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. 

19-April-2017: Impulse-Momentum

It is very easy to discover that there is relationship inside force, time, mass, and velocity. When we push an object with a larger force or a longer time, the velocity will increase larger; in the meanwhile, if the object has a larger mass, then it will gain a smaller velocity. Thus, velocity has a positive relationship with force and time, and it has a negative relationship with mass. So we can make a conclusion that Ft/M = v or Ft = Mv. However, how could we know whether it will really apply to the object? In today's lab, we study the relationship inside the impulse and momentum.

Purpose

Observe and verify the impulse-momentum theorem.

Plan

Observe two different kinds of collisions. One of them is using a stopper, which gives a longer collision and then use a larger mass to do the experiment again. The other one is using a nail to poke into clay to give us a shorter collision (as the diagram showing in the following). 

Then use the integral to calculate the impulse, and use the velocity times mass to calculate the momentum. 

Then, compare impulse and momentum. If the function applies to the objects, then the impulse and momentum should be equal. 

Last, use the functions in Newton's system to prove the relationship. 

Set up

Fasten the force probe securely to the cart so that the stopper extends beyond the front of the cart.

Set up the motion detector as shown. Level the ramp. 

Determine what quantities you will need to measure to do the experiment. 

Open a new file, and set a graph's y-axis to be momentum. 

Set the positive direction is toward left. Next, calibrate the force sensor. (Because we do not have a pulley, in this case, we need to calibrate it horizontally. Then zero it in the vertical direction.)

Next, click the collect and push the cart to collect data. 

Then, we added a 200-gram mass on the cart and did the experiment above again. 

After that, we left the 200-gram mass and switched the stopper into a nail to get a shorter collision. 

Open a new file, and set the y-axis into impulse and momentum. 

Calibrate the force probe again. Zero it after we put it down. 

Click the data and push the cart, we can get a data of inelastic collision. 

Analyze

In the first experiment, the integral of force-time is 0.4672 N*s. The beginning momentum is 0.239 kg*m/s, and the final momentum is -0.193 kg*m/s, which means the changed of momentum is 0.432 kg*m/s. The difference is 0.035, which is 7% of the data. 

In the second experiment, the integral of force-time is 0.8557 N*s. The beginning momentum is 0.378 kg*m/s, and the final momentum is -0.390 kg*m/s, which means the changed of momentum is 0.768 kg*m/s. The difference is 0.0877, which is 10% of the data. 

These two examples are quite not what we expected. The errors are too large for the conclusion. I think the main reason will be we did not level the track correctly. Moreover, when the cart bounces backward as free body diagram shown in the following, the gravity and the friction were also doing work on it. 

In the second experiment, the integral of force-time is 0.2418 N*s. The beginning momentum is 0.237 kg*m/s, which means the changed of momentum is 0.237 kg*m/s. The difference is 0.0048, which is 2% off. 

Because this time, it did not bounce backward so the error will be lower.

Overall, it is not hard to find out that all of three experiments have the largest force in the medium of collision. In my opinion, no matter the stopper or the clay will have a short time of deformation. In the stopper case, it will give the same result as spring. In the clay case, in the beginning, the contact area between nail and clay is increasing, so that the relative force of deformation will increase. In the later part, because the relative speed is lower, the speed of deformation became slower, and it gave smaller force. 

Proof

Because in this lab, we were studying objects in low speed, which means all the objects are satisfied with Newton's Laws. Then, the relationship can be 

Thus, the change of impulse is equal to the change of momentum. 

Conclusion

As the result showed in the lab, we can say that the change of impulse is almost equal to the change of momentum. However, because there are some other factors (the friction is one of the largest reasons), the result came out that the change of momentum is lower than the change of impulse. Even though, we balanced the friction force, when the cart bounces up, the part of gravity and friction force still will affect the data. 

As a result, my lab was kind of failed because the second data is over 10% error. Nonetheless, the relationship between impulse and momentum still exists. First reason is our other data came out that there was less error. Second reason is this relationship can be proved by mathematic under Newton's system. 

Therefore,