservation of linear momentum is obeyed.
Equipment:
Computer with Logger Pro software, lab pro, motion detector, horizontal track, two carts, 500 g
masses(3), triple beam balance, bubble level
Introduction:
This experiment uses the carts and track as shown in the figure. If we regard the system of the two
carts as an isolated system, the momentum of this system will be conserved. If the two carts have
a perfectly inelastic collision, that is, stick together after the collision, the law of conservation of
momentum says
Pi = Pf
m1v1 + m2v2 = (m1 + m2)V
where v1 and v2 are the velocities before the collision and V is the velocity of the combined mass
after the collision.
Procedure:
1. Set up the apparatus as shown in Figure 1. Use the bubble level to verify that the track is as
level as possible. Record the mass of each cart. Connect the lab pro to the computer and the
motion detector to the lab pro. On the computer, start the Logger Pro software, open the
Mechanics folder and the Graphlab file.
2. First, check to see that the motion detector is working properly by clicking the Collect button to
start collecting data. Move the cart nearest the detector back and forth a few times while
observing the position vs time graph being drawn by the computer. Does it provide a
reasonable graph of the motion of the cart? Remember to be aware of unwanted reflections
caused by objects in between the motion detector and the cart. Also, position the carts so that
their velcro pads are facing each other. This will insure that they will stick together after the
collision.
TABLE
carts
computer
motion
detector
track m1 m2
v13. With the second cart (m2) at rest give the first cart (m1) a moderate push away from the motion
detector and towards m2. Observe the position vs time graph before and after the collision.
What should these graphs look like? Draw an example:
The slope of the position vs. time graph directly before and directly after the collision give the
velocity directly before and directly after the collision. To avoid the problem of dealing with
friction forces (Remember, we are assuming the system is isolated.), we will find the velocity of
the carts at the instant before and after the collision.
Is this a good approximation? Why or why not?
For the velocity before the collision, select a very small range of data points just before the
collision. Avoid the portion of the curve which represents the collision. Choose
Analyze/Linear Fit. Record the slope (velocity) of this line. Repeat for a very small range of
data points just after the collision. Record this slope (velocity) as well.
4. Repeat for two more collisions. Calculate the momentum of the system the instant before and
after the collision for each trial and find the percent difference. Put your results in an Excel data
table. Show sample calculations here:
5. Place an extra 500 g on the second cart and repeat steps 3 and 4. Sketch one representative
graph showing the position vs time for a typical collision. (What do velocity vs. time and
acceleration vs. time look like?6. Remove the 500 g from the second cart and place it on the first cart. Repeat steps 3 and 4.
7. Find the average of all of the percent differences found above. This average represents your
verification of the law of conservation of linear momentum. How well is the law obeyed based
on the results of your experiment? Explain.
8. For each of the nine trials above calculate the kinetic energy of the system before and after the
collision. Find the percent kinetic energy lost during each collision. Put this information in a
separate data table. Show sample calculations here:
9. Do a theoretical calculation for ΔK/K in a perfectly inelastic collision for the three situations:
1. a mass, m, colliding with an identical mass, m, initially at rest.
2. a mass, 2m, colliding with a mass, m, initially at rest.
3. a mass, m, colliding with a mass, 2m, initially at rest.
Conclusions:
I learned that when a perfectly inelastic collision occurs, momentum is conserved. When momentum is conserved, the change in momentum is 0. For this to occur, the initial momentum must be equal to the final momentum [m1*v1 = (m1+m2)*v2]. The only way this is possible is for the velocity of the combined masses must decrease to compensate for the additional weight. If momentum is conserved, there must have been some ΔK because the velocity is going to be squared and therefore if momentum is negative, the resulting velocity had to be negative which is negated by squaring the velocity, therefore kinetic energy has to always be positive (that does not mean that you can't have a negative ΔK because ΔK = K(final)-K(initial)).
Human Power
Objective: To determine the power output of a person.
Equipment: Two meter sticks, stopwatch, kilogram bathroom scale
Power is defined to be the rate at which work is done or equivalently, the rate at which energy is converted from one form to another. In this experiment you will do some work by climbing from the first floor of the science building to the second floor. By measuring the vertical height climbed and knowing your mass, the change in your gravitational potential energy can be found with the formula: ∆ PE = mgh (where m is the mass, g the acceleration of gravity, and h is the vertical height gained) Your power output can be determined by Power = ∆ PE/∆t (where ∆t is the time to climb the vertical height h.)
Procedure:
1. Determine your mass by weighing on the kilogram bathroom scale. Record your mass in kg..
2. Measure the vertical distance between the ground floor and the second floor for the science building. This can most easily be done by using two meter long meter sticks held end to end in the stairwell at the west end of the building. Make a careful sketch of the stairwell area that explains the method used to determine this height.
3. Designate a record keeper and a timer for the class. At the command of the timing person, run or walk (whatever you feel comfortable doing) up the stairs from the ground floor to the second floor. Be sure that your name and time are recorded by the record keeper.
4. After everyone in the class has completed one trip up the stairs, repeat for one more trial.
5. Return to class and calculate your personal power output in watts using the data collected from each of your climbing trip up the stairs. Obtain the average power output from the two trials.
6. Put your average power on the board and then calculate the average power for the entire class once everyone has reported their numbers on the board.
7. Determine your average power output in units of horsepower.
Questions:
Is it okay to use your hands and arms on the hand-railing to assist you in your climb up the stairs? Explain why or why not.
No, it is not acceptable to use your hands and arms to propel yourself up the flight of stairs because then it is not your power alone that is allowing your vertical position displacement to occur. Although it is true that it takes energy to pull yourself up, you are still transferring kinetic energy into potential energy by contacting the rail of the stairwell.
Discuss some of the problems with the accuracy of this experiment.
This experiment is not fully accurate in that the recording of the change in time by stopwatch controlled by a human has the potential to be off due to observation, optical illusion, and preconceived notions of space-time orientation.
Equipment: Two meter sticks, stopwatch, kilogram bathroom scale
Power is defined to be the rate at which work is done or equivalently, the rate at which energy is converted from one form to another. In this experiment you will do some work by climbing from the first floor of the science building to the second floor. By measuring the vertical height climbed and knowing your mass, the change in your gravitational potential energy can be found with the formula: ∆ PE = mgh (where m is the mass, g the acceleration of gravity, and h is the vertical height gained) Your power output can be determined by Power = ∆ PE/∆t (where ∆t is the time to climb the vertical height h.)
Procedure:
1. Determine your mass by weighing on the kilogram bathroom scale. Record your mass in kg..
2. Measure the vertical distance between the ground floor and the second floor for the science building. This can most easily be done by using two meter long meter sticks held end to end in the stairwell at the west end of the building. Make a careful sketch of the stairwell area that explains the method used to determine this height.
3. Designate a record keeper and a timer for the class. At the command of the timing person, run or walk (whatever you feel comfortable doing) up the stairs from the ground floor to the second floor. Be sure that your name and time are recorded by the record keeper.
4. After everyone in the class has completed one trip up the stairs, repeat for one more trial.
5. Return to class and calculate your personal power output in watts using the data collected from each of your climbing trip up the stairs. Obtain the average power output from the two trials.
6. Put your average power on the board and then calculate the average power for the entire class once everyone has reported their numbers on the board.
7. Determine your average power output in units of horsepower.
Questions:
Is it okay to use your hands and arms on the hand-railing to assist you in your climb up the stairs? Explain why or why not.
No, it is not acceptable to use your hands and arms to propel yourself up the flight of stairs because then it is not your power alone that is allowing your vertical position displacement to occur. Although it is true that it takes energy to pull yourself up, you are still transferring kinetic energy into potential energy by contacting the rail of the stairwell.
Discuss some of the problems with the accuracy of this experiment.
This experiment is not fully accurate in that the recording of the change in time by stopwatch controlled by a human has the potential to be off due to observation, optical illusion, and preconceived notions of space-time orientation.
Acceleration of Gravity
Objectives:
- To determine the acceleration of gravity for a freely falling object.
- To gain experience using the computer as a data collector.
Materials:
- Windows based computer
- LabPro interface
- LoggerPro software
- Motion detector
- Rubber ball
- Wire basket
Procedure:
- Gather the necessary materials and then proceed to connect the LabPro to the computer and the motion detector to the DIG/SONIC2 port on the LabPro. Turn on the computer and load the LoggerPro software by double clicking on its icon located within the PhysicsApps folder. A file named graphlab will be used to set up the computer for collecting the data needed for this experiment. To open this file, select File, Open, and then open the mechanics folder. When this folder opens, open the graphlab file.
- Upon opening the file, you should see a blank position vs. time graph. The y-axis (position axis) should be from 0m to 4m while the x-axis (time axis) should be from 0s to 4s. These values can be changed by pointing the mouse at the upper and lower limits on either scale and clicking on the number to be changed. Enter in the desired numbers and push the Enter key.
- Place the motion detector on the floor, facing upward, and place the wire basket (inverted) over the motion detector to protection the detector from the impact of the falling ball. Check to see that the motion detector is working properly by holding the rubber ball about 1m above the detector. Click the Collect button to begin taking data and then move your hand up and down a few times in order to verify that the graph of the motion is consistent with the actual motion of your hand. After 4s the computer will stop taking data and will be ready for another trial. If the equipment does not seem to be working properly, seek assistance from the instructor.
- Give the ball a gentle toss straight up from a point about 1 meter above the detector. The ball should rise 1 or 2 m above where your hand released the ball. Ideally your toss should result in the ball going straight up and down directly above the detector. It will take a few tries to perfect your toss. Be aware of what your hands are doing after the toss as they may interfere with the path of the ultrasonic waves as they travel from the detector to the ball and back. Take your time and practice until you can get a position-time graph that has a nice parabolic shape.
- Select the data in the interval that corresponds to the ball in freefall by clicking and dragging the mouse across the parabolic portion of the graph. Release the mouse button at the end of this data range. Any later data analysis done by the program will use only the data from this range. Choose Analyze/Curve Fit from the menu at the top of the window. Choose a quadratic function [t^2+ b t + c] and let the computer find the values of a, b, and c that best fit the data. If the fitted curve matches the data curve, select Try Fit. Click on OK if the fit looks good. A box should appear on the graph that contains the values of a, b, and c. Use unit analysis to give a physical interpretation and the proper units for each of these quantities. Find the acceleration, g_exp, of the ball from this data and calculate the percent difference between this value and the accepted value, g_acc, (9.80 m/s2).
- Look at a graph of velocity vs time for this motion by double clicking on the y-axis label and select “velocity” and deselect “position”. Examine this graph carefully. Explain (relate them to the actual motion of the ball) the regions where the velocity is negative, positive, and where it reaches zero. Determine the slope from a linear curve fit to the data. Use unit analysis to find the values of m and b that best fit the data. Give a physical interpretation and the proper units for each of these quantities. Find the acceleration of the ball, g_exp, from this data and calculate the percent difference between this value and the accepted value, g_acc. Put together an excel spreadsheet for your data. Finally, select Experiment, then Store Latest Run to prepare for the next trial.
- Repeat steps 4 - 6 for at least five more trials. Obtain an average value for the acceleration of gravity and a percent difference between this value and the accepted value.
- Obtain a printout of one representative graph for position vs time and velocity vs. time and include this in your lab report. Put both graphs on a single page.
Conclusion:
The purpose of this lab was to discover through observation and measurement the constant we know to be gravity (g) of roughly -9.8 m/s^2. Human error is the source of any discrepancies in data. Considering I worked by myself, I do not think I did so poorly. Shout out to Tito for helping me set it up.
Working with Spreadsheets
Objective:
Procedure:
The point of this experiment was to implement our knowledge of physics by predicting a trend based off of given data and then graphing the data to observe the resulting curve. Analysis was indicative of a parabolic shape that was based on a position vs time graph and then by evaluating derivatives, the velocity and acceleration could be observed accurately.
- To get familiar with electronic spreadsheets by using them in some simple applications.
- Computer with EXCEL software
Procedure:
- Turn on the computer and load the Microsoft Excel software by clicking on Start, move the mouse over Programs them move the mouse over Microsoft Excel and then left click.
- Create a simple spreadsheet that calculates the values of the following function: f(x) = A sin(Bx + C). Initially choose values for of A = 5, B = 3 and C = π/3. Place these values at the right side of the spreadsheet in the region reserved for constants. Put the words amplitude, frequency, and phase next to each as an explanation for the meaning of each constant. Place column headings for "x" and "f(x)" near the middle of the spreadsheet, enter a zero in the cell below "x", and enter the formula shown above in the cell below "f(x)". Be sure to put an equal sign in front of the formula. Create a column for values of x that run from zero to 10 radians in steps of 0.1 radians. Use the copy feature to create these x values. Similarly, create in the next column the corresponding values of f(x) by copying the formula shown above down through the same number of rows.
- Once the generated data looks reasonable, copy this data onto the clipboard by highlighting the contents of the two columns and choosing edit, then copy, from the menu bar. Print out a copy of the spreadsheet and also print out the spreadsheet formulas. Be sure that your rows and columns are numbered and lettered.
- Minimize the spreadsheet window and run the Graphical Analysis program by opening the PhysicsApps icon and then double-click the Graphical Analysis icon. Once the program loads, click on the top of the x column and choose edit, paste, to place the data from the clipboard into the graphing program. A graph of the data should appear in the graph window. Put appropriate labels on the axes of the graph.
- Highlight the portion of the graph for analysis and choose analyze and curve fit from the menu bar to direct the computer to find a function that best fits the data. From the list of possible functions, give the computer a hint as to what type of function you expect your data to match. The computer should display a value for A, B, and C that fit the sine curve that you are plotting. Make a copy of the data and graph by selecting file, then print.
- Repeat the above process for a spreadsheet that calculates the position of a freely falling particle as a function of time. This time your constants should include the acceleration of gravity, the initial velocity, initial position, and the time increment. Start off with g = 9.8m/s^2, v0 = 50 m/s, x0 =1000 m and Δt = 0.2 s. Print out the spreadsheet. Copy the data into the Graphical Analysis program and obtain a graph of position vs time. Fit this data to a function (y = A + Bx + Cx^2) which closely matches the data. Interpret the values of A, B, and C. Get a printout of this graph with the data table.
The point of this experiment was to implement our knowledge of physics by predicting a trend based off of given data and then graphing the data to observe the resulting curve. Analysis was indicative of a parabolic shape that was based on a position vs time graph and then by evaluating derivatives, the velocity and acceleration could be observed accurately.
Graphical Analysis
Objectives:
Part I:
- To gain experience in drawing graphs and in using graphing software.
- Windows based computer with Graphical Analysis software
- LabPro interface
- LoggerPro software
- Motion detector
- Rubber ball
- Wire basket
- Gather the required materials and then proceed to turn on the computer. Once started, access the PhysicsApps folder by double-clicking on the folder icon. Subsequently, load the Graphical Analysis software within the folder by double-clicking on its respective icon.
- Once the program is operational, open the file functionplot by clicking on File (located at the top left-hand corner of the screen), Open, then double-click the Physics folder. If properly executed, it should show a graph of a function and the data used to create the graph. Read the instructions that appear in the text window to see how to enter your own function.
- Decide upon and choose a mathematical function of your liking to plot. Make sure that the function has the following components:
- a title
- x and y axis labels
- units
- Make sure that the graph you created has the appearance that you want. Obtain a printout of the graph for each member of your group (excluding data table) by selecting File, then Print.
- Connect the lab pro to the computer and the motion detector to the DIG/SONIC2 port on the LabPro. Load the LoggerPro software in the PhysicsApps folder. Click on File, then Open, and open the mechanics folder. Within that folder, open the graphlab file. Take a position vs. time graph for a falling ball with the motion sensor.
- When you finally achieve a nice curve, select the appropriate data range and perform a fit to the data. Confirm that the distance fallen adheres to the following relationship: d*α*g*t^n where g = 9.8 m/s2, is the acceleration due to gravity. Obtain a printout of your graph (with the fit) for each member of your group (without the data table).
- Use dimensional analysis and unit analysis to verify the equation.
Results:
Dimensional Analysis:
therefore:
Conclusions:
During the lab, I learned how to use the LoggerPro software in order to capture the velocity of a falling object, accurately detect the acceleration of gravity to be roughly around -9.8 m/s^2, and to analyze a graph to estimate values for velocity through curve fitting a linear fit around a parabolic position vs. time graph (the slope is equal to the velocity).
 |
| Figure 1 |
 |
| Figure 2 |
d
α gt2
v=(dx)/(dt)
g=(dv)/(dt)therefore:
v=m/t
g=m/t^2
d=g*t^2=[m/t^2]*t^2=m
Unit Analysis:
v=m/t
g=m/t^2
d=[m/t^2]*t^2]=mConclusions:
During the lab, I learned how to use the LoggerPro software in order to capture the velocity of a falling object, accurately detect the acceleration of gravity to be roughly around -9.8 m/s^2, and to analyze a graph to estimate values for velocity through curve fitting a linear fit around a parabolic position vs. time graph (the slope is equal to the velocity).
Drag Force on a Coffee Filter
Purpose: To study the relationship between air drag forces and the velocity of a falling body.
Equipment: Computer with Logger Pro software, lab pro, motion detector, nine coffee filters, meter stick
Introduction:
When an object moves through a fluid, such as air, it experiences a drag force that opposes its
motion. This force generally increases with velocity of the object. In this lab we are going to
investigate the velocity dependence of the drag force. We will start by assuming the drag force,
FD, has a simple power law dependence on the speed given by
1) FD = k |v|^n , where the power n is to be determined by the experiment.
This lab will investigate drag forces acting on a falling coffee filter. Because of the large surface
area and low mass of these filters, they reach terminal speed soon after being released.
Procedure:
NOTE: You will be given a packet of nine nested coffee filters. It is important that the shape of this
packet stays the same throughout the experiment so do not take the filters apart or otherwise
alter the shape of the packet. Why is it important for the shape to stay the same? Explain and
use a diagram.
1. Login to your computer with username and password. Start the Logger Pro software, open the
Mechanics folder and the graphlab file. Don’t forget to label the axes of the graph and create an
appropriate title for the graph. Set the data collection rate to 30 Hz.
2. Place the motion detector on the floor facing upward and hold the packet of nine filters at a minimum
height of 1.5 m directly above the motion detector. (Be aware other of nearby objects which can
cause reflections.) Start the computer collecting data, and then release the packet. What should the
position vs time graph look like? Explain.
Verify that the data are consistent. If not, repeat the trial. Examine the graph and using the mouse,
select (click and drag) a small range of data points near the end of the motion where the packet
moved with constant speed. Exclude any early or late points where the motion is not uniform.
3. Use the curve fitting option from the analysis menu to fit a linear curve (y = mx + b) to the selected
data. Record the slope (m) of the curve from this fit. What should this slope represent? Explain.
Repeat this measurement at least four more times, and calculate the average velocity. Record all
data in an excel data table.
4. Carefully remove one filter from the packet and repeat the procedure in parts 2 and 3 for the
remaining packet of eight filters. Keep removing filters one at a time and repeating the above steps
until you finish with a single coffee filter. Print a copy of one of your best x vs t graphs that show the
motion and the linear curve fit to the data for everyone in your group (Do not include the data table;
graph only please).
5. In Graphical Analysis, create a two column data table with packet weight (number of filters) in one
column and average terminal speed (|v|) in the other. Make a plot of packet weight (y-axis) vs.
terminal speed not velocity (x-axis). Choose appropriate labels and scales for the axes of your
graph. Be sure to remove the “connecting lines” from the plot. Perform a power law fit of the data
and record the power, n, given by the computer. Obtain a printout of your graph for each member of
your group. (Check the % error between your experimentally determined n and the theoretical
value before you make a printout – you may need to repeat trials if the error is too large.)
6. Since the drag force is equal to the packet weight, we have found the dependence of drag force on
speed. Write equation 1 above with the value of n obtained from your experiment. Put a box around
this equation. Look in the section on drag forces in your text and write down the equation given there
for the drag force on an object moving through a fluid. How does your value of n compare with the
value given in the text? What does the other fit parameter represent? Explain.
Conclusion:
The point of this experiment was to determine through observation and experimentation the effect of drag using the formula k|v|^n by evaluating the value of n with the LoggerPro curve fit analysis. There were several factors to which the experiment may or may not go accordingly; the fact that the coffee filters are very fragile and subject to physical modification and, of course, human error for collecting data, dropping the filters accurately, and fitting the data with a curve fit inaccurately or with a non-optimum value. The result of the trials my group conducted hinted us to believe that the n value was approximately 2. The surface area of the filters is also a dependence factor for the drag calculations, which is why it is very important to keep the filters in pristine condition. In conjunction with the former, we concluded that the more filters you have, the greater the drag because there is more surface area to inhibit an expedited fall.
Equipment: Computer with Logger Pro software, lab pro, motion detector, nine coffee filters, meter stick
Introduction:
When an object moves through a fluid, such as air, it experiences a drag force that opposes its
motion. This force generally increases with velocity of the object. In this lab we are going to
investigate the velocity dependence of the drag force. We will start by assuming the drag force,
FD, has a simple power law dependence on the speed given by
1) FD = k |v|^n , where the power n is to be determined by the experiment.
This lab will investigate drag forces acting on a falling coffee filter. Because of the large surface
area and low mass of these filters, they reach terminal speed soon after being released.
Procedure:
NOTE: You will be given a packet of nine nested coffee filters. It is important that the shape of this
packet stays the same throughout the experiment so do not take the filters apart or otherwise
alter the shape of the packet. Why is it important for the shape to stay the same? Explain and
use a diagram.
1. Login to your computer with username and password. Start the Logger Pro software, open the
Mechanics folder and the graphlab file. Don’t forget to label the axes of the graph and create an
appropriate title for the graph. Set the data collection rate to 30 Hz.
2. Place the motion detector on the floor facing upward and hold the packet of nine filters at a minimum
height of 1.5 m directly above the motion detector. (Be aware other of nearby objects which can
cause reflections.) Start the computer collecting data, and then release the packet. What should the
position vs time graph look like? Explain.
Verify that the data are consistent. If not, repeat the trial. Examine the graph and using the mouse,
select (click and drag) a small range of data points near the end of the motion where the packet
moved with constant speed. Exclude any early or late points where the motion is not uniform.
3. Use the curve fitting option from the analysis menu to fit a linear curve (y = mx + b) to the selected
data. Record the slope (m) of the curve from this fit. What should this slope represent? Explain.
Repeat this measurement at least four more times, and calculate the average velocity. Record all
data in an excel data table.
4. Carefully remove one filter from the packet and repeat the procedure in parts 2 and 3 for the
remaining packet of eight filters. Keep removing filters one at a time and repeating the above steps
until you finish with a single coffee filter. Print a copy of one of your best x vs t graphs that show the
motion and the linear curve fit to the data for everyone in your group (Do not include the data table;
graph only please).
5. In Graphical Analysis, create a two column data table with packet weight (number of filters) in one
column and average terminal speed (|v|) in the other. Make a plot of packet weight (y-axis) vs.
terminal speed not velocity (x-axis). Choose appropriate labels and scales for the axes of your
graph. Be sure to remove the “connecting lines” from the plot. Perform a power law fit of the data
and record the power, n, given by the computer. Obtain a printout of your graph for each member of
your group. (Check the % error between your experimentally determined n and the theoretical
value before you make a printout – you may need to repeat trials if the error is too large.)
6. Since the drag force is equal to the packet weight, we have found the dependence of drag force on
speed. Write equation 1 above with the value of n obtained from your experiment. Put a box around
this equation. Look in the section on drag forces in your text and write down the equation given there
for the drag force on an object moving through a fluid. How does your value of n compare with the
value given in the text? What does the other fit parameter represent? Explain.
Conclusion:
The point of this experiment was to determine through observation and experimentation the effect of drag using the formula k|v|^n by evaluating the value of n with the LoggerPro curve fit analysis. There were several factors to which the experiment may or may not go accordingly; the fact that the coffee filters are very fragile and subject to physical modification and, of course, human error for collecting data, dropping the filters accurately, and fitting the data with a curve fit inaccurately or with a non-optimum value. The result of the trials my group conducted hinted us to believe that the n value was approximately 2. The surface area of the filters is also a dependence factor for the drag calculations, which is why it is very important to keep the filters in pristine condition. In conjunction with the former, we concluded that the more filters you have, the greater the drag because there is more surface area to inhibit an expedited fall.
Tuesday, December 11, 2012
Balanced Torques and Center of Gravity
Objective: To investigate the conditions for rotational equilibrium of a rigid bar and to determine the center of gravity of a system of masses.
Materials: Meter stick, meter stick clamps (knife edge clamp), balance support, mass set, weight hangers,
unknown masses, balance.
Procedure:
1. Balance the meter stick in the knife edge clamp and record the position of the balance point.
2. Select two different masses (100 grams or more each) and using the meter stick clamps and weight hangers, suspend one on each side of the meter stick support at different distances from the support. Adjust the positions so the system is balanced. Record the masses and positions. Sum the torques about your pivot point O and compare with the expected value.
3. Place the same two masses used above at different locations on the same side of the support and balance the system with a third mass on the opposite side. Record all three masses and positions. Calculate the net torque on this system about the point support and compare with the expected value.
4. Replace one of the above masses with an unknown mass. Readjust the positions of the masses until equilibrium is achieved, recording all values. Using the equilibrium condition for rotational motion, calculate the unknown mass. Measure the mass of the unknown on a balance and compare the two masses by finding the percent difference.
5. Place about 200 grams at 90 cm on the meter stick and balance the system by changing the balance point of the meter stick. From this information, calculate the mass of the meter stick. Compare this with the meter stick mass obtained from the balance.
6. With the 200 grams still at the 90 cm mark, imagine that you now position an additional 100 grams mass at the 30 cm mark on the meter stick. Calculate the position of the center of gravity of this combination (two masses and meter stick). Check your result by actually placing the 100 g at the 30 cm mark and balancing this system. Compare the calculated and experimental results.
Experiment Questions:
What point in the meter stick does this correspond to?
Is it necessary to include the mass of the clamps in your calculations? EXPLAIN!
Should the clamp holding the meter stick be included as part of the mass of the meter stick? EXPLAIN!
Where should the point of support on the meter stick be to balance this system?
Materials: Meter stick, meter stick clamps (knife edge clamp), balance support, mass set, weight hangers,
unknown masses, balance.
Procedure:
1. Balance the meter stick in the knife edge clamp and record the position of the balance point.
2. Select two different masses (100 grams or more each) and using the meter stick clamps and weight hangers, suspend one on each side of the meter stick support at different distances from the support. Adjust the positions so the system is balanced. Record the masses and positions. Sum the torques about your pivot point O and compare with the expected value.
3. Place the same two masses used above at different locations on the same side of the support and balance the system with a third mass on the opposite side. Record all three masses and positions. Calculate the net torque on this system about the point support and compare with the expected value.
4. Replace one of the above masses with an unknown mass. Readjust the positions of the masses until equilibrium is achieved, recording all values. Using the equilibrium condition for rotational motion, calculate the unknown mass. Measure the mass of the unknown on a balance and compare the two masses by finding the percent difference.
5. Place about 200 grams at 90 cm on the meter stick and balance the system by changing the balance point of the meter stick. From this information, calculate the mass of the meter stick. Compare this with the meter stick mass obtained from the balance.
6. With the 200 grams still at the 90 cm mark, imagine that you now position an additional 100 grams mass at the 30 cm mark on the meter stick. Calculate the position of the center of gravity of this combination (two masses and meter stick). Check your result by actually placing the 100 g at the 30 cm mark and balancing this system. Compare the calculated and experimental results.
Experiment Questions:
What point in the meter stick does this correspond to?
Is it necessary to include the mass of the clamps in your calculations? EXPLAIN!
Should the clamp holding the meter stick be included as part of the mass of the meter stick? EXPLAIN!
Where should the point of support on the meter stick be to balance this system?
Tuesday, September 18, 2012
Vector Addition of Forces
Objective: To study vector addition by:
- Graphical means
- Using components.
A circular force table is used to check results.
Materials:
- Circular force table
- Masses
- Mass holders
- String
- Protractor
- Four pulleys.
 |
| Circular Force Table, Pulleys, Masses, Mass Holders, and String |
Procedure:
- Your instructor will give each group three masses in grams (which will represent the magnitude of three forces) and three angles. Choose a scale of 1 cm = 20 grams, make a vector diagram (figure 3) showing these forces, and graphically find their resultant. Determine the magnitude (length) and direction (angle) of the resultant force using a ruler and protractor (figure 4).
- Make a second vector diagram and show the same three forces again and find the resultant vector by components (figure 5). Show the components of each vector as well as the resultant vector on your diagram. Draw the force (vector) you would need to exactly cancel out this resultant.
- Mount three pulleys on the edge of your force table at the angles given above. Attach strings to the center ring so that they each run over the pulley and attach to a mass holder as shown in the figure below. Hang the appropriate masses (numerically equal to the forces in grams) on each string. Set up a fourth pulley and mass holder at 180 degrees opposite from the angle you calculated for the resultant of the first three vectors. Record all mass and angles. If properly done, the ring in the center of the circular force table should be in equilibrium. Ask your instructor to check your results before going on.
- Confirm your results by simulation at the website:http://phet.colorado.edu/en/simulation/vector-addition. Add the vectors and obtain a resultant. Draw a diagram showing how you verified the result.
- In your conclusions, summarize what you learned and discuss any sources of error.
Results:
 |
| Figure 1 |
 |
| Figure 2 |
 |
| Figure 3 Vector Diagram |
 |
Figure 4
Adding Vectors by
Ruler and Protractor
|
 |
| Figure 5 Adding Vectors by Components |
 |
| Figure 6 |
 |
| Figure 7 |
Given the masses that represent the magnitude of vectors (100, 200, and 160) and the angles (0, 71, and 144) of inclination for those vectors respectively, figure 3 illustrates the resultant vector and the negative of the resultant vector. These vectors are necessary because they allow us to counter-balance the force table's mass vectors and place the vectors in equilibrium as seen in figure 1.
To calculate the magnitude and angle of the resultant vector:
R_x = (100cos0) + (200cos71) + (160cos144) = 36.1
R_y = (100sin0) + (200sin71) + (160sin144) = 283
θ = 82.7
R = (283^2 + 36.1^2)^(1/2) = 285.3
According to the data we collected and the vectors calculated, the approximation of the vectors we visualized online (figures 6 and 7) were close to our exact answers.
Conclusion:
During our experiment we learned that in order to place three forces in equilibrium a fourth force was necessary to counter-balance the other three. The counter-balancing force was exactly equal to the negative of the resultant of the sum of the first three forces (calculated as vectors).
Any source of error that was present was a result of human miscalculation.
R = (283^2 + 36.1^2)^(1/2) = 285.3
According to the data we collected and the vectors calculated, the approximation of the vectors we visualized online (figures 6 and 7) were close to our exact answers.
Conclusion:
During our experiment we learned that in order to place three forces in equilibrium a fourth force was necessary to counter-balance the other three. The counter-balancing force was exactly equal to the negative of the resultant of the sum of the first three forces (calculated as vectors).
Any source of error that was present was a result of human miscalculation.
Tuesday, September 11, 2012
Acceleration of Gravity on an Inclined Plane
Objective:
- To calculate the acceleration of gravity by observing the motion of a cart on an incline.
- To acquire further computer literacy with respect to collecting and analyzing data.
Materials:
- Windows-based computer with LoggerPro software installed
- Motion detector
- Ballistic cart
- Aluminum track
- Wood blocks
- Meter stick
- Small carpenter level
 |
| LabPro Interface |
 |
| Aluminum Track, Ballistic Cart, and Wood Block |
-
 |
| Motion Detector |
Procedure:
- Connect LabPro to the computer with the LoggerPro software and the motion detector to the DIG/SONIC2 port on the LabPro. Turn on the computer and load the LoggerProsoftware by double clicking the corresponding icon located within the PhysicsApps folder. Locate and open the file named graphlab; it will be used to set up the computer for collecting the velocity and acceleration data in order calculate the lines of best fit, thereby calculating the acceleration of gravity using the formula "gSinθ=(a_1+a_2)/2". Double-click on the Mechanics folder and then proceed to open the file by double-clicking it.
- Incline the end of the aluminum track by using the wood block as a wedge. Place the wood block roughly at the 50cm mark on the track. Use the level to make sure that the track is level. Once level, calculate the angle of inclination using basic trigonometry (calculate the heights of the track at both ends, subtract the smaller value from the bigger value, and then take the inverse sin of the result as seen in Figure 1).
- Place the motion detector at the upper end of the track facing down toward the lower end. Start with the cart at the lower end of the track and gently push the ballistic cart toward the motion detector until it is just outside 50cm from the motion detector. Make sure to stop the cart before it crashes into the track to prevent damage to the cart or track.
- Start data collection after the cart leaves your hand and observe both the position and velocity graphs simultaneously by having two windows open (one above the other). Select suitable scales for both vertical and horizontal axes to best show the motion of the ballistic cart. Properly label the graphs with titles, units, and other relevant information. Repeat the trial until the graph looks like a consistent curve.
- Once an appropriate curve has been acquired, use the LoggerPro software to take a virtual snapshot of the slope of the curve v vs. t by selecting a range of times that represents the motion of the ballistic cart going up the incline. Choose the "Analyze/Curve Fit" option to fit the selected portion of the curve to a linear function of time. Repeat this process for when the ballistic cart is returning down the incline.
- Complete at least two more trials for the same angle of inclination. For each trial, take your two values of the slope and plug them into the formula: gSinθ=[(a_1+a_2)/2]. Doing this allows us to disregard the force of friction and isolate gravity as the only force affecting the acceleration of the ballistic cart.
- Repeat the experiment for a larger value of theta by either using a larger block of wood or upending the wood block you have (if materials allow) to elevate the aluminum track even further. Doing this assures that the angle of inclination increases.
- Print out copies of the two graphs from the trials in order to show the set of data from the position and velocity graphs. Show the time intervals used and the slope of the two different velocity curves on the graphs.
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| Figure 1 |
Experiment Question:
- What type of curve do you expect to see for x vs t and v vs t? Explain.
For the position curve (x vs t), a parabolic shape is the expected result because the ballistic car travels up the incline and then hits an apex at which point it descends again. Since the position function is parabolic in shape, then the velocity function must be represented by a linear function because the derivative of the position function is the velocity function.
Results:
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| Trial 1 |
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| Trial 2 |
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| Trial 3 |
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| Trial 4 |
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| Trial 5 |
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| Trial 6 |
Trials 1, 2, and 3 (as labeled in the captions of the pictures) were for the angle:
"θ=ArcSin(9.55cm/228cm)."
Trials 4, 5, and 6 (also labeled by captions) were for the angle:
"θ=ArcSin(18.65cm/228cm)."
The values of gravity for trials 1, 2, and 3 were as follows:
Trial 1: g= 8.63 m/s^2
Trial 2: g= 8.48 m/s^2
Trial 3: g= 8.52 m/s^2
The values of gravity for trials 4, 5, and 6 yielded:
Trial 4: g= 9.08 m/s^2
Trial 5: g= 9.29 m/s^2
Trial 6: g= 9.20 m/s^2
Conclusion:
Based off of the results produced by our group, the only logical conclusion that can be deduced is that the higher the angle of inclination, the more accurate the calculation of gravity will become. That is, the experimental gravity will ultimately become closer and closer to the actual quantity of gravity (g=9.80m/s^2) as the angle of inclination increases.
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