9 October 2014: Impulse Momentum Activity Lab 13

Objective:
The purpose of this lab is to verify the impulse moment theorem. This theorem proved that an impulse is equal to the change of momentum which is equal to the the integral of a force equation as a function of time. To verify this theorem, there will be three total collisions in this lab: 2 elastic and 1 inelastic.

Setup:
Since there are three total collisions there will be three different setups. The first setup will serve to observe collision forces that change with time. We clamped a dynamics cart to a rod clamped to a lab table and extended the spring plunger on the dynamics cart. We then mounted a force sensor on another dynamics cart and placed a rubber stopper over the hook of the force sensor. When the two carts would collide the stopper would collide with the plunger of the stationary cart. Lastly we placed a motion detector at the other end of the track. The set up can be seen below:

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The setup for the next collision was the same as the first except this time we added 500 g to the cart:

For the last collision, the setup differed. We left the 500 g mass on the cart but we attached a nail to the stopper on the force sensor. We then replaced the dynamics cart with a wooden block and attached a ball of clay to it at the height of the nail. Now, when the cart hits the wall, it would come to rest at the block. Pictures of the setup can be seen below:

You will notice the 500 g mass is not in the picture but it was used in the experiment. Also we thought it would be fun to make our clay into the shape of a little man.

Procedure:

For the first collision:

1) We completed the setup as mentioned above

2)  We made sure the software used to record our data was set up to record a push on the force probe as a positive force and velocity toward the motion detector as positive.

3) We calibrated the force probe using the hook screwed into the force sensor

4) Lastly it was important to be able to push the cart toward the plunger while making sure without our hands getting between the motion sensor and the cart

5) Finally, we started collecting data, pushed the cart, released it, and let it collide. We did this until we got a good set of data.

The procedure for the second collision was exactly the same as the first except this collision took place with an addition 500 g on the pushed cart.

For the final collision:

1) We completed the setup as mentioned above

2) We again made sure the software used to record our data was set up to record a push on the force probe as a positive force and velocity toward the motion detector as positive

3) The force prove was still calibrated from the first two collisions

4) Lastly, we started collecting data, pushed the cart, released it and let it stick to our clay on the block.

Results:

For the first collision:

The force is not constant. However the impulse-momentum theorem states the amount of momentum change for the moving cart is equal to the amount of net impulse on the cart. This collision was meant to test that idea. We measured the impulse acting on the cart by constructing a Force vs Time graph and taking the area under it.  We measure the change in momentum by knowing its mass and measuring its velocity before and after the collision using the motion detector. 

The mass of the cart: .432 kg

This was our Force vs Time graph:

As you can see from the graph the impulse was found to -0.4014 s*N

We can calculate change in momentum by looking at the velocity before and after the collision. At 1.66 s the velocity= .494 m/s. At 1.86 s, after the collision took place, the velocity = -.454 m/s. So:

change in momentum= mvf -  mvi =  (0.432*(-0.454)) - (0.432*0.494) = -0.409 N*s

For the second collision:

The graph below is the Force vs Time graph we were able to construct with our gathered data. We again measured the impulse acting on the cart by taking the area under the curve:

The mass of the cart + 500 g = (.434+.500) = 0.934 g

As seen in the graph the impulse was found to be -0.9374 s*N

We again can calculate change in momentum by looking at velocity before and after the collision. At 1.50 s the velocity = .515 m/s. At 1.78 s, after the collision took place, the velocity = -.462 m/s. So:

change in momentum= mvf - mvi = (0.934*(-0.462)-(0.934*0.515)= -0.913 N*s

For the final collision:

This collision was inelastic. When the moving cart hit the clay, it stuck to it instead of bouncing back like the other collision.

The graph below is the Force vs Time graph we were able to construct with our gathered data. We again measured the impulse acting on the cart by taking the area under the curve:

The mass of the cart + 500 g = (.434+.500) = 0.934 g

As seen in the graph the impulse was found to be -0.3134 s*N

We again can calculate change in momentum by looking at velocity before and after the collision. At 1.18 s the velocity = .490 m/s. At 1.38 s, after the collision took place, the velocity = 0 m/s. So:

change in momentum= mvf - mvi = (0.934*(0)-(0.934*0.327)= -0.305 N*s

Conclusion:

As you can see from the above table the collisions proved a success as the percent error was very small and the values were around the same numbers. However in the last collision, the percent error was  the highest. But it was still in the acceptable range. So in, conclusion, our collisions proved the impulse momentum theorem.

7 October 2014: Magnetic Potential Energy Lab, Lab 12

Objective:
The purpose of this lab is to verify conservation of energy applies to this system that uses magnets. Since we do not have a formula for magnetic potential energy, one will have to be derived to prove energy is conserved. Fortunately, for any system where there is non constant potential energy , the potential energy can be defined by a force F as the relationship between the two is that that the negative integral of F(r) dr is equal to potential energy. So now, all we have to do to find magnetic potential energy is to find F(r).

Set up:
A glider was placed on an air track. On one end of the air track was a strong magnet. When the glider was closest to the magnet, the glider's kinetic energy was zero and all the energy in the system was transformed into magnetic potential energy. So first we had the air track level to take initial measurements. But for the majority of the time the track was at an angle. We also attached a reflector to the glider so that its motion could be recorded easily by the motion detector at the end of the air track.

Procedure:

1) First, we leveled our air track and measured the height.

2) Next we tilted the air track. We tilted it because when tilting the track, the glider will end up at some equilibrium position where the magnetic repulsion force between the two magnets well equal the gravitational force component of the glider parallel to the track. We again measured the height from and compared it to first measurement of height.

3) We placed a glider with a strong magnet and an aluminum reflector on the air track while it was tilted and measured the distance between the fixed magnet on the air track and the opposing magnet of the glider.

plotted a graph of F vs r. We assumed that the relationship of a power law such that F=A(r^n)

4) We then discovered the values of A and n from a curve fit of our graph. From this we could develop a function of potential energy (U(r)) for the interaction between the magnets.

5) Next we leveled our track again and placed the glider on the air track close to the fixed magnet and ran the motion detector, We set the motion detector to record 30 measurements/second. We then ran the motion detector and gave the cart a gentle push towards the fixed magnet and were able to record the glider's speed and distance from the magnet.

6) With this data we were able to prove energy was conserved.

Results:

These are the results we achieved when we were simply trying to plot a graph of F vs r. We obtained r by measuring the distance between the two magnets, mass of the glider by using a scale, and theta by using an angle measuring device on a classmate's phone. With this information we were able to get the gravitational force on the glider. 

Now that we knew Force and r were were able to graph a Force (N) vs r(m) graph:

With our force and distance r measurements we created the above graph and assumed the graph was a power law graph. Therefore we assumed force was defined by the function F=Ar^B and with our power fit we were able to find that A=3.98*10^-5 and B=-2.71.

We can now find U(r) or potential energy using these values:

Since we just derived the formula for magnetic potential energy, we still had to show energy was conserved. We collected data from running the motion detector and pushing the glider towards the magnet while the track was level which included distances from the magnets and the velocity at which the glider approached the magnet. Using these values and the above equation for magnetic potential energy we were able to create graphs that proved energy was conserved:

You'll notice we were able to graph Kinetic Energy (yellow), Potential Energy (purple), and Total Energy (red). The graph of KE was calculated by .5 x (mass of glider) x (velocity recorded by motion detector). PE used the above formula and was calculated by 0.0000233 x ((distance recorded by motion detector)-(distance magnet is from motion detector))^ (-1.711). And lastly TE was calculated by (KE)+ (PE). 

Conclusion:

As you can see from the above graph, the dip in KE almost mirrors the hill in PE. And although the graph of TE is not completely level, it does not fluctuate wildly. In general, it can be said energy was conserved. The graph is also effective because when KE= 0, the PE is at its maximum value. At this poing, all KE has been transferred to magnetic potential energy as hypothesized. 

7 October 2014: Conservation of Energy Lab 11

Objective:
The purpose of this lab was to study the conservation of energy and to understand that energy is transferred in different ways. To prove energy is conserved, the kinetic energy of the mass, kinetic energy of the spring, potential energy of the mass, potential energy of the spring, the elastic energy of in the spring, gravitational potential energy of the spring, and lastly the total of all the energy must be found.

Set up:

As you can see from this picture, we attached a spring to a set up made of rods and clamps that was high enough so that it could hang freely. We then attached known weights to the bottom of the spring. And lastly we had a motion detector on the ground to record oscillation and movement.












Procedure:
Before all else, certain data had to be collected:

Mass of Spring: 0.114 kg

Mass of Hanging Weight: 0.5 kg

Length of Spring Unstretched: 0.731 m

After these measurements were known, it would be easier to create a graphs using Logger Pro. We had to create a graph for  kinetic energy of the mass, kinetic energy of the spring, potential energy of the mass, potential energy of the spring, the elastic energy of in the spring, gravitational potential energy of the spring, and lastly the total of all the energy must be found. To calculate all these, we used the motion detector and Logger Pro to measure the velocity and position at certain times. However, finding some of the equations needed proved a bit trickier. Luckily, the professor helped us to find equations that would help us find all we needed. These is the help he provided: 

Here I have provided all the equations used in this lab:

Here are the equations (in order) for Kinetic Energy of the Mass, Potential Energy of the Mass, Elastic Energy of the Spring, Potential Energy of the Spring, Kinetic Energy of the Spring, and Total Energy.








As seen, in these equations, we must find k (spring constant of the spring) to find EE. To do this, we mounted the spring and mass to a force sensor and created a Force vs Position graph using Logger Pro. This is the graph that resulted. The slope of the graph is our k.

k=20.86

 Now that we knew everything that we needed to know, we created all our equations on Logger Pro that allowed us to create graphs of all our energies. The resultant graphs looked like this:

(Note (In order from bottom): Kinetic Energy of Mass= orange, Potential Energy of Mass= purple, Eleastic Energy of Spring= red, Kinetic Energy of Spring= blue, Potential Energy of Spring= green, Total Energy= yellow.

To prove that energy was conserved we must take a look at our the values of Total Energy (TE: dark puple):

As yo can see, the total energy was around 3.3-3.4 J with some but not drastic deviations. Here is a look at the total energy graph:

Conclusions:

Through the graph of Total Energy vs Time, it was apparent that energy was constant and conserved. Although there were slight fluctuations found in the table of Total Energy, the discrepancies were not drastic as they never were above 0.1 J. This means that no new energy entered the system and no energy left the system thus proving what we set out to prove: the principle of the conservation of energy.

2 October 2014: Work- Kinetic Energy Theorem Lab 10

Objective:
The purpose of this lab is to find the relationship between work and kinetic energy. Our experiment aims to prove that the total amount of work is equal to the total kinetic energy at a certain position.

Materials:
1) a cart
2) a block
3) a spring
4) clamps and rods to secure a force sensor
5) a smooth track on which to push the cart
5) a force sensor
6) a motion sensor
7) Logger Pro

Set Up:

For this experiment, we placed a the cart on top of the track and the motion sensor at the end of the track. (The motion sensor will measure kinetic energy) We then placed a block on top of the cart, Lastly, we attached a spring to a force sensor secured with a set up made of clamps and rods.  (Force senor will measure the force of the spring.)










Procedure:

First we had to calibrate the force sensor using a hanging known mass. Next we pulled the cart along with the spring and zeroed the motion detector so that when we let go of the cart the motion detector would record positive distance and not a negative distance.

Since all the work done will be completed by the spring when it snaps back after being pulled, all the work will be done in a short amount of time. This is why it is important to collect data at 30 samples per second.

We also recorded the mass and the cart together which was 0.681 kg. This mass would allow us to create a kinetic energy graph. Finally we were able to proceed with our experiment

We stretched the spring by pulling the cart back, started recording data, and let go of the cart. Then we analyzed the data we collected.














Results:
After releasing the cart, Logger Pro was able to create a Force vs Position graph for us and we were able to create a Kinetic Energy vs Position graph manually. We then took the integral of the Force vs Position graph because that calculates work and compared it to the Kinetic Energy vs Position graph. The resultant graph looked like this: (Note: Kinetic Energy vs Position = purple, Force vs Position = blue)

As you can see from the graphs, the integral of the Force vs Position graph closely equals the value of the Kinetic Energy graph. This demonstrates what we set out to prove about the work- kinetic energy theorem. The work done by the spring equals the kinetic energy of the cart.

Conclusions:
Even though the values are never the exactly the same, they come very close to each other. There could have been some discrepancies which could account for the difference in values. However as seen by the following table, there was not that much error between both values.

30 September 2014: Work and Power Lab 9

Objectives:

The purpose of this lab was for physics students to gain appreciation of their high metabolism, In this lab, the amount of work required to walk up a flight of stairs, to run up a flight of stairs, and to pull a heavy bag up the same distance will be calculated. Power will also be calculated.

Setup:

The only thing needed was this stair case located outside the physics building. Physics students were to first walk all the way up the stairs and time themselves. Then they would run all the way up the stairs and time themselves.

For the last part of this activity, a pulley system was set up on the railing of the second floor and physics students were to pull a bag of 5 kg up the distance they had just walked and run and time themselves.






Procedure:

The first thing we did was to calculate the height of one step and then count how many steps there were in total. With this information, we could find the height of the distance we were to walk and run.

We found that each step was 16.5 cm and there were 26 steps. So the height was 4.29 m.

We then walked up the steps and timed ourselves. Then we ran up the steps and timed ourselves.

My time was 10.29 s when walking and 8.65 s when running.










Then we pulled a 5 kg bag using a pulley system already set up for us and timed ourselves.

My time was 10.99 s.
















After everyone in the class had finished, we went inside to calculate our results.

Results:
Here are my results from running and walking up the stairs:

Running Work: 2672 J, Walking Work 2672 J, Running Power 309 Watts, Walking Power:260 Watts

 Here are my results from the pulley system:

Work 210 J, Power 19.1 Watts

Conclusions:
As you can see pulling the bag took the least amount of work and power. But to burn more calories, walking or running up the stairs would be best.

From this experiment I learned to appreciate my high metabolism. My favorite cheesecake ever is the Lemon Raspberry Cheesecake from Cheesecake Factory. This cheesecake however contains 730 Calories. If I consider every 1 J= 4186 Calories and that from taking the stairs to the second floor, I burn (2672/4186)=.64 Calories, I would have to climb 1140 flights of stairs to completely burn off that cheesecake. That would be like climbing the stairs in the empire state building 11 times!!