01-June-2015: Physical Pendulum Lab

Lab 20: Physical Pendulum Lab

Purpose:
The purpose of this experiment was to derive expressions for the period of various physical pendulums and to verify the predicted periods by experiment.

Part 1:

Apparatus:

 

The apparatus of the first part of the experiment consisted of two stands, with a knife edge attached horizontally. The edge was used to hold the ring in place as it oscillated. A photogate was attached to one of the stands. A tape marker was attached to the ring so that the photogate would track its movement. In this case, the photogate measured the period of the oscillating ring. In addition, we also used a computer with LoggerPro to collect and interpret the data from the photogate.

 

Procedure:

First, we calculated the moment of inertia of the ring. We treated the ring as a hollow cylinder rotating about its center. Then, we used the parallel-axis theorem to find the moment of inertia from its new pivot point.

The dimensions of the ring and equations used can be seen below. 

• I(ring)= 0.00328 kg*m^2

Next, using the calculated moment of inertia and torque, we were able to determine the period of the ring, while oscillating at small angles.

First, we drew and RBD of the system and setup its respective equations. In this case, we used Torque=I*alpha.

Next, we solved for alpha and noticed that it took the form : acceleration=-constant*displacement

From here, we can interpret it as: acceleration=-Omega^2*displacement

Finally, we solved for omega and inputted into T=(2*pi)/Omega.

As you can see, our theoretical value for the period of the oscillating ring is 0.7162 seconds.

Next, we verified the predicted period by running the experiment. During the experiment, the ring was oscillating at small angles and the photogate was used to measure the period. The result can be seen below:

To recap:

• Theoretical Period:0.7162 sec
• Experimental Period:0.7211 sec

Conclusion:

The percentage error for this experiment was 0.7%. This shows us that the theoretical period is accurate. Furthermore, it also proves that the method used to obtain the period is accurate. The discrepancy between the two values can be the result of uncertainty in the dimensions of the ring. In addition, there can also be an uncertainty from the data recorded by the photogate. Fortunately, these errors are small enough that they are negligible for this experiment.

 

Part 2:

 

Apparatus: The same setup was used for the second part of the experiment. However, for this part of the experiment, we used the following physical pendulums:

• Isosceles triangle, base B, height H, oscillating about its apex (Trial 1)
• Isosceles triangle, base B, height H, oscillating about the midpoint of its base (Trial 2)
• Semicircular plate of radius R, oscillating about the midpoint of its base (Trial 3)
• Semicircular plate of radius R, oscillating about a point on its edge, directly above the midpoint of the base (Trial 4)

Trial 4

Trial 3

Trial 1

Trial 2

Procedure:

First, we measured the appropriate dimension of each shape. Next, we used electrical connectors and a pin to setup the pivot points for each object. This allowed the object to oscillate freely. Like in Part1, we attached a tape marker to each object so that the photogate would pickup its movement.

 

Next, we derived the center of mass of each shape and used this information to derive the moment of inertia of both the triangle and semicircle from each orientation.

Center of Mass of Semicirlce

•  Semicircle Center of Mass: 4R/(3*pi)

Next, we derived the moment of inertia of the semicircle about the midpoint of its base. Then, we used the parallel axis theorem to determine the moment of inertia of the semicircle about its new pivot points.

 

• I(cm): 1.20*10^-5 kg*m^2
• I (trial 3) = 1.88*10^-5 kg*m^2
• I(trial 4)= 2.45*10^-5 kg*m^2

Next, we determined the periods of the oscillating semicircle. First, we drew a RBD of the system and set up the sum of torques equation. Then, we followed the same steps as in Part 1 and setup the equations in the following format:

acceleration=-constant*displacement & acceleration=-Omega^2*displacement

From here, we solved for omega and inputted into T = (2*pi)/omega

Trial 3

Trial 4

As you can see, we solved for the period of the semicircle in both orientations.

Next, we conducted the experiment as shown above and recorded the period of the oscillating semicircle. The results can be seen below:

Trial 4

Trial 3

To recap:

 

Next, we found the periods of the oscillating isosceles triangle. The same steps were used as before. First, we found the center of mass of the triangle. Then, we found the moment of inertia of the triangle about its center of mass. Next, we used the parallel axis theorem to find the moment of inertia about its new pivot points.

 

• Triangle Center of Mass: (1/3)H

Above you can see how we derived the moment of inertia of the triangle about its apex.

Next, we used the parallel axis theorem to find the moment of inertia about its center of mass. Then, we used the parallel axis theorem once more to determine the moment of inertia about its base. These derivations can be seen below:

 

Next, we determined the periods of the oscillating triangle. First, we drew a RBD of the system and set up the sum of torques equation. Then, we followed the same steps as in Part 1 and setup the equations in the following format:

acceleration=-constant*displacement & acceleration=-Omega^2*displacement

From here, we solved for omega and inputted into T = (2*pi)/omega

Trial 1

Trial 2

 

Finally, we ran the experiment as shown above and recorded the period of the oscillating triangle. The results can be seen below:

Trial 1

Trial 2

To recap:

Trial 1:

• Theoretical Period: 0.748 sec
• Experimental Period: 0.784 sec
• Percentage Error: 4.8%

Trial 2:

• Theoretical Period: 0.640 sec
• Experimental Period: 0.661 sec
• Percentage Error: 3.2%

Conclusion:

The margin of error for the second part of the experiment is low. This shows us that the theoretical periods are accurate. Furthermore, it also proves that the method used to obtain the periods is correct. The difference in the values can be the result of uncertainties present during the experiment. In this experiment there is uncertainty in the mass and dimensions of both the semicircle and triangle. Lastly, there could also be a source of error if the physical pendulums were not pivoted correctly. Since the margin of error is low, we can assume that these factors are negligible in this experiment.

 

5-20-15: Conservation of Energy/ Conservation of Angular Momentum Lab

Lab 19: Conservation of Energy/Conservation of Angular Momentum

Purpose:
The purpose of this lab was to use the principles of conservation of energy and angular momentum to determine the max height of a swinging mass system.

Apparatus:

 

 

The apparatus used for this experiment consisted of a metal stand and a meter stick pivoted near one of its ends on a rotational sensor. Tape was wrapped around the other end of the meter stick. A piece of clay was also wrapped in tape and placed on a stand made of paper clips. The clay was strategically placed so that it would collide with the meter stick at the bottom of the swing. Both the clay and the meter stick were wrapped with tape so that the clay would stick to the meter stick and essentially create an inelastic collision.

 

A camera was also used during the experiment to record the collision and to determine the final height of the system. Furthermore, LoggerPro was used to analyze the video.

 

Procedure:

First, we recorded the mass of the clay and meter stick and measured the distance from the meter stick's center mass to the pivot point.

• Mass of clay = 19.4 g
• Mass of meter stick = 85.6 g
• Distance from CM to pivot = 0.487 m

Next, we setup the apparatus as seen above and positioned the camera so that we could see the end of the meter stick throughout the whole swing.

 

Now for the actual experiment. The meter stick pivoted near one end was released from a horizontal position. Right when it reached the bottom of the swing, the meter stick collided inelastically with a piece of clay. Then, the meter stick and clay rotated together to a final position. Video was recorded of this process. The video was used to determine the max height of the system.

 

Furthermore, we determined the max height of the system through use of the principle of energy and the principle of angular momentum.

 

Finally, we compared the theoretical max height to the experimental value.

 

 

Data:

 

Theoretical Value



As stated before, the theoretical height was found through use of the principle of energy and the principle of angular momentum.

First, we determined the meter stick's new inertia since it was not being pivoted from its center. This was done by the parallel axis theorem.

• Distance from CM to pivot = 0.487 m
• Mass of meter stick = 85.6 g

I(stick)= (1/12)ML^2 + M(.487)^2
I(stick)= 0.0274 kg*m^2

Next, we calculated the inertia of the clay. We treated this inertia as a point mass.

• Distance of clay from pivot= 0.9987 m
• Mass of clay= 19.4 g

I(clay)= 0.0194 *(0.9987)^2 = 0.0193 kg*m^2
Finally, we calculated the Inertia of the system.

• I(system)= I(stick)+I(clay)= 0.04675 kg*m^2

Now, with a calculated inertia for the stick and an inertia for the system, we can determine the max height of the rotating system.

First, we calculated the angular speed at the bottom of the swing. This was achieved based on the principle of conservation of energy. Next, we calculated the angular speed of the system after the inelastic collision. This was done through the principle of conservation of angular momentum.

In the picture above, you can see our calculated values of inertia as well as the angular speed of the stick before and after the collision.
Notice that after the collision in angular momentum, we used the inertia of the system and not the stick.

• Angular Speed at the bottom of the swing = 5.46 rad/s
• Angular Speed after the collision = 3.2 rad/s

Next, we symbolically solved for the max height of the system.

As you can see in order to find the max height of the system, we must find theta.

 

Fortunately, this was achieved through the principle of conservation of energy.

First, we established the pivot point as our GPE=zero mark. Then, we setup equations for the change in the GPE of the stick and the GPE of the clay. Finally, we took the sum of GPE of clay and meter stick and set it equal to the KE of the system. From here, we were able to solve for theta. Finally, we plugged theta into out equation for h and determined the max height of the system.



For our experiment, theta= 53.06 degrees.
Therefore, our theoretical max height of the system is 0.399 m

Experimental Value

In this portion of the lab, we captured video of the rotating system and analyzed the footage on LoggerPro.

In the picture above you can see how we tracked the movement of the clay after the collision up until the max height. Each blue point represents the clay's location at a given time interval. In order for LoggerPro to give us an appropriate max height of the system, we set a scale in the video using the meter stick. This was very easy since we already knew the length of the meter stick. Next, we established the origin at the bottom of the swing. From here, LoggerPro was able to plot the clay's movement and plot it on a graph. We adjusted the graph, so that it only gave us movement in the y-direction.

As you can see above, we used the Examine feature on LoggerPro to determine the max height of the system.

• Experimental Max Height = 0.387 m

Uncertainty/Error: The sources of uncertainty/error for this experiment are the following:

• Uncertainty in the measurements of the masses of the clay and meter stick.
• Uncertainty in the measurement of the distance between the pivot and the clay and the distance between the pivot and the meter stick center of mass.
• Uncertainty in plotting the points during the analysis of the video.
• Friction present during the experiment

Conclusion:

When comparing our theoretical value (0.399m) to our experimental value (0.387m), we can see that they are fairly close. In fact, there is only a 3% error. This helps us conclude that the sources of error or uncertainties present during the experiment are negligible to the point that it does not affect the outcome of the experiment. Furthermore, it was expected for the experimental value to be less than the theoretical value. Anything higher would raise some red flags, since the theoretical value is seen as the limit for the experiment. Lastly, we proved that the method of determining the max height of swinging system through the conservation of energy and the conservation of angular momentum is correct.

5-13-2015: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque

Purpose: The purpose of this experiment was to experimentally determine the moment of inertia and the frictional torque of a large metal disk rotating about a shaft in its center.

Apparatus:

 

 

 The apparatus for this experiment consisted of a large metal disk on a central shaft. The disk and central shaft rotate about a center axis. The disk and shaft are held up by metal stands attached to a metal plate. Calipers were used to measure the diameter of the large disk and the central shaft. The mass of the disk and shaft was stamped on the side of the large disk.

In addition, a camera was used to calculate the deceleration of the rotating part of the apparatus. The video was then analyzed on LoggerPro.

Lastly, we used the setup shown above to verify the accuracy of the calculated moment of inertia and frictional torque of the system.

 

Procedure (Part 1): Calculating the Inertia of the apparatus

 

The dimensions of the large disk and central shaft were measured using calipers. In this case, we treated the central shaft as two separate cylinders. To recap, now we have three cylinders: a large disk and two identical cylinders. Since the total mass of the system was given, we calculated the mass of each cylinder by first calculating each cylinder's volume. We then calculated the percentage of volume it occupied compared to the total volume of the three cylinders and we determined the appropriate ratio for each cylinder. We then used that ratio compared to the mass of the system to determine the mass of each individual cylinder.

 

Below you can see the dimensions and volume calculations of each cylinder.



As you can see, we determined that the total volume of the three cylinders was 0.0005982 m^3
Next, we found that the large disk represents 86.82%  of the total mass. So if the total mass was  4.887 kg, this means that large disk weighs 4.243 kg. Now, the two cylinders represent 13.18% of the total mass. This means that each cylinder weighs 0.322 kg.

Finally, we can find the Inertia of the system since we now have the mass and radius of each cylinder. The moment of inertia was found using the equation shown above.

• Inertia of the system: 0.0214784 kg*m^2

Procedure (Part 2): Determining the Angular Deceleration and the Frictional Torque of the system
In this part of the lab, we calculated the angular deceleration of the disk. First, we put a piece of green tape on the side of the disk to use as a marker. Next, we captured video of the spinning disk. Then, we analyzed the video on Logger Pro. We tracked the marker as it completed one rotation. We also set the scale of the video by inputting that diameter of the disk on LoggerPro. From here, LoggerPro calculated the velocity of the marker in the x and y direction.

Video Capture

Next, we created a calculated column on LoggerPro to find Vtangential. Then, we created another calculated column to find omega. Finally, we graphed omega vs. time and took a linear fit of the graph; as the slope of the graph equals the angular deceleration.

• Vtangential: sqrt( (Vx^2) + (Vy^2) )
• Omega: Vtan/radius of disk

Angular Deceleration =  -0.07691 rad/sec^2

 

Lastly, the Frictional Torque of the system can be determined using:                                                     Tf = Inertia*Angular Deceleration=  (0.0214784 kg*m^2)*-0.7691 rad/sec^2

In this case, the frictional torque is -0.0165 N*m

 

Procedure (Part 3):

To determine the accuracy of our values for angular deceleration and frictional torque, a 500-gram cart was attached to the central shaft of the apparatus using a long string. The cart was then placed on an inclined slope and we timed how long it took for the cart to travel one meter.

 

First, we solved the problem symbolically. Then we inputted the following data:

• mass of cart= .5 kg
• Angle of incline= 40 degrees
• Frictional torque= -0.0165 N*m
• Inertia= 0.0214784 kg*m^2
• radius= 0.0156 m

In this case, the cart traveled 1 meter in 6.71 seconds. We used this as a benchmark for our actual experiment.

The setup for our experiment was the same but our angle of incline changed.

All the necessary values can be seen below.



 

As you can see, first we drew and FBD of the cart. Then we setup our Fx ,Fy and Torque equations. We solved symbolically for a and then inputted the data. Ultimately, we predicted that the cart would travel 1 meter in 6.2 seconds.

Next, we actually timed the cart as it traveled 1 meter. We conducted three trials and the average time came out to be 7.7 sec.  As you can see our answers are ways off, we assume that the difference can be the result of slow reaction time when using the stopwatch in the experiment.

If we compare our theoretical value of 6.2 seconds to our benchmark, our value seems accurate. The benchmark experiment had an inclination of 40 degrees and the time it took to travel 1 meter was 6.71 sec. In our theoretical experiment, the inclination was 51.7 degrees and the time it took to travel 1 meter was 6.2 sec. It makes sense that a steeper incline would make the cart travel faster, which would result in a faster time.

Conclusion:
In this lab, we determined the inertia, angular deceleration, and frictional torque of a rotating system. We then tested these values for accuracy and came to the conclusion that the uncertainties are very small. However, the experiment in calculating time resulted in a 19.85% error. After going over our work multiple times to catch any mistakes, we came to the conclusion that the biggest source of error is the result of slow reaction time when using a stopwatch. Other than that the methods used in calculating the inertia, angular deceleration, and frictional torque of a rotating system are systematically correct and any uncertainty is very minimal to the point that it will not affect the final result.

5-13-2015: Finding the moment of inertia of a uniform triangle about its center of mass

Lab 17: Finding moment of inertia of a triangle

Purpose: To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

Apparatus:

The apparatus shown above is the same apparatus we used for the Angular Acceleration Lab. For this lab, we used the same steel disks and large torque pulley as in the previous lab. The only difference being that this time we added a triangular plate to the rotating mass centered about its center of mass. A holder was added to the rotating stand to hold the triangle in place. Lastly, the same hanging mass was used in this experiment as in the Angular Acceleration Lab.

Furthermore, we adjusted the settings in LoggerPro so that it would record the rotary motion of the system and we also set the sensor settings to 200 counts per revolution.

The mass of the disks, torque pulley, hanging mass, and triangle were measured .
The diameters of the torque pulley and disks were measured.
The dimensions of the triangle were also measured.

Procedure:
First, we derived the moment of inertia of a uniform triangle about its center of mass. We did so, using the parallel axis theorem. This method is faster and easier because the limits of integration are easier if we calculate the moment of inertia around a vertical end of the triangle and then use the parallel axis theorem to find the inertia around the triangle's center of mass.

Inertia around the  vertical edge

Inertia around the center of mass

 Measurements of Disks and Torque Pulley and Triangle:

• Top Steel Disk: d=12.630 cm  m=1355g
• Bottom Steel Disk: d=12.630 cm  m=1348g
• Larger Torque Pulley: d=5.36 cm  m=36.3g
• Hanging mass: m=24.6
• Triangle: m=453.3g

Like in the Angular Acceleration Lab, we will run the experiment and record the angular acceleration of the rotating mass. Then, we will use the same derived equation that we used before to find the inertia of the system. However, this time we only need the inertia of the triangle; therefore we will run an experiment without the triangle to find the moment of inertia of the disk. Then, we will run a separate experiment with the triangle in place. Finally, we could find the inertia of the triangle around its center of mass by taking the difference between the two values. The last step is to compare the experimental and theoretical values for the moment of inertia of the triangle around its center of mass. This process was used to find the Inertia of the triangle in two orientations:

• Orientation 1: Height=14.83 cm, Base=9.85 cm
• Orientation 2: Height=9.85 cm. Base=14.83 cm

For this lab, we setup the apparatus so that only the top disk would rotate. Like before, we would turn the air on. Then we wrapped the string of the hanging mass around the torque pulley so that the hanging mass was at its highest point. Here, we would start the measurements and then release the mass. Next, we used the angular velocity graphs to measure the angular acceleration as the hanging mass moved up and down. The angular acceleration was found using the slope of the angular velocity graphs.

In the picture above, the positive slope is the mass going down and the negative slope is the mass going up. Using both angular accelerations, an average angular acceleration was calculated. This process was repeated for the triangle in both orientations. We used this method to find the alphas listed below:

• Alpha Avg for disk w/o triangle = 2.17 rad/sec^2
• Alpha Avg for disk w/Triangle in Orientation1= 2.005 rad/sec^2
• Alpha Avg for disk w/Triangle in Orientation2= 1.826 rad/sec^2

Orientation 1:
The picture below shows the derived equation we used to find the inertia of the rotating mass.

•  Inertia of the disk w/o triangle is 0.00296 kg*m^2.
• Inertia with the triangle is 0.00321 kg*m^2.

If we take the difference of these values, we find that the Inertia of the Triangle around its center of mass is 0.00025 kg*m^2

Next, we found the theoretical value for the moment of inertia using:
(1/18)Mb^2, m=0.4533 kg and b=.0985 m.

• Theoretical Value= .000244 kg*m^2

If we compare the experimental and theoretical values, we find that there is a 4% Error

Orientation 2: Here, we followed the same process shown above.

 In the picture above we show that in this new orientation, the equation for Inertia around cm is still the same. In this case, mass still equals 453.3g but the base is now 0.1483 cm.

• Theoretical Value of Inertia around cm = 0.000554 kg*m^2

Alpha for disk w/ Triangle

• Inertia of the disk w/o triangle is 0.00296 kg*m^2.
• Inertia with the triangle is 0.00354 kg*m^2.

To find the inertia of the triangle around its cm, we still take the difference between the inertia of disk and the inertia of the disk w/ triangle.

• Experimental Value of Inertia around cm= 0.000564 kg*m^2

If we compare the experimental and theoretical values, we find that there is a 1.85% Error.

Conclusion:
The theoretical and experimental values for the moments of inertia around the triangles center of mass for both orientations are very similar. In the first orientation, where the shorter leg is parallel to the floor, the percentage error is 4%. In the second orientation, where the longer leg is parallel to the floor, the percentage error is only 1.85%. This shows us that the method upon acquiring the Inertia around the cm is accurate. The very small percentage errors can be attributed to the uncertainty in the measurements of the triangle, torque pulley, and disks. Furthermore, error can also be found in the different calculated angular accelerations of the experiment.

5-4-2015: Angular Acceleration

Lab 16: Angular Acceleration Part 1 & 2

Purpose:
The purpose of the experiment was to apply a known torque to an object that can rotate and measure its angular acceleration. Using this information, we can find a measured value for the moment of inertia of the rotating object.

Apparatus:

Apparatus used during experiment

 The apparatus shown above consisted of a rotational stand, rotational sensor, a hanging mass and a nearly frictionless pulley. The metal disks and torque pulleys are placed on the stand and are allowed to rotate. The disks can rotate together or independent of one another through the use of running air through a hose. A hose clamp is used to divert the air. If the clamp is open the disks rotate independent of each other and if it is closed the disks rotate together. On the sides of disks you can find 200 marks, alternating between black and white. The rotational sensor will use these marks to measure the angular velocity of the disks. Lastly, we have a frictionless pulley that allows for the hanging mass to travel freely.

Furthermore, we used LoggerPro to record and analyze the data of the experiment. First, we setup a Rotary Motion sensor for the apparatus and adjusted the sensor settings to 200 counts per rotation.

Calipers were also used to measure the diameters of the disks and pulleys used during the experiment.

Part 1: Angular Acceleration

Procedure:
 During this part of the lab, we conducted six experiments. Each with the same apparatus but sometimes using different disks, torque pulleys, and hanging masses.

 

Measurements of Disks and Pulleys used in the experiment: d= diameter, m=mass

• Top Steel Disk: d=12.630 cm  m=1360g
• Bottom Steel Disk: d=12.630 cm  m=1348g
• Top Aluminum Disk: d=12.630 cm  m=465g
• Smaller Torque Pulley: d=2.81 cm  m=10g
• Larger Torque Pulley: d=5.36 cm  m=36.3g
• Hanging Mass: Varies

Next, LoggerPro was setup as stated above and we connected the rotational sensor to the computer.     For the first five experiments conducted, the hose clamp was left open so that the disks would rotate independently of each other but a pin was placed in the stand so that only the top disk would rotate. For the sixth experiment the hose clamp was closed so that both disks would rotate.

Now the actual experiment. First, we would turn the air on. Then we wrapped the string of the hanging mass around the torque pulley so that the hanging mass was at its highest point. Here, we would start the measurements and then release the mass. Next, we used the angular velocity graphs to measure the angular acceleration as the hanging mass moved up and down. The angular acceleration was found using the slope of the angular velocity graphs.

In the picture above, the positive slope is the mass going down and the negative slope is the mass going up. Using both angular accelerations, an average angular acceleration was calculated. This process was repeated for all six experiments.

Essentially, what we are doing during this lab is seeing how changing one component of the experiment effects the angular acceleration of the system.

Expts 1,2, and 3: Effect of changing the hanging mass
Expts 1 and 4: Effect of changing the radius and which the hanging mass exerts a torque
Expts 4, 5, and 6: Effect of changing the rotating mass

The data table below shows the components of each experiment: mass of the hanging mass, torque pulley used, which disk is rotating, angular acceleration when mass goes up/down and the calculated average.

Furthermore, we conducted a separate experiment during experiment 5 to ensure that the given value of the angular acceleration of the system was correct. We achieved this by measuring the linear acceleration of the hanging mass. During expt. 5, we placed a motion sensor on the floor so that it measured the velocity of the hanging mass. We then took the slope of the velocity graph and found the acceleration.

 acceleration of hanging mass = 0.165 m/s

Now, we can relate this acceleration to the angular acceleration from expt 5 using the following relationship:  a = alpha*radius of torque pulley

 

From expt 5, Alpha = 6.588 rad/s^2 and radius of large torque pulley= 0.0268 m

Using these values, a= 6.588*0.0268= 0.176 m/s while expt value a= 0.165 m/s

As you can see the values are fairly close, this shows us that the given angular acceleration from the experiment is correct. The small discrepancy can be the result of friction present during the experiment as well as the uncertainties in the measurements.

 

Conclusion:  

If we look at experiments 1 through 3, we notice that the hanging mass does effect the angular acceleration of the disks. When the hanging  mass doubles from 24.6g to 49.6g the angular acceleration also doubles from 1.13 rad/s^2 to 2.28 rad/s^2. When the mass triples from 24.6 to 74.6g, the angular acceleration also nearly triples from 1.13 to 3.56 rad/s^2. Therefore, we can assume that the hanging mass is proportional to the angular acceleration of the disks.

 

In experiments 1 and 4, we can see that the radius of the torque pulley effects the angular acceleration of the disks. Here, the radius was nearly doubled from 1.40 cm to 2.68 cm and we see that the angular acceleration also doubles from 1.13 to 2.19 rad/s^2. Therefore, the radius of the torque pulley is proportional to the angular acceleration of the disks.   

 

Lastly, in experiments 4 through 6, we can see that changing the rotating mass has an effect on the angular acceleration of the system. From experiments 4 and 5, the mass changes from 1360g to 465 g nearly three times lighter. Here, the angular acceleration nearly triples from 2.196 to 6.588 rad/s^2. Furthermore, when the mass was almost doubled from 1360g to 2708g, the angular acceleration was almost twice as slow, from 2.196 to 1.107 rad/s^2. Therefore, we can assume that the rotating mass is inversely proportional to the angular acceleration.

 

 

Part 2:

Procedure: Using the data from Part 1, we found the moment of inertia of the disk in each experiment using a derived equation. We compared these values to a theoretical value of the moment of inertia of each disk.

The derived equation for the moment of inertia can be seen below, as well as the standard moment of inertia equation for a disk. All moments of inertia are in kg*m^2

Experimental and Actual moments of inertia for experiments 1 and 2

Experimental and Actual moments of inertia for experiments 5 and 6



Experimental and Actual moments of inertia for experiments 3 and 4

 

Next, we found the percent error in each experiment by comparing the experiment value of inertia to the actual value of inertia.
Percentage Error:

• Experiment 1: 9.96 % Error
• Experiment 2: 9.93 % Error
• Experiment 3: 5.90 % Error
• Experiment 4: 7.75 % Error
• Experiment 5: 3.99 % Error
• Experiment 6: 7.96 % Error

As you can see, the percentage error varies between the experiments. However, the percentage error is less than 10% so we can assume that the values are accurate and the discrepancy between the values is the result of uncertainties. The uncertainties may come from the measurements of disks, such as the mass and radii. Furthermore, the apparatus used for the experiment was not entirely frictionless.

 

Conclusion:

As stated above, we found the moments of inertia for each experiment and compared them to the theoretical values of inertia of the disks. We found that uncertainties exist within the experiment, since our values did not entirely equal each other. The uncertainties can be found in the measurement of the diameters and mass of the disks and torque pulleys. Uncertainty can also be found in the measurement of the angular acceleration of the rotating mass and the linear acceleration of the hanging mass. Finally, friction present in the system could also slightly effect the experimental values. If we were to reduce the uncertainty, better equipment would be essential as well as a larger data pool to minimize errors.



27-April-2015: Ballistic Pendulum Activity

Lab 15: Ballistic Pendulum Lab

Purpose:  To find Vo+/-dVo of a projectile in a ballistic pendulum.

Apparatus:  The ballistic pendulum contains a protractor that begins at the end of the block. A lever will move along with the block and give us a reading of an angle once the block stops.

In the apparatus shown above, a spring gun shoots a ball into the block. The block then moves along the protractor and stops at its max height. During this process, the lever moves along with the block and stops at the max height. This angle will help us determine the displacement of the block.

Procedure:
Measured Data:

• Mass of block(m2) = 80.9+/-0.1 grams
• Mass of ball(m1) = 7.63+/-0.01 grams
• Length of String (L) = 20 +/-0.1 cm

Next, we ran the experiment to find the angle the block moved along the protractor.

• Measured Theta = 17+/-0.5 degrees

We used this reading to determine the height displacement of the block.

We used the setup shown above to determine the height displacement of the block (h).
In this case: h = L-Lcos(theta)

• Calculated height = .0087 m

The next step is to find Vo. In order to find Vo, we used conservation of momentum and energy equations.

First, we solved our conservation of momentum equation for Vf. We then plugged Vf as "v" into our conservation of energy equation and solved for Vo.

Finally, we plugged in our known values into our equation for Vo and solved for Vo.
Known values:

Mass of block(m2) = 80.9+/-0.1 grams

Mass of ball(m1) = 7.63+/-0.01 grams

Length of String (L) = 20 +/-0.1 cm

Measured Theta = 17+/-0.5 degrees

Calculated height = .0087 m

Vo = 4.80 m/s

 

Now, we must solve for the propagated uncertainty in Vo since each of our values contains some uncertainty. As you can see, Vo depends upon m2,m1,L, and theta. Therefore, Vo(L,theta, m1,m2). The equation for dVo can be seen above.

dVo is the sum of the product between the partial derivative of a given in respect to Vo multiplied by that variable's uncertainty.

The picture above shows the partial derivative for each variable in respect to Vo.

Uncertainties for each variable:

• dL = .001 m
• d(theta) = .00873 degrees
• dm1= .00001 kg
• dm2= .0001 kg

Using the known values and equations stated above, we solved the partial derivative of each variable.

• dVo/dL = 16.98
• dVo/d(theta)= 16.07
• dVo/dm1= -575.12
• dVo/dm2= 54.24

Finally, we add the products of each partial derivative multiplied by that variable's uncertainty to get dV0.

• dVo = (16.98*.001)+(16.07*.00873)+(575.12*.00001)+(54.24*.0001)
• dV0 = 0.17 m/s

Therefore, the initial velocity of the projectile was 4.80 +/- .17 m/s

 

Conclusion:

In this lab, we found the initial velocity of a projectile in a ballistic pendulum. We solved for Vo using the principles of conservation of momentum and energy. Furthermore, the sources of error and uncertainty for this lab were accounted for in finding dVo. In finding dVo, we accounted for the uncertainty in all of the measurements: mass, length, theta.  We found that the uncertainty of Vo of the projectile was 3.5%. This percentage error is very low and shows us that our value of Vo is accurate.