28-Nov-2016: Lab 22 Physical Pendulum Lab

 1. Title: Lab 22 Physical Pendulum Lab
     Name: Qiwen Ye (Sherry)
     Partner: Matt
     Date: 28-Nov-2016

2. Purpose
The gold of the experiment was to determine the expression for the period of the ring and the semicircle disk pendulum and verified the predicted periods by experiment.

3. Introduction/Theory
Since we tried to find the period of the physical pendulum, we need to find the moment of inertia of an object first in order to set up the torque equation.
Part 1 - A ring of finite thickness and a little notch cut out at the top
Assumed the point of pivot was exactly half way between the inner and outer radii and that the notch did not impact how uniform the mass distribution was.

First, derived the expression for the moment of inertia of the ring by using parallel axis theorem, then set up the torque equation as shown:

The radius of the ring should be consider the outer and inner of the radius; therefore, the radius of the ring was:

So, the period of the ring in pendulum motion was:

Part 2 - A semicircular disk of radius R

At first,  calculated the moment of inertia of the semicircular disk that was rotating its midpoint of the diameter:

Then, calculated the center of mass of the semicircular disk by using integral method:

After we found the center mass of the semicircle, we used the parallel axis theorem to calculated new moment of inertia of the semicircular disk, including it was rotating about the midpoint of its base and rotating about a point on its edge, directly above the midpoints of the base.

1) Oscillation about the midpoint of its base 

2)oscillation about a point on its edge, above the midpoint of the base. 

The angular velocity and angular acceleration, rotating its midpoint of its base:

The angular velocity and angular acceleration, rotating a point on its edge, directly above the midpoint of the base.

4. Apparatus/Experimental Procedure

Part 1 - A ring of finite thickness and a little notch cut out at the top

Placed the ring as shown, and attached a thin piece of making tape to the base of the ring. Set up a photogate below the ring such that, for small oscillation, the marking tape passed through the photogatte. Used LoggerPro (opened Pendulum Timer.cmbl) to determine the period of oscillation of the ring.

Part 2 - A semicircle disk of radius R

Attached a thin piece of masking tape to the bottom of each object and attached a pin to the top of both side of the plate of each object as shown:

1) oscillating about the midpoint of its base

2) oscillating about a point on its edge, directly above the midpoint of the base

5. Measured Date

For the ring and the semicircle, measured the outer and inner of the radius of the ring and the radius of the semicircle:

6. Calculated Result/Graph

By using the photogate, we got the graph of period vs. time in LoggerPro. The Y-Intercept (b) was the experimental value of the period of the ring or the semicircle.

Part 1 - A ring

Part 2 - A semicircle

1) Oscillating about the midpoint of the base

2) Oscillating about a point on its edge, directly above the midpoint of the base.

7. Explanation

After we got the experimental value of the period in each part, we calculated the theoretical value of the period.

Part 1 - A ring

Part 2 - A semicircle

Compared the experimental and theoretical value of the period in each shape.

8. Conclusion

In this experiment, we calculated the period of an physical pendulum of the ring and the semicircle by using the moment of inertia, the center of mass and the parallel axis theorem of the object to set up equation of Newton's second law. Our range of error and uncertainty is between 0.19 to 1.24%. For the semicircle, we attached a pin to it as a pivot clip; however, we don't need to consider the mass of the pin when we calculate the moment of inertia because the radius of the pin to the pivot point and the mass of the pin are too small. The masking tape should be placed exactly in the center of the base of the semicircle because it would effect the value when we are measuring the moment of inertia. If the masking tape is not in the center of the base, the distance from the center of mass to the masking tape will become bigger, it means Md^2 become larger that the moment of inertia of the object will become larger.

23-Nov-2016: Lab 21 Mass-Spring Oscillations Lab

1. Title: Lab 21 Mass-Spring Oscillations Lab
    Name: Qiwen Ye (Sherry)
    Partners: Matt
    Date: 23-Nov-2016

2. Purpose
In this experiment, we were trying to find the factors that affect the period of an oscillation of a spring. Also, we determined the relationship between the spring constant of the spring, the mass attached the spring and the period of the system.

3. Theory
According to the equation of the period of oscillation:

We found that the period of the an object was relate to the mass of an object and the its spring constant. Therefore, we separated the lab into two parts.
Part 1 - Same mass, different spring constant
We collected the data of period for same mass, where we varied spring constant k. We used the same mass of the system and different spring constant to find the relationship between the spring constant and the period of the system. We used five different spring that they had different spring constant, and let the mass of the an object and spring be the same.
Part 2 - Same spring constant, different mass
We collected the data of period on one spring, where we varied mass of the system. We used the same spring and five different masses to find the relationship between the period and the mass of the system.

4. Procedure
For part 1, we did the same experiment by five different groups with five different springs. Here was the set up of our group:

Placed the spring as shown, let the mass of the spring system=115g because all fie different groups need to have the same mass of the spring system.
For the mass of the spring system could calculate in that way:

Then, we used Logger Pro to find the position without mass and find the position with mass 115 gram as shown:

For part 2, used the same set up, but this time we varied the hanging mass on one spring: 20g, 40g, 60g, 80g, and 100g. Also, used the graph of position vs. time from Logger Pro to find its period because the period was the time of an object to complete one oscillation. For the Logger Pro set up, used the motion sensor to detect the movement of the spring, and then used the graph of position vs. time to find its period. 

5. Measured Date

Par 1, first, weighted the spring, then calculated the hanging weight that we need to make the mass of the whole system equal to 115g. Here was the process how to find the hanging mass. Our hanging mass was 110g.

6. Calculated/Result

Part 1 - same m, different k.

For the spring constant, we measured the height of the spring without any hanging mass (only the weight of the spring) and measured the height of the spring with hanging mass 110g (mass of whole system was 115gram). Then, used that formula we could find the spring constant.

We did the same experiment with different springs by different groups, here was the data we got from the other groups. 

Part 2 - same k, different m

In this part, we determined the period of the system on one spring (spring constant=2.5N/m) by using the equation of the period of oscillation. 

First, used the motion sensor to record the movement of an hanging mass that we could get the graph of position vs. time. 

Then, used 10 times oscillation to find the period as shown:

For the theory value of period, we used the equation of the period of oscillation, here was the example how we calculated the theory value of period in 20g:

7. Explanation/Analysis

We determined the graph of  period vs. oscillating system mass, and period vs. spring constant by entered the data into Logger Pro to create those graph. 

1) period vs. oscillating system mass

The period increased while we increased the mass attached the spring because when the spring of mass getting bigger it need more time to go back to its equilibrium.

2) period vs. spring constant 

The period decreased while we increased the spring constant because the spring with big k need less time to go back to its equilibrium, and the spring with small k need more time to go back to its equilibrium. (assume both equilibrium was the same)

8. Conclusion

In this experiment, we studied the mass-spring system by finding the relationship between 

the period, the spring constant of the spring, and the hanging mass. While we increase the spring constant of the spring, the period will decrease. While we increase the mass attached the spring, the period will increase. In part 2, we calculated the theory value of the period and compared to the experimental value, the different percent is less than 2%. It means that our experiment is successful. The factors cause our value different may be the spring constant of the spring is not exactly correct, or the air assistance affect the mass-spring oscillation.   

21-Nov-2016: Lab 20 Conservation of Linear and Angular Momentum

1. Title: Lab 20 Conservation of Linear and Angular Momentum
    Name: Qiwen Ye (Sherry)
    Partners: Matt
    Date: 21-Nov-2016

2. Purpose
In this experiment, we determined the angular velocity of  a rolling ball by using conservation of linear and angular momentum theorem and compared the theoretical and experimental values.

3. Introduction
In all the experiments so far, we had looked at the angular motion of an object rotating about some internal axis through its center of mass. This was an important type of rotation because any motion can be analyzed in terms of the translation of the center of mass and the rotation about the center of mass. However, in some instance it's more convenient to think in terms of rotation about a point external to the object. In this experiment, we would do that, investigating the conservation of angular momentum about a point that is external to the rolling ball.

4. Procedure
We did the lab together in the class, and Professor Wolf help us to set up the equipment (as shown in Figure 8.1) and record the data. Used the aluminum top disk, and mounted the ball center on top of the small torque pulley by using a gray-capped thumbscrew.

Determined the moment of inertia of the disk and ball catcher. Also, measured the mass of the ball.

Placed the ramp on the edge of a table, as shown. Determined the horizontal velocity of the ball as it rolls off the end of the ramp. First, marked a starting point on the ramp. Release the ball from this starting point and noted where the ball strikes the floor, and put the carbon paper on the floor. Measure the distance L and h. The horizontal velocity of the ball was then equal to L/t where t=2h/g.

5. Measured Data

Here was the measurement about the mass and the diameter of the ball; the diameter of the pulley.

6. Calculated Result

After we measured the data, we could find the moment of inertia of the system.

Then, used the conservation of angular momentum to calculate the angular velocity. 

7. Explanation
Compared the different percent between experimental and theoretical value for angular velocity, both different percent were less than 10%.

By the table, we found that once the radius of the ball getting bigger, the angular momentum would become larger.

8. Conclusion
In this lab, we studied the conservation of the linear and angular momentum theorem by calculating the angular velocity of the rolling ball. Also, we determined the relationship between the radius of the ball and the angular velocity that if the radius gets larger, the angular velocity becomes bigger.

21-Nov-2016: Lab 19 Conservation of Energy/Conservation of Angular Momentum

1. Title: Lab 19 Conservation of Energy/Conservation of Angular Momentum
    Name: Qiwen Ye (Sherry)
    Partners: Matt
    Date: 21-Nov-2016

2. Purpose
In this experiment, we need to use conservation of energy and conservation of angular momentum to predict how high a clay-stick combination should rise, then capture the experiment on video in order to compare actual results with predictions.

3. Theory
From the conservation of energy theorem, we could find an angular velocity of a meter stick before it hit the clay. Then, for the conservation of angular momentum, we could find the angular velocity after it hit the clay. Meanwhile, the clay stuck on the meter stick and they were moving together, it means that they have the same angular velocity. Last, used the conservation of energy theorem again to determent how height the clay-stick combination should go up.

4. Apparatus/Experimental procedure
Set up a system that released the meter stick, which rotated about one end (20 cm away from the end of the meter stick) from a horizontal position. After the meter stick reached the bottom of its swung it collides in elastically with a blob of clay. The meter stick and clay would stick together and continue to moving to its maximum height. Then, captured the experiment on video by camera and used Logger Pro to analysis the system movement in order to find the experimental maximum height.

5. Measured Data
By using the Logger Pro, we described the movement of the meter stick from horizontal position to its maximum height. Used a blue dot to record the movement of the meter stick in order to find what is it maximum height. While we put the dot to record the movement, it would create a table about the the height of the meter stick in its rotation.

According to that table, we found that the maximum height of the clay after collision was 0.2857 m. 

Here was the measurement about the meter stick and the clay:

6. Calculated Results

By using the conservation of energy and conservation of angular momentum theorem, we could find the maximum of the system in the rotation. Used the measurement to calculate angular velocity just before the stick hit the clay, when it was in its vertical position. Then, used the conservation of angular momentum equation to find the angular velocity hit. We got the maximum height of the clay was 0.3079 m.

7. Explanation
Compared to the theoretical and experimental value of the maximum height, we found that the different percent was 7.75.

8. Conclusion
In this experiment, we studied the conservation of energy and conservation of angular momentum to calculate the angular velocity before the meter stick and the clay collided and after collided. Also, determined the momentum of inertia of the meter stick and the clay while they were rotating about near one end of the meter stick. Used the conservation of energy again to find the maximum height of the combination of the system. The different of the experimental and theoretical value was 7.75%. The friction between the meter stick and the apparatus may cause the error percent. Or, we marked a dot of the system movement in Logger Pro, we may not that in the center. The center of mass of the clay may not have been exactly at the end of the meter stick.

14-Nov-2016: Lab 17 Finding the moment of inertia of a uniform triangle about its center of mass

1. Title: Lab 17 Finding the moment of inertia of a uniform triangle about its center of mass

    Name: Qiwen Ye (Sherry)

    Lab Partners: Xavier

    Date: 14-Nov-2016

2. Purpose

In this experiment, we determined the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientation of the triangle by using the parallel axis theorem. `

3. Theory

The parallel axis theorem states that

Because the limits of integration were simpler if we calculated the moment of inertia around a vertical end of the triangle, we could calculate that moment of inertia and then get Icm from:

Then, we used an equation (like above shown) derived from the previous lab to find the moment of inertia in each system. This equation uses hanging mass, radius of pulley, and average of acceleration.

4. Apparatus/Experimental Procedure
We mounted a holder and disk, and the upper disk floated on a cushion of air. A string was wrapped around a pulley on top of an attached to the disk and goes over a freely-rotating "frictionless" pulley to a hanging mass. The tension in the string exerted a torque on the pulley-disk combination.

Then, set up Logger Pro in Rotary Motion by 200 counts per rotation. Recorded the angular acceleration of the system in order to determine the moment of inertia of the system.

After we measured the moment of inertia of the system without triangle, we could mount the triangle onto the disk-pulley-holder system and measured its acceleration again and determined the moment of inertia of the triangle in vertical and horizontal orientation.

Vertical Triangle

Horizontal Triangle

5. Measured Date/Graph
hanging mass: m=0.0245kg
radius of pulley: r=0.0141m
the mass of triangle: M=0.4568kg
base of triangle: B=0.0987m ; H=0.1485m

Those three graphs we got from Logger Pro to find the average acceleration in each system. The slope of graph was its acceleration.

No Triangle: velocity vs. time

Vertical Triangle: velocity vs. time

Horizontal Triangle: velocity vs. time

6. Calculated Results
Through the above graphs, we got the acceleration in each system.

Here was the example, how could we calculated the average acceleration:

After we calculated the average acceleration in each system, we could calculate the moment of inertia in each system by using the equation we derived from the previous lab, and we used the moment of inertia of triangle system minus the moment of inertia without triangle to find the moment of inertia of the triangle in each of the two orientations. 

We could also calculate the theoretical value of the moment of triangle in each of two orientations by using parallel axis theorem:

7. Explanation

After we got the value of theoretical and experimental of the moment of the triangle in vertical and horizontal, we compared both values by finding the difference percent.

8. Conclusion

In this experiment, we studies how to use the parallel axis theorem to find the moment of inertia and how to use the torque to find the moment of inertia of an object. For the moment of inertia of the triangle, we compared the theoretical and experimental values, it shows that the difference of uncertainty was 18.12% in vertical triangle; 3.26% in horizontal triangle. Even though the horizontal triangle have a small different percent than the vertical one, our experiment still successful. The factors cause our theoretical and experimental values different are: the mass of the triangle plate may not completely in uniform density; the air being released to spin the rotating disks may have fluctuated that will cause the acceleration fluctuating; or our measurements a little bit error.