31-May-2017: Conservation of Energy/Conservation of angular momentum

We learned the conservation of linear momentum before, which is when the collision happens the momentum always conserved. Also, when the object did not have any transformation or any other force on it, the energy will also conserve. So will it be similar to the object when it is rotating? Today, we set up a lab to study on it.

Purpose

Verify the conservation of energy and angular momentum.

Plan

We release a meter stick, pivoted at or near one end, from a horizontal position.
When the meter stick reaches the bottom of its swing it collides inelastically with a blob of clay. The meter stick and clay continue to rotate together to some final position.

Then, measure the appropriate masses, etc.
Make a prediction for how high the clay stick combination should rise by the following functions.

Then capture the experiment on video and compare your actual results with your predictions.
If the energy and angular momentum are conserved, it should give us the same result as what we calculated by the theorem.

Set up & data

We set up the apparatus as follows. (Notice: there are some nails at the end of the rule to make sure that the clay would stick with it, and the pivot point is at the 10cm.)

Measure the weight of ruler and the clay. We got the rule is 97.39g, and the clay is 19.89g. 

Use the iPhone's slow motion camera to collect the video when the ruler started to drop down till it reached the highest point. 

Next, use the Logger Pro to get the data. 

And we get it would reach 37cm. 

Analyze

Through the functions 

We predicted the height of the clay should raise 0.394 m 

Compare with the actual value, 

(0.394 - 9.3743)/0.394 ≈ 5%

which is pretty close. 

Thus, we said the energy and angular momentum is conserved during the similar process as linear. 

Conclusion

This lab verified that the conservation of energy and angular momentum as it only give us the 5% difference in the result. Due to some measure uncertainty (like the capture of video might be not so exact, the measure of the ruler's and the clay's weight), there is some acceptable error on it. Moreover, like the linear conservation, there might be some friction force on the ruler, and the inertia may not be so accurate since the ruler is not uniformity. 

Overall, this lab is good enough to proof that the energu and angular momentum are conserved during each part of collision. 

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

In the past, we have already learned how to calculate the moment of inertia of a triangle by the parallel theorem.

Then the inertia will be

Because the horizontal center is in 1/3 of the base from the left side, 

Today, we tried the formula in the real triangle to see whether it works.

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.

Plan

Set up the apparatus as lab on May 8th, which is

Because it can reduce the friction as much as possible by the flowing air.  

Rotate it first to get the inertia of the part without the triangle. 

Then use the same step as that lab to calculate the total inertia (which calculates the angular acceleration first, then use functions to calculate it; again, to reduce the error we use the average of the angular acceleration). 

And then put the small part component on the top to hold a standing triangle. 

Lie down the triangle and repeat the step above.

Compare the calculated value with the real value to see whether it fits. 

Set up & Data

Set up the apparatus as following, connect them to the laptop, and set up everything in Logger Pro. 

Rotate it without the triangle to get the inertia of the rest part. And we got

Weight the small mass and the triangle. We got the data. 

Rotate the triangle by release the small mass. And we got the data. 

After lie down the triangle, rotate it and collect the data again. 

Analyze

By the formula above we can first calculate the inertia of triangle.

Then use the other formula to calculate the actual inertia, then subtract them to get the triangle's inertia.

Compare them

Both of the difference are within 5%, so those are acceptable data, which implies that the parallel theorem works in these situations. 

Conclusion

In general, this lab confirmed the parallel theorem works in different situations by getting almost the same (since there are only 5% off). However, there still may be some problems on measures, which will give us some uncertainty. Moreover, we did not think about the deceleration on the apparatus. Even though this apparatus is very frictionless, we can find out every time the hanging mass goes up and down, the acceleration decreases a little bit, which implies that there is some frictional torque in it. It might slightly influence the result. Moreover, we only did the experiment once. There might be some unexpected factors (like someone accidentally hits the table).

To get a more precise result, we could consider about friction/resistance and repeat the experiment a few more times. 

22-May-2017: Momentum of Inertia and Fictional Torque

In the past, we have already verified that the calculate inertia meets with the actual inertia. In theorem, because at any moment, rotational friction is always static friction, which implies that it should not consume any energy. However, from small marble to car, we know that it still consume some energy. Today, we did a lab on a rotational disk to study the rotational friction (or frictional torque).

Purpose

Find out the frictional torque of a spinning disk.

Plan

First, calculate the disk's momentum of inertia by its shape and mass.

Then, capture a video that the disk is rotating and decelerating.
Use the Logger Pro to find out the acceleration of the disk.
By the function,

we could know the frictional torque. 

Then, use another model which is the same disk and pull by a car. 

Predict the time that it need to slide for a distance by functions. 

Compare it predict time and actual time to see whether the frictional torque is correct. 

Set up & Data

1. Make appropriate measurement of the rotating part of the apparatus and determine its moment of inertia. 

2. Spin the apparatus. Use video capture determine its angular deceleration as it slows down. Calculate the frictional torque acting on the apparatus. 

which says the angular acceleration is -0.4395 r/s^2

3. Connect the apparatus to a dynamics cart. Set up the apparatus as follows. The cart weights 572g. 

4. Let the cart slide down for 1 meter, and record the time. And we got 6.92 s. 

To record the most accurate time, our team (4 people) collected the time together, and we use the average of them.

Analyses 

For the inertia, we can get from the data of shape and mass. 

Then, total inertia is 0.0213858583075759 kg*m^2. 

So by the function 

we know the frictional torque is -0.00939908472617959 N*m

Solving the functions above, we can get the angular acceleration is 

in this case is 2.590426854535 r/s^2

Then by function 

We can calculate the time is 

Plug in the number, and I get 7.03504746291719s, which is only 2% off. 

Conclusion

This result comes out that the predicted value is pretty close to the actual value. And because the time is collected by people, there might be small different from actual value. Moreover, there are some uncertainty in the measurement. One thing really important is that while the cart was sliding down, there are also some frictional torque on the wheels, which is ignored. 

In general, the result confirmed what we thought, which is there is some frictional torque while the disk is spinning, and the magnitude will not change even though we apply different force. It is some kind similar to kinetic friction. 

8-May-2017:Angular Acceleration

In the past, we calculate the inertia of a disk, which is

but is there some other factors that we might mistakenly ignore. Today, we did a lab on different disks to see whether the real situation is matched what we calculate. 

Purpose

Calculate the disks' momentum of inertia to see whether it matches what we calculate.

Plan

We set the lab into two part. The first one is to collect data in different situations; the other one is using the data we collected to calculate the momentum of inertia.
First, we have to know some basic information about the disk like diameter and mass.
Then, set up the apparatus as follows.

Connect it with Logger Pro, then collect the data in different situations. 

Then, analyze the force on the system. 

On the disk and block, 

Then, we can know when it is going up

When it is going down

Solve these functions, we got 

Then, considering the friction force, we can get the functions from torque

After plugging everything, we can get 

Eventually, compare the data we get from the lab and calculation to see whether they are fit. 

Set up & Data

In the first experiment,
1. Measure each of the followings to significant figures
              Including the diameter and mass of the top steel disk, the bottom steel disk, the top aluminum disk, the smaller torque pulley, the larger torque pulley, and the mass of hanging mass.

2. Plug the power supply into the Pasco rotational sensor.

3. Set up the computer.
4. Make sure the hose clamp on the bottom is open so that the bottom disk will rotate independently of the top disk when the drop pin is in place.
5. Turn on the compressed air so that the disks can rotate separately.
6. With the string wrapped around the torque pulley and the hanging mass at its highest point, start the measurements and release the mass.
7. Change the hanging mass, torque pulley, and the top disk, collect data.
    Experiment 1, 2, and 3: effect of changing the hanging mass
    Experiment 1 and 4: effect of changing the radius and which the hanging mass exerts a torque
    Experiment 4, 5, and 6: effect of changing the rotation mass

In the second experiment,
1. by functions we calculate above, we can plug them into Excel to get inertia.
2. Calculate the inertia by the formula.

Analyze

From the table, we can know the percentage of difference are 

All of them are within 10%, which means the actual inertia is the same as calculating inertia if there is no other friction force and measure error. 

Conclusion

Through this lab, we could know the calculation of inertia is the same as the actual inertia. In general, we could know that the hanging mass or the magnitude of force does not effect the inertia, the change of radius and mass are exactly the same as we expected. The most error might be we do not know about the friction torque of disk, so we can only assume they are the same. Moreover, even though we use the air on the pulley, it still spun, which might take a part of the energy.