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. 

3-May-2017: LAB: Ballistic Pendulum

This time, we tried to apply the information we learned to some things about our daily lives. Let's think about when a bullet was fired. It is hard to measure the velocity directly by measuring the distance and the changed of time. However, we can measure them indirectly. For example, we can calculate by the momentum of the system combine by the gun and bullet. In the lab, we picked up two ways to calculate it.

Purpose

Determine the firing speed of a ball from a spring-loaded gun.

Plan

We have two ways to determine the speed of a firing ball.
The first one is to detect it through energy conservative and momentum conservative. We set up an apparatus as following.

When the ball hit the block, we have the function

So as long as we know the speed of the block after the collision, we can know the speed of the ball. 

To calculate the speed of block, we can find how much energy does it have. By the function

We know if we could know the maximum height of the block, then we could find out the energy. 

Therefore, by measuring the angular of the bar, we can get the speed of the ball. Moreover, because it is also hard to measure the height directly, we would measure the angle of bar use the triangle to find the height. 

Another plan is using the most original way, which is fired the ball horizontally.

We can measure the distance it falls down and the distance it goes. To measure how far it goes, we place a piece of carbon under, because it is also hard to Then by the functions

We can calculate the speed easily. 

Set up & data

Experiment 1:

Grab the apparatus that professor has already made for us. 

Measure the mass of the ball and block. 

Level the apparatus including the whole plate and the block. 

Pull back and lock the spring into position, place the ball, and zero the angle. 

Fire the ball and collect the angle. 

Repeat least two steps, until we get enough data. 

Experiment 2:

Then, grab the apparatus on some heights. (We choose on the height of three books, which is 20.5 cm)

Level the apparatus. 

Pull back and lock the spring into position. 

Fire the ball and collect data. (Make sure that the apparatus will not move)

Analyze

(Notice: all the data were converted into international unit before being plugged in)

In the first experiment, 

because the have the functions

We can solve the initial speed is 

Then, plug all the data,

We can get the average speed is 4.88 m/s.

In the second experiment, 

from the functions

We can solve the initial speed is 

Next, plug in the data

And calculate the average is 5.17 m/s

Uncertainty

In the first experiment, we measure 4 different data, and it is acceptable to have some uncertainty. 

By doing some derivative, we can get the uncertainty of the first speed is 

Plug in the data 

In the second one, by doing the derivative we can get

Then, the data tells

Comparing with those data, we can briefly say that the data is fitted within uncertainty, which means the data is good. 

Conclusion

In this lab, we measure the fired ball in two different ways. Except we could make sure how much the initial speed is, we can also compare which way is better for measuring data. 

Moreover, there are some differences between these two data. The first reason will be we actually did experiments in two different days (which might cause a slight change of apparatus). It might also be because the apparatus did not level. We can found out that the beginning data of the first experiment is much closer to the second data (which has less uncertainty). One possible reason is after we fired the ball twice or three times, the apparatus became less leveled. Another reason might be when we measured the distance of the second experiment, we did not measure it parallel with the trajection (which will cause the distance a little bit larger).

24-April-2017: Collisions in two dimensions

In the previous study, it is easy to know that momentum are always conservative in one dimension since we can prove it with the function

With the Newton's third law, they always have the same magnitude but opposite force, and the time will be equal. Thus, their momentum is conservative. However, does it still work in two dimensions or more dimensions. Today, we did a lab about the collisions in two dimension.

Purpose 

Look at a two-dimensional collision and determine if momentum and energy are conserved.

Plan

Find a smooth plane, and level it (as following) to decrease the influence of friction force and gravity.

Fix a camera with slow mode on the top.
Then, we collected two videos when a steel ball hit a steel ball, and when a steel ball hit marble ball.
Next, use Logger Pro to analyze the data of balls, and get the velocity.
Plug them into the functions of momentum and energy to see whether they are conservative.

Set up

Gently set up the stationary ball on the leveled glass table.
Then, level the table.
Aim the rolling ball so that it hits the side of the stationary ball, and collect the video by iPhone.

Experiment 1

Before collision

 After collision

Experiment 2

Before collision

After collision

Plug the videos into Logger Pro. Set the frame rate to 120 and advance the movie 4 frames after adding a new point.
Collect the points and set the length of the table to get the data of velocity.

Experiment 1

Moving steel ball's x and y-direction displacement  

Motionless steel ball's x and y-direction displacement

Experiment 2

Steel ball's x and y-direction displacement

Marble ball's x and y-direction displacement

Weight the marble ball and the steel balls.

Analyze

For momentum, we can divide them into x-direction and y-direction.
And use the function

For energy, we can calculate the velocity of each ball the directly use the function

For the first one

For the second one

By doing the subtraction, we can know that the differences and the changing percentage are

Experiment 1

Experiment 2

Thus, we can roughly make an assumption that momentum is still conservative because all the most momentum change is within 7%; while the energy is not since both experiment's energy changed over 30%.

Conclusion

For the collision in two dimensions, it still obeys the law of momentum conservative, since most change is with 7%. Except we will make some mistakes on collecting data (it is hard to avoid), the main reason for that is when the collision happens, there is a part of the energy which turns into heat. Moreover, because the balls are not smooth, they might be spinning after collision, and the span would make a larger friction with the plate, which will produce more heat.
In fact, even in three dimensions, the collision still has momentum conservative. Because we are detecting the objects with low speed, which implies that they follow the Newton's Laws. Then, for each direction,

then, the momentum will not be created or disappear. 

However, there is still some energy consumed since both experiment's energy is changed at least 30%. It is easy to know that there is some friction force between balls and plate, which will produce some heat (cost energy). Moreover, when they collided, they would also produce some heat because they are not elastic, and they will comsume some energy in deformation. 

In this lab, there are several factors that maybe affect the data. 

The first one is whether the table is well leveled. If it is not leveled, the gravity will participate as an external force and affect the impulse and momentum. Moreover, it always has some friction force that will influence the momentum as an external force. 

The second one will be the points collection. It is hard to hit the same place of the ball over and over again. We might hit the point next to it, which might affect the velocity. 

The third one will be the camera. There is a possibility that the camera is not leveled. If it is not, then it will slight influence the result because the distance measure will be different from the parallel one.