3-Dec-2014: Semicircle Oscillation

Purpose:
The purpose of this lab is for students to find the period of half a circular disk. Also to use calculus to find the moment of inertia of a semicircle.

Apparatus:
With a semicircle cut out, we hooked it from the middle of the long side and middle of the top side. Then used Logger Pro to measure the periods. To start the oscillation, we tapped the side of the semicircle and the equipment measured the tape that passed the equipment.

Explanation:

We first started with finding the moment of inertia of the semicircle, which resulted in 1/2MR². We also found that the center of mass for the object was 4R/3π. So the inertia we found using calculus was the inertia at the parallel axis, and depending on if the semicircle will be hooked from the top or bottom. If the object is hooked from the top of the semicircle, then the distance away from the center mass is the radius minus the distance of the center mass. If the object is hooked from the bottom of the semicircle, then the distance away from the center mass is the center of mass. There are two results at the end: the actual for the pivot at the top of the semicircle was .694s and the theoretical was .613s; the actual for the pivot at the bottom of the semicircle was .746s and the theorectical was .714s. The percent error were 4.5% and 13.2%.

Conclusion:

To find the theoretical period of the semicircle, we needed the inertia of the object when pivoted at a certain point on the object. Then we compared the actual to the theoretical, and the percent errors were quite large. This is because the circle was not cut so evenly into a semicircle, and also because the point of pivot wasn't exactly at the center of the object. There is also the fact that the object had different amount of ink from a previous project, so that could have shifted the center mass.

20-Nov-2014: Oscillation Lab

Purpose: 

The purpose of this lab is for students to experience the relationship between the spring constant and the mass hanging on the spring. Also, to find the equation of the relationship with changing mass and changing spring constant.

Apparatus: 

With 4 different groups, each group gets a spring with different mass and spring constant. Each group has to have add mass to the spring up to .109kg before beginning the experiment. We hung our spring onto a rod that was clamped onto another rod which was clamped to the table. The motion detector should be right below the hanging mass which is hanging at the bottom of the spring. With Logger Pro set up, the data should collect the distance traveled, giving the amplitude of the spring. After collecting the data for the oscillation of the spring with the extra weight added on the hanging mass, we needed to do 3 more trials with different masses. This show give us 2 different graphs

Explanation: 

Our mass of the spring was 11.1g and the spring constant was 16.89N/m. The other spring masses and spring constant is shown in the picture below. Using these data, we graph period vs spring constant and come up with a power slope equation to predict a period for another spring constant. With the trials for different masses, we added 50g and 70g to the already hanging mass, so the total was 159g and 179g. To find the period, we used Newton’s 2^nd law to derive an expression for the period of the mass-spring system. The period for the original trial, with the hanging mass of 109g, was 0.814s. The period for the second trial, with the hanging mass of 159g, was 0.983s. The last trial, with the 179g, has a period of 1.734s. With these data, we should come up with a logarithmic curve to predict another trial with a different hanging mass. The equation for the graph Mass(M) vs Period (T) was 2.955*10^-4 x^1.651. The equation for the graph Spring Constant (K) vs Period (T) was 2.455 x^-.4309.

Mass (M) vs Period (T)

Spring Constant (k) vs Period (T)

Conclusion:

The class came up with the same graph for the Spring Constant (k) vs Period (T). The line was power fit and it gave that nice inverse slope. The Mass vs Period graph was our own, which came out not so well because we finished up our lab another day with another spring. This graph is misleading and not accurate.

20-Nov-2014: Conservation of Linear and Angular Momentum

Purpose:
The purpose of this lab is for students to experience with rotation about a point external to the object and investigate the conservation of angular momentum about that external point.

Apparatus:
First part, we are going to roll a ball off a small ramp and measuring the distance the ball traveled. With the distance traveled and the height ramp off the floor, we can solve for the velocity of the ball as it leaves the ramp.

Second part, we attached a hanging mass onto the rotational apparatus and found the angular acceleration of the spinning disk. After find the angular acceleration, we had to find the inertia of the system.

Last part, we attach the ramp to the rotational apparatus, as shown below. The ball catcher will be attached onto the aluminum disk and the ramp should be angled so that the ball will roll right into the ball catcher at a right angle. As the ball rolls down and into the ball catcher, the disk should start spinning and this causes rotational acceleration.

Explanation:

For the first part of the lab, we solved for velocity using the conservation of energy. The traveled a height of 0.146m before entering kinetic energy, which gives us a velocity of 1.69m/s.

Then we attached a hanging mass of .0247kg onto a pulley which was settled in between the aluminum disk and the ball stopper. After collecting data from the apparatus, the angular acceleration was about 5.6365rad/s². To find the Inertia of the spinning ball catcher and disk, we used Newton's 2nd law, which needed the angular acceleration for. The moment of inertia came out to be 0.001065kg*m².

The last part of the lab was to put everything together. With the apparatus set up, we released the ball, and the ball rolled into the ball catcher. The disk started to move due to the motion of the rolling ball. This situation was an inelastic collision, which we set up the equation to find the final angular velocity. The final angular velocity was 1.945rad/s, and the actual angular velocity was 1.775rad/s. The percent error was 8.5%. 

Conclusion:

For the first part of this lab, we needed to find how fast the ball will travel when it leaves the ramp. The second part of the lab is to find the moment of inertia of the spinning ball catcher and disk. The last part is to find the angular velocity and compare it will the theoretical value. The percent error was 8.5%, which means this lab experiment is acceptable.

18-Nov-2014: Momentum with Inelastic Collision

Purpose:
The purpose of this lab is for students to use their knowledge in collision and momentum in order to find a certain variable, height after the collision.

Apparatus:
The ruler is set up so that it can swing most of its body below the pivot point. There will be a small piece of clay sitting right below the tip of the ruler so that when the ruler hits the piece of clay, the clay will stick to the ruler, causing an inelastic collision. Use Logger Pro to record the motion of the ruler to get the highest point the ruler reaches.

Explanation:

First we calculated the theoretical height using momentum, but we needed the angular velocity before the collision, so we used conservation of energy to find the angular velocity. With the angular velocity we came up with an equation of the angular velocity after the collision but right before the system lifts. We used the conservation of energy once again to find the height of the system. With the equation simplified, we ended up with the theoretical height of 0.6539m.

With Logger Pro set up, we recorded the motion of the ruler swinging into the small piece of clay, then swing up a certain height after the inelastic collision. The actual height was 0.6656m, which gives us a percent error of 1.7%.

Conclusion:

Before the collision, the moment of inertia was just the ruler, but after the collision, the moment of inertia was ruler and clay, which we used the parallel axis theorem. When the ruler collide with the clay, the collision was inelastic and this process slowed the angular velocity down. However, the clay was so light that the difference when the ruler swung by itself was not so different. The percent error is less than 5% so we can consider this lab experiment acceptable.

13-Nov-2014: Moment of Inertia of a Triangle

Purpose:
The purpose of this lab is to find the moment of inertia of a triangle using calculus and comparing the theoretical results to the actual.

Apparatus:
We mounted the triangle on a holder and disk, attached the air tube to a supply of air, and attached a pulley
between the disk and triangle with a string wrapped around the pulley. On the other end of the string is a hanging mass of 25 g. With the system released, the hanging mass should pull on the pulley and the disk should be spinning on on a cushion of air, for frictionless rotation. The triangle spins at a certain angular acceleration.

Explanation:
My partners for this lab was Kenji Karuhaka, Ivan Contreras, and Henry Shih. We derived an equation for the moment of inertia, show below.

The equation of inertia

The graph above shows how we obtained the angular acceleration using Logger Pro. According to Logger Pro, our acceleration was around 1.8173 rad/s². We came up with an expression to evaluate the actual moment of inertia of the spinning triangle. The calculation below shows the actual and the theoretical moment of inertia. The theoretical inertia was 5.609*10^-4, and the actual moment of inertia was 3.35*10^-3, which gives us a significantly large percent error of almost 500%.

Conclusion:

We first derived an equation for the theoretical moment of inertia of a right triangle spinning at the center mass. Then we expressed the actual moment of inertia using Newton's 2nd law. In the end, our percent error was massively large. The large percent error was due to the fact that we didn't take the inertia of the disk into account when calculating for moment of inertia. With the mass of the disk added into the equation, moment of inertia should be larger, and would giving us a significantly smaller percent error.

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.