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.