Vector Addition of Forces

Objective: To study vector addition by:

1. Graphical means
2. Using components.

A circular force table is used to check results.

Materials:

1. Circular force table
2. Masses
3. Mass holders
4. String
5. Protractor
6. Four pulleys.

Circular Force Table, Pulleys,
Masses, Mass Holders, and String

    

Procedure:

1. 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).
2. 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.
3. 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.
4. 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.
5. 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.

Acceleration of Gravity on an Inclined Plane

Objective:

1. To calculate the acceleration of gravity by observing the motion of a cart on an incline.
2. To acquire further computer literacy with respect to collecting and analyzing data.

Materials:

1. Windows-based computer with LoggerPro software installed
2. Motion detector
3. Ballistic cart
4. Aluminum track
5. Wood blocks
6. Meter stick
7. Small carpenter level

LabPro Interface

Aluminum Track, Ballistic Cart,
and Wood Block





Motion Detector

Procedure:

1. 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 LoggerPro
   software 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.
2. 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).
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.

Figure 1

Experiment Question:

1. 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:

Trial 1

Trial 2

Trial 3

Trial 4



Trial 5

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.