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, 

17-April-2017: Magnetic Potential Energy

In our daily lives, we can observe that when we are pushing a magnet to another, we are doing work on it. In the meanwhile, the other magnet is giving us a larger and larger force while approaching. But how exactly the work is done during the process? Today, we did a lab on magnetic potential energy and detected whether the energy inside magnets were conservative or not.

Purpose

Find out the relationship between distance and potential energy, and verify whether conservation of energy applies to this system.

Plan

Because we do not know what is the relationship between force and distance, we have to detect that first. Thus, we divided the lab into two parts. 

Part I:

Find the relationship between magnetic force and distance by the apparatus (We used a glider on an air track with a magnet attached to its end. Another magnet is also attached to one end of the air track) as following (by the air track, we can lower the friction force).

Then, place the track in different angle. When we turned the air on, the cart would be balanced somewhere eventually. Draw the body diagram,

By the function,

we can know the value of magnetic through the angle.
Using the caliber, we could measure the distance between two magnets.
Then, after we collected a few data, we could roughly know the relationship between magnetic force and distance by logger pro. 

Use the integral to calculate the relationship between magnetic potential energy and distance.

Part II:
Level the track.
Then, add a motion detector in the front to detect the distance and velocity. (But because it is hard to detect directly the distance between two magnets, we used the target into top instead of that.)
Collect the data, use

to calculate kinetic energy and magnetic potential energy.
Compare the energy to see whether it is conservative.

Set up & Data

Part I:
Set up a frictionless cart with a strong magnet on one end approaches a forced magnet on one end approaches a fixed magnet of the same polarity as following.

Rising the left side by adding some books.
Then, we collected the angle by phone and turned on the air track.

After the cart came to rest, we turned off the air track and measure the distance between.
Repeat the step for several time to get different data. Eventually, we took 7 data because the beginning five did not seem to be perfect.

Part II:
We leveled the track first (by detect whether the cart can keep the constant speed).
Then, add a motion detector on the right side, and turn it on.
After a slightly push, we collected the data of velocity and distance.

Analyze

Part I:

Through plugging all the data we got to Logger Pro and doing the power fit, 

we got the relationship between distance and magnetic force was 

Then, after doing the integral, we got the function of potential energy was

Part II:

Using the functions

We can get the graph of U(t) and KE(t), and we sum the up to get the graph as below. 

From the graph, we can know that the total energy always staying around some certain number. So we could make a conclusion that the system of kinetic energy and magnetic potential energy is almost conservative. 

Conclusion

Firstly, there are many factor that we do not know about magnet (since many groups got different answer for the relationship), so we can use our experiment's data to calculate the relationship of force and distance, which is

Then, we could get the U(x) by integral.
Secondly, because the total energy is almost the same, so we could say that the energy between kinetic energy and magnetic potential energy is conservative.
However, we can easily observe that the total energy is actually going down. Maybe it is caused by the air resistance or some other friction that made some part of kinetic energy into heat, and the heat is also the reason why would the cart will stop eventually. (My group's kinetic energy is just reducing too fast that maybe some other factors are also influenced.)

10-April-2017: Work-Kinetic Theorem Activity

When we are running, during the accelerate part, we could tell we are accelerated by the friction from the ground. However, are we ever noticed that how much of work done by friction is converted into kinectic energy? Today, we did a lab on the relationship between work and kinetic energy. 

Purpose

Verify the work done by force is equal to the kinetic energy that object are gained. (by energy conservation)

Plan

Experiment 1:

1. Set up an apparatus as the following diagram. 
   
2. If the theory works, then the kinetic energy should be equal to the work done by the tension. 
3. Detect the distance and velocity by motion sense. 
4. Use a force sense to measure the tension force. 
5. Then, calculate the integral of force, to get work. 
6. Compare the work done and the kinetic energy. Find out whether it is equal or not. 
7. Repeat the experiment to find out whether the results come out the same conclusion. 

Experiment 2:

1. Set up an apparatus as the following diagram. 
   
2. This time, we use a changing force (spring) to check whether the work done by force is still equal to kinetic energy. 
3. To calculate the work, let us calculate the spring constant first. 
4. Detect the distance by motion sense. 
5. Use a force sense to measure the spring force. 
6. Calculate the spring constant by pulling the cart to different position. 

Experiment 3:

1. Using the appartus in the last experiment. 
2. Weight the cart. 
3. Detect the distance and velocity by motion sense. 
4. Use a force sense to measure the tension force. 
5. Calculate the integral of force to get the work, and compare it with the kinetic energy. 

Set up

Experiment 1:

Set up the apparatus as following and connect everything to Logger Pro. 

Then, calibrate the force sense with hanging no mass and hanging 500g (4.9N) mass. 

Next, weight the car. (We got 1183 gram in total)

Add a new column in Logger Pro, which is kinetic energy. (KE = 0.5 * "mass" * "velocity" ^ 2)

After we push the car, we get the data. 

Then, redo the experiment to get more data. 

Experiment 2:

Set up the apparatus as following and connect everything to Logger Pro (make sure that the motion detector can see the cart, and it is set to "Reverse Direction"). 

Then, pull out the cart to get the data. 

Experiment 3:

Using the apparatus in the last experiment. 

Measure the weight of cart. (The weight is 0.531 kg)

Pull the cart for a distance. After the cart get rest, let it go. 

Add a new column in Logger Pro, which is kinetic energy. (KE = 0.5 * "mass" * "velocity" ^ 2)

Integral the Force - Distance graph to get the work. 

Analyze

In the first experiment. We did it twice. 

In the first one, the integral of force - distance (work) is 0.196 J. The kinetic energy is 0.196 J. 

In the second one, the work is 0.244 J. The kinetic energy is 0.234 J. 

The result came out that they are very close; in other word, the work done is almost converted into kinetic energy. 

In the second experiment, we could find out the spring constant by the function

It means the slope of Force - Distance function is the spring constant. Then, it came out that the constant is 3.271 N/m. 

In the third experiment, even though there are some mistakes in the end part, we can use the beginning part to continue our experiment. 

In the first one, we got the work is 0.230 J, and the kinetic energy is 0.227 J. 

In the second one, we got the work is 0.397 J, and the kinetic energy is 0.377 J. 

The result almost came out that the work done on the cart is equal to the kinetic energy. 

The reason why there is some error is mainly because the spring also has mass. When the cart is accelerated, the spring are also accerlated, which will take a part of kinetic energy. 

Overall, the work done by force is close to the energy it has gained. 

Conclusion

From the experience above, we can say that the work done by force is equal to kinetic energy the cart gained (if there is no other source to convert the energy). 

5-April-2017: Work and Power

When talking about work and power, the first thing came to my mind is driving. When we step deeper on the gas pedal, the engine gives a larger power for the car. Then, as a result, the car receives some more work to transfer into kinetic energy and goes faster. Today, we did three activities outside the classroom to find out the power we do daily and compare them with other machines.

Purpose

Calculate the power of daily action, and by the function see how much work do we do on daily. Then, compare it with the power of some heating objects.

Plan

1. Do some "workout" outside the classroom. Calculate the work through the change of the height, the weight, and g. Then, divide the work by time to get the power.
2. First "workout" is lifting a known mass up by a measured distance. Use a stopwatch to time for lifting the backpack from the ground to the bottom of the backpack being level with the top of the balcony. Then, use a ruler to measure the height of one level of stair. Multiply the measurement with the number of stairs to get the height of one floor.
3. Second "workout" is walking up the stairs. Use a stopwatch to time. Detect the angle of stair with phone.
4. Third "workout is running up the stairs. Use a stopwatch to time.
5. Use the function

to calculate our work done on the progress of "workout", and then divide it by time to get our power. 6. Notice that when we walk/run up the stairs, there will be a kinetic energy in our body. So using the functions

to add kinetic energy.

To do

We got out of classroom to the balcony to set up the pulling machine as following.

We wore a pair of gloves to protect our hands. Then, with help from the other student, we measure the time we pull up.
The result came out that I spent 10.2s to pull up a 6 kg bag. And there are 26 17-centermeters stairs.
As walking up, I can time myself with phone.
I spent 15.5s to walk up. And the angle is around 45º.
Then, in similar step, I measure I need 4.5s to run up stair.

Analyze

The height of stairs is 26*0.17m=4.42m
For the first part, I calculated my power is 6*9.8*4.42≈260J.
Then we power is 260/10.2≈25.5 W.

For the second part, because my weight is 95 kg, my work done for gravity will be 95*9.8*4.42≈4115 J. Because my velocity is 4.42/(cos45º*15.5)≈0.40m/s. Then, kinetic energy will be 0.5*95*0.4^2≈7.7J. Total work is 4123 J.
Then my power is 4123/15.5≈266 W.

In the third part, using the same function as last part, and I get my total work is 4207J, and my power is 4207/4.5≈939 W.

Uncertainty

The base uncertainties are ∆h = 0.01m, ∆t = 0.1s, ∆θ = 1º

By the function

We can get

Then, for the first part, the uncertainty will be ±0.256 W

For the second and third part, by the functions

Then, the uncertainty will be 

Then, the uncertainty for those part will be ±2.08 W and ±24.3 W. 

Compare

A typically microwave has 1100W. 

Then I need to climb 1100/939≈1.17 times of stairs. 

If I cook two potatoes for 6 minutes (360 seconds), the total work is 360*1100 = 396000 J. 

Then, I need to complete 396000/939≈422 times stairs. 

For a 100% water heater (require 10 minutes and 12.5 MJ energy to heat 10-minutes shower's water), the power is 12.5 * 10^6 / 600 ≈ 20800 W. Then it needs 20800/100 = 208 men to heat the water. (A normal man can produce 100 W constantly.) If I am the only one to produce the energy, it will take 12.5 * 10^6 / 100 = 125000 s to produce the energy.

Conclusion

Because my weight is pretty large, the work I done everyday is  more than normal people. However when comparing with the heater, I am just doing a small part of work. 

3-April-2017: Centripetal force with a motor

This time, we used another way to verify the function of centripetal force.

Purpose

By using the apparatus as following to find whether the angular speed also works on the function. 

Plan

1. Set up an apparatus as following. 
   
2. Through the diagram
   
   We can get the function 
   
   Then, it turns out to be 
   
3. Then, we can derive angular speed from measurements of R, L, H, and h to get r and tanθ. 
   
   
   
   
4. Though the function
   
   We can measure angular speed with a stopwatch. 
5. Compare calculated angular speed and actual angular speed (from the period). 
6. If the absolute value of calculated angular speed minus actual angular speed is smaller than the uncertainty, then the function is verified. 

Set up

Professor gave the apparatus as following. (And the diagram shows under.)

For the period, we used a stopwatch to measure 10 times period and then divided it by 10 to get the period. 

When we measure the h, we used a small piece of paper. 

Every time the small object passed through, we rose the paper until it hit the object. Then, we measure it by a ruler. (so as other measurements)

Then, we collected the data in different angular velocity as follow. 

Analyze

By the given information, we can calculate the angular speed is 

Uncertainty

By the function above, we can calculate the uncertainty of angular speed. 

After plugging into Excel, I got

Most data is in the range of uncertainty, which means the experiment works. 

Conclusion

From the uncertainty, we can say those data is acceptable. Therefore, we can say the centripetal force function

does work in the different situation.