PHYS4A/ Fall 2014; Periods of Oscillations

Purpose: The goal of the experiment was to calculate the moments of inertia for a triangle and a semi circle. These moments were then used to calculate a theoretical period of oscillation for both objects.

For this experiment, we had to cut a triangle and a semicircle out of a foam and calculate theoretical periods of oscillation under small angles and then relate that to experimental values found with loggerpro. Unfortunately, the triangle data was lost ( I really have no idea where it went) and I only have the semi circle data.

To start the prep for the experiment, we had to find the moment of inertia of the semicircle at the center of mass using calculus. By now, we have done these kinds of calculations many times and you should be able to follow the math below.

Once the moment of inertia at the center of mass was found, we then were able to once again use Newton's 2nd Law to find an angular acceleration of the object. This value was then related to the formula

(angular acceleration) = -(angular velocity)^2 * (displacement)

(alpha) = -(omega^2) * (theta)

Since all we had to look for was the constant before the displacement (theta) and relate that to the equation above, we could figure out an angular velocity value. This value was then used to calculate a period using the equation

(Period) = 2pi/angular velocity

T=2pi/(omega)

 From the calculations above, we were able to find a theoretical period value of 0.889s. We then went over to the setup and use loggerpro with a photogate that was able to record the period of the oscillations. From loggerpro, we got an experimental period of 0.640s. When compared to our theoretical value, we had a 7.72% error. This could be attributed to the pivot not being exactly center or the width of the hanging tape breaking the photogate being too large and recording shorter periods due to a sooner break in the light.

And this brings an end to the PHYS4A Fall 2014 Semester. Not too sure if Prof. Wolf did this on purpose but the final experiment was similar to the very first experiment we did with oscillations in a photogate.  The way we started was the way we ended but more knowledgeable about physics and how the world around us works.

PHYS4A/ Fall 2014; Mass-Spring Oscillations

Purpose: The purpose of this lab was to calculate and observe the changes in period of springs with different spring constants and as hanging mass changed for a specific spring.

For this experiment, the room was split into two groups each with a set of springs in order to lessen the amount of time to do the experiment. Each subgroup was given a spring of which we needed to calculate the spring constant. Using a motion sensor and hanging mass on the spring, we were able to calculate our spring constant by comparing the unstretched length of the spring and the stretched length as the hanging mass was added. We calculated a spring constant of 2.330 for our spring and shared that with the class in the data pool.

Spring Oscillation Set-Up

While the other groups were figuring out their spring constants, we then used several different masses on our spring and measured the period of oscillations. Once the data was recorded, we were able to create the graphs in the form below to look for the relations in the periods of the springs.

Graphs we were asked to analyze

Our Graphs of Period vs hanging mass and Period vs spring constant

 Once all the groups were able to get their constants and periods with a set mass, we were able to create a period vs spring constant graph with data from the class. After that graph was created, we were able to compare that with an experimental period which we calculated in loggerpro and compared that to what the theoretical period turned out to be. As you can see, the two period graphs were fairly close meaning error was not that large. Once this graph was analyzed, we then graphed the multiple hanging masses on our spring and made a similar analysis.

Based on the data, we were able to see that springs with a higher spring constant had a shorter period as the force of the spring resisted stretch. We also noticed that if a heavier mass was put on a spring, the period also got longer. These observations made sense both logically and mathematically using the Period formulas.

PHYS4A/ Fall 2014; Conservation of Linear and Angular Momentum

Purpose: Prof. Wolf does an experiment to show that linear and rotational momentum are both conserved.

In this experiment, Prof Wolf performs an experiment while the rest of us observed and made calculations. For the first part of the experiment, a ball rolls down a ramp and we are asked to calculate the rolling velocity of the ball and the launch velocity of the ball under constant acceleration and through slipping. Calculations below show that with the static friction causing the ball to roll, launch velocity was slightly higher.

Velocity calculations

Angular acceleration was measured by Wolf as we were doing calculations

Once the velocity was calculated, the moment of inertia had to be calculated using the same method as the previous lab with moment of inertia. This was needed so that conservation of momentum could be used in the next part as it would be an inelastic collision and energy would not be conserved. Since Prof. Wolf had already attached a hanging mass to the disks and run the experiment to find angular acceleration, all we had to do was use that and Newton's second law to find the moment of inertia like below.

Moment of inertia of the disks calculations

Prof. Wolf and the lab set-up

Prof. Wolf then ran the experiment to make better use of time as we waited for groups to get their values. The ball was released from the top of the ramp and the ball was launched into a metal trap on the spinning disk which then started to spin due to rotational momentum being transferred into the disk.

Calculations for omega

 Once the experiment was over, using conservation of momentum, we were able to calculate a theoretical omega value once the collision occurs. This omega value was then compared to the graph below where the position in radii was recorded vs time. The slope of this graph was the angular velocity of the spinning disk and the value of 1.775 rad/s was compared to our calculated value of 1.95 rad/s.

The values were fairly close and shows that momentum was conserved throughout the experiment since we were able to use the conservation to calculate what we needed for the final goal.

PHYS4A/ Fall 2014; Angular Momentum

Purpose: Using angular momentum and the concept of conservation of momentum, we will predict how high a ruler will swing after an inelastic collision with a piece of putty.

For this experiment, we split the problem into two parts in order to show that momentum is conserved in a rotational collision. To do this, we had set up a ruler and putty scenario where the ruler would start at an angle and be released from rest so that it would hit a piece of putty resulting in an inelastic collision. The goal would then be to calculate the momentum and show that the momentum was conserved throughout the process.

Ruler and putty set-up

The above set-up was created with a camera set across from the ruler centered on the location that the putty ended up at the peak of the rotation. Once we were set-up, the ruler was lifted and released from rest to hit the putty and pick it up. The camera was used to record a video of the motion and loggerpro was used to analyze the data. 

LoggerPro's view of the collision and plotting the points frame by frame of the path of the putty.

When analyzing the video, we had to put a point on the path of the putty, frame by frame. Once all of the points were plotted, we were able to see a fairly smooth curve along a circle. Once the video was finished analyzing, we calculated a theoretical final height for the putty by splitting the experiment into two intervals. For the first interval, we used conservation of energy starting at the initial point at rest to just before the ruler hit the putty.

Total Energy = Potential Energy = Rotational Kinetic Energy

mgh = (1/2)Iw^2

By using conservation of energy we were able to calculate the angular velocity just before the ruler hits the putty. This angular velocity was then used to calculate for the angular velocity just after the collision using conservation of rotational momentum. Initially, the rule was moving at some angular velocity. Then, once it makes contact, its mass changes and it has a new angular velocity. Since the collision was inelastic, energy was not conserved but momentum was. This new omega was then used in the next part of the calculations to finally come up with final height.

All the messy work in our calculations (which Prof. Wolf called beautiful)

Final part of the calculations where we find the height.

The last part to find the height was by using conservation of energy once again. Using the angular velocity of the moment right after collision as the initial speed, we were able to compare that to when the ruler and putty stopped at the end of the path. By using conservation of energy again, this time starting with rotational kinetic energy and ending with gravitational potential, we were able to calculate the height of the putty to be 0.303m.

Knowing this height, we went back to loggerpro and set an axis at the origin of the collision, then using the analysis data it was able to give us a final height. Since both the calculations and loggerpro were both using the same axis, the values could be directly compared to each other. From loggerpro, we got a value of 0.2325m where the theoretical was 0.303m. The experimental value being lower does make sense due to frictional forces such as air drag or friction at the pivot. Therefore, we were able to prove that momentum is conserved in a rotational collision.

PHYS4A/ Fall 2014; Moment of Inertia Pt. 2 (Triangle)

Purpose: To find the moment of inertia of a solid triangle that is revolving about its center on a spinning disk apparatus.

For this experiment, we will be using the same apparatus that we used in the angular acceleration experiment done, about two labs ago. First, we calculated the theoretical moment of inertia for a triangle using calculus.

Deriving the moment of inertia for a triangle about its center.

To find the moment of inertia about its center, we first had to find the moment of inertia about the end of the triangle. The parallel axis theorem was then used to move the axis of rotation to the middle of the triangle. Once the moment of inertia was found at the center of mass, the experiment was done on the air ring to obtain values that can be compared to the theoretcial value.

The triangle was then mounted on the spinning air disk and the same process used to find the angular acceleration was used. However, this time, moment of inertia of the disk was found. These moments were recorded on the board seen above and compared to the theoretical value also found above. The experiment was done twice with the long side up and the long side down to show that the moment of inertia at the center of mass should not be different. This was also proven as the moment of inertia was very close to each other.

The same method as the angular acceleration lab was used but this time, to calculate the experimental moment of inertia

The moment of inertia when compared to the theoretical was very off due to it being a magnitude off. This may be due to the way the measurements were taken as the expression for the theoretical was correct.

PHYS4A/ Fall 2014; Moment of Inertia Pt. 1

Purpose: This experiment was done in order to find the moment of inertia of a spinning disk and to use that info in order to predict how long it would take for the cart to reach the bottom of the track.

For this experiment, a metal wheel was set up on a chair. A cart on rollers was set at an angle on a frictionless track and laid against the chair. For the experiment, we needed to calculate the moment of inertia for the spinning disk apparatus. This required the moment of inertia for not only the disk, but the two ends as well.

Calculations for the moment of inertia of the disk, two ends, and dimensions of the apparatus.

Once we found the moment of inertia, we were able to use Newton's Second Law to calculate the to find the translational acceleration of the disk.

Once the moment of inertia was calculated, acceleration was calculated

Once translational acceleration was calculated, kinematics were  used to determine the time that it
would take the cart to reach the bottom of the ramp which we calculated to be 8.27s. The experiment was then actually done in order to measure the time it took to go down the ramp by timing it with a stopwatch. 

Kim and Ivan timing the fall of the cart

It was recorded that it took 10 seconds to reach the bottom. This value was compared to the theoretical value of 8.27 s with a percent error of   17.3%. This could be due to the calculations not including frictional torque which would have given us a larger time needed. Thus the calculations do conceptually match the experimental data.

PHYS4A/ Fall 2014; Angular Acceleration

Purpose: This experiment was done to calculate the angular acceleration of disks of different masses and see how it affects the angular acceleration of an object.

For this experiment,  a hanging mass is attached to a pulley and a rotating disk. 

Experimental Set-Up

The bottom disk is a solid steel disk which has air flowing over its surface. The top disk is either an aluminum disk or a steel disk. The two disks can be made to either spin independently or together by allowing air to flow between them or directing the air through the holes. By using these techniques and understanding that torque in this case is equal to the radius multiplied by the force (tension).

Torque = r* T 

The set-up with the hanging mass

By changing different aspects of the experiment, we are able to see whether acceleration increases or decreases. Running the experiment with different parameters and recording all of the info, it is apparent that as the hanging mass increases, the angular acceleration increases. When the diameter of the torque pulley changes angular acceleration increases proportionally. The same is seen if the mass of the disks increase.

The angular acceleration  graphs were found using logger pro and taking linear fits of the angular velocity over time graphs like the one below. When looking at the acceleration, due to the orientation of the set-up, negative acceleration was the acceleration as the hanging mass moved up and the positive acceleration is when the hanging mass moves down.

Angular velocity vs Time for Trial 1

In the above graph, velocity is either increasing or decreasing at a steady rate meaning that angular acceleration is constant. This was done for each graph with the acceleration as the hanging mass went up, went down, and the two accelerations averaged to fill in the table above. The same thing was done for all six runs with similar graphs. The average acceleration is used because by taking the average, you are removing systematic error due to friction, making the results more accurate. The observations taken earlier support the equation given where

Torque = (radius)*(Tension) = (Moment of Inertia)*(angular acceleration)

or 

Tr = r*T = I(alpha)

This matches with the above observations where the radius of the of the pulley either got bigger or smaller and when the hanging mass changed which changed the tension in the string. The mass of the disks were changed to get different I's or moments of inertia. As masses increased, the moment also gets larger about the center of mass. Dividing torque by a larger I gives a smaller angular acceleration which was observed. This showed our experiment supports the theory and the experiment was a success.