24-Apr-2017: Lab 14: Impulse-Momentum Activity

Lab 14: Impulse-Momentum Activity

Joel Cook

Nina Song

Lynel Ornedo

Eric Chong

Lab performed April 19, 2017

In this lab, we are attempting to prove the impulse-momentum theorem that the net impulse acting on an object is equal to the change in the momentum of the object.

Introduction:

To test the impulse-momentum theorem, we used a force sensor and collided two carts to measure the force exerted during the collision. We know that the force exerted during the impact is not constant. Impulse is the product of a force and the time interval over which the force is acting. Momentum is the product of mass and velocity. To calculate the momentum, we will measure the mass of the cart and force sensor and use a motion sensor to measure the velocity. We will collide the carts, and verify that the impulse is equal to the change in momentum.

Procedure:

Part 1:

For the first experiment of the lab, we attached a cart to a ring stand and positioned the plunger of the cart to collide with the force sensor of the second cart. The second cart was placed on the track with the force sensor attached to the cart with a rubber stopper replacing the hook of the sensor. A motion sensor was positioned at the end of the track in order to measure velocity of the cart. The three photos below show the apparatus as described.

Logger pro was used to record the measurements from the force and motion sensors. The track was leveled on the table. The force sensor was calibrated and both sensors were zeroed prior to experimentation. 

To gather the proper data, we plotted force vs time and velocity vs time. The motion sensor was setup such that motion toward the sensor is positive. Collection was initiated and the cart on the track was gently pushed toward the stationary cart and allowed to collide and rebound toward the sensor.

Part 2:

For the second portion of the lab, we repeated the procedure above with 500 grams added to the cart. The same types of data were collected in the manner previously described.

Part 3:

For the last portion of the experiment, we replaced the stationary cart with clay attached to a block of wood, as seen in the last picture above. The rubber stopper of the force sensor was replaced with a sharpened screw to ensure an inelastic collision, in which the cart will collide with the clay and "stick".

The type of data collected was the same as the previous portions and collected in the same manner.

Data and Analysis:

Part 1:

For the first part of the experiment, the following graphs were recorded.

As seen in the photo above, the velocity as the cart approached the stationary cart and the force was zero. At the point the carts collide, the velocity goes to zero and the force on the cart increases. The cart bounces off the second cart, the force goes back down to zero and the velocity is positive as the cart moves toward the sensor. The magnitude of the velocity after the collision is smaller than before the collision. 

As shown below, the integral of the force vs time graph will give us the impulse for the collision. To compare the impulse to the change in momentum, the velocity of the cart before and after the collision were used, as shown in the photos below.

As stated previously, we will confirming the impulse-momentum theorem by comparing the impulse calculated from the graph of force vs. time and comparing this value to the change in momentum.

J = pf - po

J = m*vf - m*vo

J = 0.641kg (0.306 - (-0.325))m/s

J = 0.404 Ns

The impulse from the integral of the graph above shows a value of 0.4132 Ns which is within 3% of the value we calculated from change in momentum. 

Part 2:

As previously described, the second part of the lab was the same as the first part but with 500 grams added to the cart. The first photo below shows the graph of velocity vs time and force vs time for the collision. The integral of the force vs time graph was performed to calculate the impulse during the collision. The last two photos show the velocity before and after the impact used to calculate the change in momentum.

As the photo shows, the impulse calculated from the integral of force vs time for the collision was 0.8575 Ns. The impulse from change in momentum was calculated as previously shown and gave us a value of 0.8934 Ns, which is within 5% of the value calculated from the integral.

Part 3:

The following graph shows the velocity vs time and force vs time for the inelastic collision with the clay. The velocity is negative as the cart moves toward the end of the track and goes to zero at the collision. The force on the cart goes up during the collision and is slightly negative at the point that the cart is attempting to bounce back but is stuck in the clay.

The integral of the force vs time graph during the collision gives us the impulse. The velocity right before the collision was used to calculate the initial momentum and the final momentum is zero when then velocity is zero.

As the photo shows, the integral calculated the impulse as 0.3154 Ns. Change in momentum was calculated as previously shown but final momentum was zero because the final velocity was zero. The value calculated was 0.3096 Ns, which was within 2% of the integral value.

Conclusion:

As the values show in each of the parts of the lab, our calculated change in momentum and the impulse from the integral of the force vs time graph were consistently within 5% each other. This is sufficient to confirm that the impulse exerted during the collision was equal to the change in momentum and the theorem is accurate.

There were errors that existed in our experimentation in this lab. First, it is possible that our track was not perfectly level which would have affected the velocity of the cart. We attempted to mitigate this by leveling the track and using a velocity value immediately before the collision and after the collision. In addition, the cart and track are not perfectly frictionless which would have affected our values. Finally, the instruments used are not perfectly accurate but are certainly sufficient for our purposes. We adjusted the force sensor to record 200 data points per second to attempt to make the force vs time information more accurate. 

In summation, the lab was successful and our theories and predictions were correct.

17-Apr-2017: Lab 13: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy

Joel Cook

Lynel Ornedo

Eric Chong

Nina Song

Lab performed April 17, 2017

We do not have an equation for the magnetic potential energy that exists between two magnets repelling each other (same polarity). In this lab we will attempt to model the magnetic potential energy between a pair of magnets to verify that energy is conserved in our experiment.

Procedure: In this lab, we used an air glider and air track. By pushing air up through holes in the track, the glider and move along the track with almost no friction. The photos below show the track, glider, and magnets.

With the magnets repelling each other, a force exists as a function of the separation distance between the magnets. The force changes as the separation distance changes. To determine the potential energy, we will need to find the negative integral of the force, F(r), from infinity to r, where r is the separation distance. We will assume that the force is zero when the separation distance goes to infinity.

To find the integral, first we need to determine F(r). We measured the mass of the cart and used calipers to measure the separation distance between the magnets. If we raise the end of the track opposite the magnets, the force exerted by the magnet to separate the magnets is equal to the x component of the force gravity is exerting on the cart. An smart phone application was used to digitally measure the angle at each position.

F = mgsinΘ

The track was raised to various angles. The angles were measured and recorded, as well as the separation distance between the magnets at that angle. The force at each angle was calculated from the previous equation. Next, the a graph was created for Force vs Separation Distance. Using logger pro, a power fit of the line created shows gives F(r).

Now that we have a function for separation force as a function of separation distance, we can verify the conservation of energy in our system. First we positioned a motion sensor at the end of the track  (as shown above) and determined the the relationship between the distance from the motion sensor to the cart and the separation distance. With the air to the track turned off, the distance was measured using the motion sensor and logger pro. Calipers were used to measure the separation distance and a calculated column was created for the actual separation distance based on the distance the motion sensor is reading. 

Separation distance = (Distance from cart to sensor) - (Difference between sensor distance and actual separation distance)

Using separation distance, a calculated column was created for U(r), magnetic potential energy as a function of separation distance, from the integral of F(r).

Next, with the air track leveled, the air was turned on to the air track. The cart was pushed toward the motion sensor and allowed to "bounce" back. The motion sensor was then used to plot the speed of the cart as it travelled on the track. The speed of the cart was used to plot kinetic energy of the cart by creating a calculated column KE = 1/2*mass*speed^2.

We expect that the energy in the system will be conserved. Graphs were plotted for KE, PE, and total energy to compare the values and verify that energy was conserved.

Data:

The following table shows the data for the separation distances that were gathered at various angles:

Θ (degrees)	r (mm)	F (N)
1.7	28.0	0.100
2.2	24.6	0.129
3.5	21.3	0.206
4.2	19.7	0.247
5.1	18.5	0.300
7.2	16.3	0.423
11.6	13.0	0.678

The mass of the cart was measured as 0.344 kg using a digital scale. As previously stated, F was calculated as the following sample shows:

F = mgsinΘ

F = 0.344 * 9.8 * sin(1.7)

F = 0.1000

The following graph was created for Force (N) vs Separation Distance (m).

As the graph shows, a power fit of the graphed data points shows the following function:

F(r) = 1.864 * 10^(-5) r ^ (-2.421)

The uncertainty in the first term was +/- 6.337 * 10^(-6).

The uncertainty in the exponent was +/- 0.08114.

To determine magnetic potential energy as a function of separation distance, the negative integral of the function F(r) from 0 to r was taken as shown:

The photo above shows the three graphs that were created from pushing the cart towards the motion sensor and letting it "bounce" off the other magnet. The first graph shows position (m) vs time (s). 

The second graph shows KE, PE [U(r)], and Total Energy (PE + KE) vs position (m). As the graph shows, as the cart moved toward the end of the track, the potential energy increases until the "collision" and then decreases. The kinetic energy is approximately constant and then decreases until the "collision" and then increases again to a value slightly lower than the initial kinetic energy. The total energy is approximately constant.

The third graph shows KE, PE [U(r)], and Total Energy (PE + KE) vs time (s). As the graph shows, the potential energy increases until the collision at about 1.7 seconds and then decreases. The kinetic energy decreases until the collision (KE is zero at collision) and then increases again as speed increases. The total energy is approximately constant.

Analysis/Conclusions:

As the graph shows, the model we developed for magnetic potential energy as a function of separation distance for our pair of magnets was accurate. The kinetic energy of the system as a function of time, or position, changed in equal magnitude to the potential energy and the total energy was approximately consistent. The conservation of energy in our system proves that our model was accurate. This also confirms that our method for developing force as a function of separation distance was accurate.

Although uncertainty exists in our lab, the experiment was successful. First, it is not possible that our track be completely frictionless. As the glider moves along the track it is cutting streams of air that may affect the speed of the glider. From our graph of velocity vs time it is evident that our glider is losing some speed as it travels. Second, the most accurate data points used to calculate F(r), were the points at a larger angle. This is due to the fact that when the glider collides with the magnet in the second portion, it has a smaller separation distance. Therefore, the larger angles will produce a smaller separation distance and will thus be comparable to the force during collision. Finally, errors and uncertainty exist inherently in our measurements. Angles were measured to the 1/10th of a degree and distance was measured with digital calipers to the 1/10th of a millimeter. As shown in the graph previously, uncertainty in our function of F(r) was calculated by logger pro and reported in the lab earlier. Even though all these errors and uncertainties exist, and possibly others, our results were sufficiently accurate and successful in confirming conservation of energy in our system.