20 May 2015. Conservation of Angular Momentum

Purpose
to observe the conservation of angular momentum (from kinetic to potential) of a meter-stick and a clay.

Apparatus:

A set up shown in the picture. There is this meter-stick that is going to be released parallel to the floor, and purposedly aim to hit a clay in the floor. We are going to measure the kinetic energy that is going to be transformed to potential.

Procedure

1. We set up the apparatus as follow.
2. Then, we align the camera right in the middle of the impact.

3. Weight the mass of the clay and the ruler.

4. Record all the data.

Data Analysis:

• Now, for the experimentally part, we can analyze the video capture, set the origin, and add one point series for the height.

h = 0.2904 m

• For the true value: For the first section, we use conservation of energy to calculate the angular speed right before the impact of the clay.

• Then for the impact, we can find the omega final (ruler+clay).

• Then, we use conservation of energy again to determine the final height.

so the height is 0.401 m

• So the percent error is: (0.401-0.2904)/0.401*100% = 27.58%

Conclusions

• We manage to find the height using the law of conservation of energy and angular momentum.
• The percent error is quite huge, around 25%.
• These errors are quite not surprising, for this reason:
• The fact that the energy is really conserved in this collision. It transformed to heat and sound, not only the height.
• Uncertainties in measuring length and mass of the ruler and clay.
• Rounding significant figures.

13 May 2015. Inertia of Triangle.

Purpose

Part 1
to find the moment of inertia of a triangle being rotated at the center of mass (twice, with the width and the length on the bottom) experimentally.

Part 2
to find the moment of inertia of a triangle being rotated at the center of mass with calculation (true value) and compare it to the experimental value.

Apparatus:

Angular acceleration apparatus. This apparatus is built of pair of polished disks with a tiny hole that will accept air from the machine, to remove friction and allowing the disks to turn with minimal force. We will use a light hanging mass with a string tied around a torque pulley connected to the disks to provide torque, then measure the angular acceleration experimentally.

However, in this lab we used a disk that did not push air through it, plus now, we will attached triangle on top, as the picture shown.

Procedure and Data analysis

for this lab blog, the procedure and data analysis will be described individually under one part section. Since there are two parts of this experiment.


Part 1 (find the moment of inertia experimentally)

Procedure

1. Set up the apparatus so that it can measure the angle like our previous lab.
2. We need to measure the width, length, and weight of any apparatus used for data record.

3. We record the data of the angular acc. without any triangle mounted on it.
4. Then, with the triangle in the horizontal direction.
5. And the vertical direction.

Data Analysis

• First. recall that by measuring the angular acceleration of the system, we can determine the moment of inertia of the system.
• When the disk is spinning by itself. we will get the moment of inertia of this disk.
• Then, if the triangle is spinning, we will get the moment of inertia of the triangle plus the disk.
• So, the moment inertia of the triangle is the difference between moment of inertia of these measurements.
• Data recorded:

• And the radius of the large torque pulley is 0.025 m.

• We have, let's call it, three expt.: the disk by itself, the triangle in horizontal, and vertical.
• Now, we have data of angular position, angular velocity, and angular acceleration. The graph of angular acceleration vs time is useless due to the poor timing resolution of the sensors. So to get the angular acceleration we need to plot velocity vs time, and the slope is our answer! (Recall the kinematic equation ωt = ωo + αt)
• But,  we realize that there is some frictional torque happening here (even though we definitely minimize friction using air) as we plot the velocity vs time, one of the slope is higher when the mass is going down, meaning the acceleration is greater, and the acceleration is slower when the hanging mass is going up. (see the graphs)

Disk by itself

vertical triangle

horizontal triangle

• Therefore, for the angular acceleration, we need to find the average of these data.
• This the list of experiments that we did:

• So now, let's calculate it. Using the same concept from the last angular acc, lab, we can find the moment of inertia as follow.

• Now, let's simplify this equation as the mgr and mr^2 is always going to be the same, and calculate the moment of inertia.

Part 2 (find the "true" value of moment of inertia)

Data analysis

• First. we can find the moment of inertia of the triangle being rotated at the edge.

• Now, by the parallel axis theorem, we can find Icm.

• With L being the length of the triangle or the height of the triangle
• Now, we can calculate the moment of inertia of the triangle in vertical and horizontal direction.
• And the percent error.

Conclusions

• We manage to find the moment of inertia experimentally and the average percent error is approximately 5 %, so the result is pretty good.
• There are some small errors in this experiment that comes from;
• Some uncertainties when calculating the radius and others.
• Uncertainties from weighing the weight.

11 May 2015. Inertia and Frictional Torque.

Purpose

1. To find the frictional torque of an apparatus designed by finding the moment of inertia of the wheel.
2. From that, we will calculate the inertia for the wheel, and then use it to determine how long a cart hanging off the wheel by a string would take to go a meter-long ramp.

Apparatus:

The wheel. This is the steel cast in one piece with the cylinder axle protruding on both sides. Imagine the object as three pieces: a solid disk, and two cylinders on the sides. The total mass of this is 4.840 kg.

Procedure and Data analysis

for this lab blog, the procedure and data analysis will be described individually under one part section. Since there are two parts of this experiment.


Part 1 (finding the friction torque)

Procedure

1. We need to measure the radius and all the mass needed to find the inertia.
2. Then we also need to measure the angular deacceleration, which is also caused by the friction torque.
3. To find this, we need to do video capture.
4. We line up the camera, then do add point series at every same point when the wheel has finished rotating one full rotation.
5. Note that, if there is no friction, the wheel will turn infinitely, but of course it doesn't. So, there must the angular deacceleration that we mentioned.

Data Analysis

• Now, we let's find the inertia first.
• We know that the whole mass is 4.840 kg (written there). We need to divide this mass as three pieces: a solid disk, and two cylinders on the sides. Let's do this.
• We already measure all the radius of this disk as follow.
• Then we calculate the volume of each one, and compare it to the mass. Remember that mass and volume has a linear relationship.

• So, we got our small cylinders masses as .653 kg and the solid disk 4.187 kg.
• And the total inertia is the combined inertia of both disk as follow.

• Next, let's find the angular deacceleration. When we do the video capture analysis (explained in the procedure section), we will have the delta t time, when each disk is finished rotating one full rotation. And from that we can determine the angular speed, which will be slower each time due to friction.
• We plot omega vs time to find the slope as our angular acceleration.

• Next, we transformed this line equation into:

• Yes! We finally found our torque friction in this wheel.

Part 2 (finding the time when a cart attached to this wheel by string will travelled 1m)

Procedure

1. Set up the ramp as the picture shown.

2. Now, attached a cart to the smaller wheel with a string.
3. Do not forget to weigh the mass of the cart and measure the angle of the ramp.
4. Then give a gentle push to the wheel (not to hard, not to soft), making the cart goes down.
5. When the cart stars going down, hit the stopwatch, and press it again when the cart reaches 1 m long.
6. Do this three times and compare it with the true calculated value.

Data Analysis

• Now, let's calculate the calculated true value.

• Now, from the calculated value, we should get 7.34 s for the cart to go 1-m long.
• Let's calculate the percent error.

• The percent error is under 5 percent.

Conclusions

•  We manage to calculate the friction torque.

• And from there, calculate the time of an object that is attached to the wheel by considering the equation of the torque, angular acceleration, and inertia.

• There are some small errors in this experiment that comes from;

• When we add point series, we did not really click it precisely. This will alter the velocity and the position of angular movement.

• Friction from the ramp, but we ignore that.

• Some uncertainties when calculating the radius and others.

• Uncertainties when calculating the time with a stopwatch.

4 May 2015. Angular Acceleration.

Purpose

Part 1
to observe the component of angular acceleration by applying different radius, torque, and mass.

Part2
Using the data recorded from part 1, we are going to calculate the moment of inertia and compare it to the true value.

Apparatus:

Angular acceleration apparatus. This apparatus is built of pair of polished disks with a tiny hole that will accept air from the machine, to remove friction and allowing the disks to turn with minimal force. We will use a light hanging mass with a string tied around a torque pulley connected to the disks to provide torque, then measure the angular acceleration experimentally.

Base Theory

• We know that the the equation for torque is: 
• τ = Fd = Iα

• Using the apparatus described, we are going to observe whether this equation really applies to the real life. Analyzing the apparatus used, we can modify the equation and say that the torque F times d is equal to the Tension times radius.

Procedure and Data analysis

for this lab blog, the procedure and data analysis will be described individually under one part section. Since there are two parts of this experiment.


Part 1 (observing the equation of the torque)

Procedure

1. Set up the apparatus so that it can measure the angle as the picture shown.

2. There is 200 marks on the top disk, and this is where the sensor rotary motion come in. We need to set up 200 counts per rotation for the rotation, or else the apparatus going to read 360 counts per rotation as the default set up.
3. To make sure that it gives reasonable data, we also put a motion sensor below the hanging mass and record the data.

Seems reasonable....

4. There are several different aspect that are going to be tested that will be explained in Data Analysis section.
5. To play with the top disk or the combination of the bottom and the top, we need to play with the air hose on the side.
6. We are going to record the data six times with changing properties of angular acc.

Error fix

Our group forgot to set the 200 counts per rotation before we took the data. But, no worries, we can modify the data that we have so that it reads 200 counts per rotation.

What the sensor reads is 360 counts per rotation, so it actually reads: (400/360) = 1.11111 slower than it should be. But, remember that this is in degrees. To convert it to radians we times it to 2pi and get 6.981317 slower. That means we need to times the radians to this number to get true value. Since we can't change the value of the radians, we can change the velocity (which is what we concern about when doing the analysis), which is derived from the radians data and set up the column as follow:

Done! This will fix everything.


Data Analysis

• Now, we have data of angular position, angular velocity, and angular acceleration. The graph of angular acceleration vs time is useless due to the poor timing resolution of the sensors. So to get the angular acceleration we need to plot velocity vs time, and the slope is our answer! (Recall the kinematic equation ωt = ωo + αt)
• But,  we realize that there is some frictional torque happening here (even though we definitely minimize friction using air) as we plot the velocity vs time, one of the slope is higher when the mass is going down, meaning the acceleration is greater, and the acceleration is slower when the hanging mass is going up. (see the graphs)

Expt 1: hanging mass, small pulley, top steel

Expt 2: 2 x hanging mass, small pulley, top steel

Expt 3: 3 x hanging mass, small pulley, top steel

Expt 4: hanging mass, large pulley, top steel

Expt 5: hanging mass, large pulley, top aluminium

Expt 6: hanging mass, large pulley, top steel+bottom aluminium

• Therefore, for the angular acceleration, we need to find the average of these data.
• This the list of experiments that we did:

• Comparing expt 1,2,3 (the tension changing represented the weight of hanging mass)

• Comparing expt 1,4 (the radius changing represented by the size of torque pulley)

• Comparing expt 5,6 (the mass (inertia) changing represented by the type of disk or rotating mass)

Part 2 (comparing the moment of inertia)

Procedure

1. We are going use the data from part 1.
2. We are going to measure the radius and the weight needed to find the inertia.

Data Analysis

• Now, we have data to calculate the experimental value of inertia as follow:

• And we derive the equation to calculate the moment of inertia (handout):

• Now, comparing it to the real value...

• The percent error is

Conclusions

•  the equation τ = Fd = Iα   turns out to be very reasonable.
• We have a very little error in determining the moment of inertia (under 5%), but when we are determining the moment of inertia of the two disks welded together (aluminium plus friction) we have 28.5 % error which is huge. This is probably due to friction in the axis that is not calculated. If we got 1.6 as our angular acceleration, then we got our inertia value, but we got 1.1. So, probably friction is the main reason of this error.