4-Nov-2014: Moment of Inertia

Purpose:
The purpose of this lab is to find the moment of inertia of a large metal disk attached with two smaller cylinders, which spins together. Also, to find the time it takes to travel a certain amount of revolutions.

Apparatus:
There are two parts to this lab. First part is finding the moment of inertia of the disk and cylinders. To do that we find the volume of the disk and cylinders and the mass of the whole object. We calculated the moment of inertia by timing how long it takes to make a certain amount of revolutions before coming to a stop. Then, we used certain equations to find the torque for friction. With the torque found, we can move on to the second part of the lab. We placed a track at an angle of 36 degree above the horizontal and the apparatus at the top of the track. With one end of a string tied to one of the smaller cylinders of the apparatus and the other end a cart, we can release the cart and find the time it takes to travel a meter down the track. The time it takes should be the time we predicted, if not then close to that time.

Explanation:
First we calculated the moment of inertia of the disk and cylinder, and to do that we found the volume by measuring the dimensions of the disk and cylinder. The disk had a radius of 0.1 m and a height of  0.0157 m; the cylinders had a radius of 0.0157 m and a height of 0.0571 m. The whole object had a mass of 4.808 kg, so the moment of inertia of the disk came out to be 0.018 kg*m².

Finding inertia

After finding inertia, we measured the time for the disk to stop, and found the angular acceleration with the equation in the image below. The average angular acceleration was 0.493 rad/s², and with our inertia calculated earlier, we found the torque for friction to be 0.008874 kg*m². 

Finding angular acceleration

Finding torque for friction

To find how long it would take for the cart to travel a meter when attached to the spinning cylinder, we need to find acceleration of the whole system, cart and cylinder together. We solved for acceleration symbolically and came up with an acceleration of 0.0258 m/s², and 0.0264 m/s². We used kinetics to solve for time and came up with   8.76 secs and 8.56 secs. The actual time was 8.69 secs, which gave us a percent error of 1.5% and 0.8%.

Conclusion:
We found moment of inertia and calculate angular deceleration to calculate the frictional torque. With the frictional torque, we can calculate the acceleration of the whole system, cart and cylinder. Using the acceleration of the whole system and kinetics, we found the theoretical time for the cart to travel one meter, and compared to the actual, we were at least 0.8% off, which is almost accurate.

28-Oct-2014: Angular Acceleration

Purpose:
The purpose of this lab is for students to explore torque and angular acceleration. By applying a known torque to a rotating object, we can measure the angular acceleration of the spinning object.

Apparatus:
With the Pasco rotational sensor connected to Logger Pro, we set up Logger Pro to only read the top disk. We also connected the air hose to the air supply, and turned the compressed air so that the disks can rotate separately. We attached the string to a hanging mass and a wrapped the other end around the torque pulley. The measurements start once we release the hanging mass at its highest  point, which should start pulling the pulley and spin the disks. The graphs reads angular velocity as the mass moves up and down.

Explanation:
With the apparatus all set up, we start the first 3 experiments with the small torque pulley with a diameter of 0.025 m and a mass of .001 kg. For trial one, we had the hanging mass at 25 g and only to top steel disk was spinning in this trial. The rotation was 1.069 rad/s going down and 1.213 rad/s going up, which averaged 1.141 rad/s. For trial 2, the hanging mass was doubled and only the top disk was spinning. The rotation was 2.096 rad/s going down and 2.495 rad/ s going up, which average 2.296 rad/s. The third trial had a hanging mass of 75 g and only the top disk was spinning at 3.274 rad/s going down and 3.507 rad/s going up. The average angular acceleration was 3.391. For the next trial, Trial 4, the hanging mass was 25 g, and the torque pulley was larger, with a diameter of 0.05 m and a mass of 0.036 kg. The top steel disk was spinning only, which spun at a rate of 2.072 rad/s going down and 2.258 rad/s going up. The average was 2.156 rad/s. The fifth trial has the same set up as trial four except for the fact that the top disk was aluminum instead of steel, making the spinning disk lighter. The angular acceleration 5.669 rad/s going down and 6.627 rad/s going up, which gave an average of 6.148 rad/s. The last trial, again, has the same setup as trial four, except for the fact that there are two spinning disks. The top steel and bottom steel are spinning together, which means more mass. The angular acceleration was 1.049 rad/s going down and 1.156 rad/s going up, and the average was 1.103 rad/s. The picture below shows most of the explanation from above.

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

Trial 6

Conclusion:
According to trials 1 to 3, as the mass of the hanging mass increases, the angular acceleration increases. This is true because the heavier the hanging mass, the more force there is accelerating downwards, which gives more tension to the pulley, therefore giving more torque to the disk. Also, according to trials 1 and 4, the larger diameter of the torque pulley, the larger the angular acceleration. This is true because the larger the diameter of the pulley the more leverage there is for the hanging mass to torque the disk. Lastly, according to trials 5 and 6, the larger the mass that is spinning, or torquing, the smaller the angular acceleration. This is true because inertia depends on mass so the more mass an object has, the less angular speed the object has, and vise versa.

14-Oct-2014: Collision in Two Dimensions

Purpose:
The purpose of this lab is for students to observe two-dimensional collision and determine if momentum and energy are conserved.
Apparatus:
The camera, attached to a pole, looks down at the glass board, and is able to view the entire board. By placing two balls on the board, the camera should be able to record the motion of both balls along a
xy-plane. The camera should be set up properly: shutter should be zero, reduce exposure, and increase gain so that camera can see the motion of both balls clearly. One ball should be set in place, while the other ball comes into contact with it at an angle, so that both balls should move at different angles.

Explanation:
With the apparatus all set up, we recorded to different trials: steel on steel and marble on steel. The first trial was steel on steel. One steel ball was set in place, and the second one was aimed to hit the stationary ball at an angle. With the camera recording, we rolled the ball out and both balls moved at different angles away from each other and at different speeds. We used the recording and traced the paths of both steel balls, and came up with position and velocity in x-direction and y-direction for both steel balls. We set the coordinates so that the origin is where the rolling ball first moved. The x-axis is aligned with the motion of the rolling ball so that the y-direction had no value until the balls collided.

By finding the slope of each line before the collision and after the collision, we are able to find the average velocity of each line. With the velocity, can find kinetic energy in the x and y-direction. The energies need to be consistent throughout the procedure of this lab. By plotting the energies on one graph we can see that the kinetic and potential energy has a slope closely to zero, which means that energy was not lost through the procedure.

Also, with the velocity, we were able to calculate the momentum of this trial. The initial momentum for both x and y-direction were both slightly larger than the final momentum, by a percent difference of 0.409%. During the collision, the rolling ball was spinning vertically, which is believed to have lost some energy in the spin.

The second trial was marble on steel, where the marble is the stationary ball, and the steel ball is the rolling ball. Since the steel ball is heavier than the marble ball, the motion of the marble after the collision will be more faster in general in comparison with the first trial. Knowing that the marble ball has a faster velocity, we can still assume that the energies are conserved in this procedure. We graphed the position of x and y-direction vs time, just like in trial 1, and found the velocity of each different motion.

 With the velocity, we can find the kinetic energy  for both balls. From the graph, we see that the energy has a slope of zero, which means that the energy was conserved.

After calculating the momentum, we can conclude that momentum was conserved with a percent difference of 13.8%, which is quite large since the value is higher than 10%. This is due to the difference in the mass, and the spin in the steel marble after the collision; the spinning was a cause of energy lost.

Conclusion:
In this lab, we discovered that energy and momentum was conserve by graphing kinetic energy and calculating the momentum of the initial velocity to the final velocity. There were lost in energy and momentum when the steel balls was spinning after the collision.

9-Oct-2014: Impulse

Purpose:
The purpose of this lab if for students to observe impulse through elastic and inelastic collision.
Apparatus:
There are two parts to this lab experiment. The first part is to find impulse through elastic collision, and to do that, we need the help of spring collision. We place the track on a level surface, with a cart at one end and a motion detector at the other end. The cart held in place at one end of the track has a spring connected to the cart, and is released in order for the collision to happen. Then, we place a force sensor on another cart, that is placed anywhere on the track. The second part is where we add mass to the cart in motion. The impulse should be larger since there is more mass. The third part is where we take the spring cart away and replace it with a wooden stock that has clay pasted on the wooden stock. Also, the force sensor needs a nail in place of the rubber stopper. With these equipments, we should produce an inelastic collision.

Explanation:

For Part 1 of this lab, we first calibrated the force sensor to read zero, and set Logger Pro to record force and distance. Giving the cart, on the track with a mass of 440g, a light push, the cart ran across the track and into the spring. With the proper data collected, we graphed force vs time and velocity vs time. Using the force vs time graph, we integrated the curve created by the change in force throughout a period of time, and came out with an impulse of 0.3136 N*s. Using the velocity vs time graph, we searched for the initial velocity, right before the collision, and the final velocity, right after the collision, and used the impulse equation to find the momentum of this system, which came out to be 0.292 N*s. The percent error was 6.89%, which is acceptable. The error was due to which instantaneous velocity we picked on Logger Pro. If we were to pick two different velocity, we may be able to decrease the percent error, but Logger Pro can only count to a certain decimal point. Also, the collision was not perfectly elastic, so there could have been some energy lost during the collision.

For Part 2 of this lab, we added mass to the cart, and the total mass came out to be 940g. With the same procedures as Part 1, the force vs time graph gave us a impulse of 0.7786N*s. Using the velocity vs time graph, we came up with an momentum of 0.8366N*s. The percent error was 7.45%, which is acceptable. The cause of the error was the same as Part 1, but only more heavily, due to the increase in mass. With the extra 500g, there were more energy lost during the collision.

For Part 3 of this lab, we replaced the cart with spring with the wooden stock and clay and the rubber stopper with a nail. Same as Part 1, we pushed the cart lightly, and the cart ran across the track and into the clay, stopping the cart completely. Graphing the force vs time , we can integrate it to find the impulse, which came out to be 0.1869N*s. Graphing the velocity vs time, we were able to find the initial velocity and final velocity. The mass of the cart was 0.44kg, so we calculated the momentum of the system to be 0.1892N*s. The percent error was 1.23%, which is almost zero, and considerable acceptable. This trial had less error because the purpose was to exert all the energy out of the cart and into the clay, so the only error came from Logger Pro.

Conclusion:

In order to find impulse and momentum, we had plot force vs time graph and velocity vs time graph. The impulse was found by integrating the force vs time graph, and the momentum was found by finding the initial velocity and final velocity and using the momentum equation. The impulse and momentum were almost equal to each other, which means that this system follows true with the impulse-momentum theorem. The error we found in this lab was due to Logger Pro's accuracy and the lost in energy in the collision. Other than that, the system was accurate.

7-Oct-2014: Spring Collision

Purpose:
The purpose of this lab is explore the conservation of energy in magnetic forces. Also, to find the equation of forces between two magnets.
Apparatus:
There are two parts to this experiment. The first part is to find an equation for the forces between the magnets. By tilting the track up at the other end of the track, we find the cart moves closer to the track, and with the force coming from the mass of the cart, we can find the force exerted on the magnets, and plot a force vs. position graph to find the equation. The second part of this lab is to determine if the energy throughout the experiment was conserved. To do this, we leveled the track and attached a motion detector at one end of the track. We then pushed a frictionless cart with a magnet at one end of the cart to the end of the track with a magnet attached to that end. With the magnets pushing off each other the cart should move the other direction.
.

Explanation:

For the first part of the lab, we raised the track at several different heights. For each certain heights we can observe the distance between the magnets decreases as the height of the cart increases. We took eleven different measurements, and with the mass of the cart, we can measure the force of the magnets by using the angle between the track and table. With the measurements forces and distances, we graphed a force vs distance chart, and found the equation of the forces, 1.285x10^-7r^-4.893. We had to cross-out four points to obtain a perfect logarithmic curve.

For the second part of this lab, we leveled the track and placed the motion detector at one end of the track. With Logger Pro set up, we pushed the cart, with a magnet in the front of the cart, towards the magnet where the motion detector is. The cart starts with an initial velocity; once the magnets come close to contact, the frictionless cart slows down and eventually, for an instance, hit zero velocity before moving in the opposite direction. Once the magnets exert force off each other, potential energy increases and kinetic energy decreases. To find the kinetic energy, we used the equation 1/2mv², and for potential energy, we integrated the equation we found in part one of this experiment, which gave us -3.3x10^-8r^-3.893. We plot the kinetic energy, potential energy, and the total energy on one graph and the results were as expected. The total energy should show close to a linear line while the kinetic energy and potential energy shifts at the point were the magnets come close together. With these results, we can assume that energy was somewhat conserved. But the point were the magnets almost meet, there was a lost in energy and this was recorded by Logger Pro. We believe that the lost in energy was due to how the cart lifts a little.

Conclusion:

To assure that the system of this apparatus has conserved energy, we first found an equation for the magnetic potential energy. To do this, we raised on end of the track and found the distance between the track; the higher the track was lifted, the smaller the distance was between the magnet. With this equation we were able to confirm that some energy was lost but most of the energy was conserved.

7-Oct-2014: Spring Energy

Purpose:
The purpose of this lab is for students to experience with the spring energy, and to discover how the energy is transferred from potential to kinetic and vise-versa.
Apparatus:
We calibrated the force sensor and set it up on a stand facing down. We hung a spring on the force sensor with weights hanging on the spring. We set the motion detector at the bottom of everything to measure the distance the spring moves. After the set up was complete, we zeroed every measurement on the Logger Pro.

Explanation:

With the apparatus set up, I pulled the mass down, which stretched the spring, until the mass was right above the motion detector. I released the mass once Logger Pro was ready and it recorded the force and distance it traveled. Using the velocity and mass, we found the kinetic energy of the system. Using the mass, gravity and position, we found the potential energy of the system. Using the potential energy we found previously, we set the potential energy equal to the elastic potential energy, giving us k constant to be 16.68. The distance for the potential energy was the distance traveled plus the height of the spring unstretched. On the other hand, the distance for the elastic potential energy was the distance unstretched minus the distance read by Logger Pro. The distance unstretched was found by finding equilibrium height, 0.625m, subtracting the distance distance from the equilibrium to the unstetched, 0.294m. The distance unstretched for the elastic potential energy is 0.331m, and the distance unstreched for the potential energy was the position read by Logger Pro plus 0.75m. Using the mass of the spring, 0.045 kg, and the velocity, we found the kinetic energy of the spring; and with the distance traveled by the mass, we found the potential energy of the spring. When the potential energy of the spring reaches it's max, the kinetic energy of the spring will reach a value of zero. When the potential energy of the spring reaches a value of zero, the kinetic energy should reach it's max. Adding both the potential energy of the spring and the kinetic energy of the spring, we should get a small range of numbers, which we called total energy. When we graph the total energy, the plot was close to a straight line.

The instructions of the lab.

Finding the change of position from equilibrium

All calculations are done on Logger Pro.

Conclusion:

With the spring pulled away from equilibrium position, we found that the energy throughout the spring transfers from kinetic energy to potential energy and back to kinetic energy and so forth. With the mass attached to the end of the spring, we find that the spring had more energy and traveled more further from equilibrium point. Adding all the energy together, we find that the total energy was close to constant, which means that the energy was close to conserved. The reason why the energy was not so conserved was because the gravity pulling the hanging mass and the mass of the spring is another constant energy.