22-May-2017: Lab 17: Finding the moment of inertia of a uniform triangle about its center of mass

Lab 17: Finding the moment of inertia of a uniform triangle about its center of mass

Author: Tian Cih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: May 22th, 2017

Purpose of the lab: measure Icm of a right triangle experimentally and compare with theoretical values (1/18 * M * B^2, done in video problems).

Theory: 1/18 * M * B^2 

By measuring angular accelerations (apply linear fit to velocity vs. time graphs) for both ways (up and down), we use the formula given in Lab 16 to determine the moment of inertia.
I = mgr / [(|alpha-up| + |alpha-down|) / 2] - mr^2 for both situations of naked apparatus and apparatus + triangle plate.

First we measure MoI for naked apparatus. Then we put a triangle onto this, and measure again. Then we reverse this triangle and do the same process.The subtraction of MoI combined and MoI "naked" is the MoI of the triangle plate. That is, Icm of a right triangle = I(apparatus + triangle) - I(apparatus)

Apparatus/Experimental setup: 

Holder and disk, triangle plate, hanging mass with string and "frictionless" pulley

Measured data:
Mass of triangle plate = 455 grams
Mass of hanging mass = 25 grams
Base of triangle plate = .098 meter
Height of triangle plate = .149 meter

Angular acceleration graphs for naked apparatus:

 Angular acceleration graphs for apparatus and vertically-placed plate combined:

 Angular acceleration graphs for apparatus and horizontally-placed plate combined:

Analysis:

Conclusions:

In this lab, we can see MoI don't match very well. The result is smaller than what we computed.The measurements of radius, lengths or heights contribute little to the inaccuracy. There are many possible reasons to cause this error.

22-May-2017: Lab 18: A Lab Problem - Moment of Inertia and Frictional Torque

Lab 18: A Lab Problem - Moment of Inertia and Frictional Torque

Author: Tian Cih Jiao

Lab Partners: Weisheng Zhang, Kitarou

Date: May 17th & 22th, 2017

Purpose of the lab: 
The purpose of this lab is to determine the moment of inertia given a large metal disk on a central shaft. After we have the results, we calculate how long it should take for the cart to travel 1 meter from rest. Compare predicted values with experimental values and hopefully we will see that these figures match.

Introduction/Theory:

Part 1:

What we are given is the total mass of disk plus shaft. Since the effectively rotating part is the disk only without shaft, we need to make measurements to calculate the actual volume of the rotating part (see how much it matters comparing to the total volume) so that we know how much mass the disk occupies. In order to determine angular acceleration, we look at any arbitrary point at the edge of the disk.
Frictional torque
Part 2:
We connect the apparatus with a 500-gram(hopefully) dynamics cart. Before we carry out part 2, we calculate how long it should take for the cart to travel 1 meter from rest along the track assuming that it is leaned 40 degrees above the ground (predicted value).

Apparatus/Experimental setup:
Apparatus: Metal disk on a central shaft, calipers, iPhone for video capture, tape for labeling any arbitrary point chosen at the edge of the disk, MacBook with LoggerPro, cart, air track, string, books or something else to stabilize the angle of the track leaning above the ground.

For part 1, we need to make measurements to the disk and the shaft (diameter/length). By calculating the volumes separately (we see shaft as solid cylinders and the rotating part as a solid disk) and distributing the total mass equally, we can have a great approximation of MoI of the rotating part of apparatus.
A video is captured for future analysis on angular acceleration of the disk as it slows down. We trim the video and only need the last part of the video. but when we plot out the dots, it is pretty obvious that the distance between the dots starts to increase, which is an indication of a slow-down motion.

Then we get a graph of omega vs. time and we linear fit it to get the angular acceleration.
Part 2: Angular acceleration (alpha-f) is obtained by applying linear fit to omega vs. time graph. Since total torque minus frictional torque leads us to the "actual" acceleration (see calculations ini the following sections), we can thus have a great idea of how long it will take for a 0.5 kilogram cart to slide along the track for 1 meter.

Measured data:

Mass of the disk + shaft = 4.802 kilograms
Diameter of the disk = 19.594 centimeters
Width of the disk = 2.37 centimeters
Length of the shaft for each side = 5.65 centimeters, 5.54 centimeters
Diameter of the shaft for each side = 3.13 centimeters
Volume of the disk = pi * (19.594/2)^2 * 2.37 = 714.64 cm^3
Volume of the shafts = pi * (3.13/2)^2 * (5.65 + 5.54) = 86.1 cm^3
Calculated "actual" mass for the rotating part = 4.802 * 714.64/(714.64+86.1) = 4.286 kilograms

Analysis:

Here is the calculation of the prediction:

Conclusion:
In this lab, we use camera to find out the angular acceleration, then use the calculated acceleration, we compute the predict time. Then we use cart to find the real time and average it, which is very close.

20-May-2017: Lab 16: Angular acceleration

Lab 16: Angular acceleration

Author: Weisheng Zhang

Lab Partners: Tian Cih Jiao, Kitarou, Nina Song, Joel Cook, Eric Chong

Date: May 10th & 15th, 2017

Purpose of the lab:
The purpose of the lab is to measure the angular acceleration when a known torque is applied to a rotatable object. After figuring out the angular acceleration, we use the collected data to find moment of inertia in these cases.

Theory/Introduction:

Part 1:

For the first part, we use control the changing to get the result of this lab.

For EXPT 1, 2, 3, we change the mass of hanging mass.

Then for 1 and 4, we compare the change of radius which can change the torque.

Then 4, 5, 6 are for figure out the effect of changing the rotating mass.

Part 2:

For the convenience of calculation, we let counterclockwise direction be positive and the other way be negative.

A counterclockwise torque caused by the tension in the string is speeding up the hanging mass to descend, while the clockwise friction torque is slowing down the mass. Therefore,

Torque of string - Torque of friction = MoI (Moment of Inertia) of disk * angular acceleration down

After the hanging mass reaches its possible lowest point, both of the torques start act in the clockwise direction, thus slowing down the disk.

-torque of string - torque of friction = MoI of disk * angular acceleration up

Then what we need to do is calculating data.

Summary of apparatus/experimental procedure:
Pasco rotational sensor, Macbook Pro with LoggerPro, Lab Pro Kit, disks of different materials, torque pulleys, hanging mass and hose clamp are used for this lab.
Set up the apparatus. Due to the poor timing resolution of the sensors, we can not take advantage of the default angular acceleration vs. time graph. Instead, we need to use the graphs of angular velocity to measure the angular acceleration as the mass moves down and up. Hose clamp is used to control the number of disks on operation. It is open for one disk operation and closed for two disk operation. We can turn on the compressed air so that the disks can rotate separately. It is important to make sure that the disk can rotate independently when needed.

Measured data:
Mass of the top steel disk = 1357 grams
Diameter of the top steel disk = 126.1 millimeters
Mass of the bottom steel disk = 1348 grams
Diameter of the bottom steel disk = 126.1 millimeters
Mass of the top aluminum disk = 466 grams
Diameter of the top aluminum disk = 126.1 millimeters
Mass of the smaller torque pulley = 9.99 grams
Diameter of the smaller torque pulley = 24.6 millimeters
Mass of the larger torque pulley = 36.59 grams
Diameter of the larger torque pulley = 49.7 millimeters
Mass of the hanging mass supplied with the apparatus = 24.62 grams

Here are the graphs from LoggerPro:
EXPT 1 (hanging mass only, small torque pulley, top steel disk)

 EXPT 2 (2 * hanging mass, small torque pulley, top steel disk)

 EXPT 3 (3 * hanging mass, small torque pulley, top steel disk)

 EXPT 4 (hanging mass only, large torque pulley, top steel disk)

 EXPT 5 (hanging mass only, top aluminum torque pulley, top aluminum disk)

 EXPT 6 (hanging mass only, large torque pulley, top steel + bottom steel disk)

Analysis:
This is the calculation from one of our member.

Conclusion:
Angular acceleration increases when torque increases, radius increases or MoI decreases. This conclusion is made through comparison experimental sets mentioned in previous parts.

1-May-2017: Lab 16:

Lab: Ballistic Pendulum

Author: Tiancih Jiao

Lab Partners: Nina Song, Roya Bijanpour, Joel Cook, Eric Chong, Kitarou Chen, Lynel Ornedo, Weisheng Zhang

Date: 05/02/2017

Purpose of the lab: The purpose of the lab is to determine the firing speed of a ball from a spring-loaded gun.

Theory/Introduction: A spring-loaded gun fires a ball into a block in this case. The block is supported by four string. The ball is launched so that it "sticks" with the block moving together upward through some angle. Eventually, they will stop at some point. The angle is measured by the angle indicator, where it points at an angle after having a hard contact with the block-ball mass.

Let the mass of the ball be m and the mass of the block be M. The first part of the motion is associated with conservation of momentum. m * V0 + M * 0 = (m+M) * Vf.

Therefore, Vf = m * V0 / (m+M).

For the second part, we should conservation of energy to compute.

(m+M)gh=0.5*(m+M)V^2

then drop (m+M) each side, we will get V=sqr(2gh)

Summary of apparatus/experimental procedure: 

First we need to measure both m and M. Level the base of the apparatus and the block so that the measured angle becomes accurate enough. Pull back the string in order to load the firing ball into position and place the angle indicator to zero degrees. Once the ball is launched, record the maximum angle the indicator reaches. Repeat this process five times to get an average.

Measured data:

m = 7.59 grams

M = 78.9 grams
L = 21 centimeters

1st attempt: 22.5 degrees

2nd attempt: 24.2 degrees

3rd attempt: 27 degrees

4th attempt: 24 degrees

5th attempt: 23.5 degrees 

then V average = 24.24 degrees.

Calculation shown below: 

Then for the second part, we use projectile motion to test the horizontal velocity of the ball which is actually the V0, and compare it with V0 we got in the first part.

V = 5.86 which is not very close to the velocity we calculated in the first part. But it is also possible, because there might some errors in this lab such as air resistance. Also, we did the first part one week ago, so there was one week interval that the spring might have changes in the time period. Those are all the errors that might affect the result.

Conclusion: 

In this lab, we use two different method to find out the initial speed of ball from the gun. Then we compare two results and find that there are still many errors in this lab.