Phys4A Physical Pendulum Lab Kevin Nguyen

Kevin Nguyen
09 June 2017
Lab Partner: Jose Rodriguez
Physical Pendulum Lab

Purpose: The purpose of this lab is to find the expressions of the period for physical pendulums. Then verify the expressions by comparing the predicted periods to the experimental periods

Theory:

In this lab, we are finding the value of the period for the triangle with the axis of rotation on the apex and the half circle with the axis of rotation on the midpoint using the concept that

 net torque = Inertia of shape x angular acceleration

We are using the first and third shapes.

 We then derived the inertia and center mass (from the axis of rotation) of the shapes.

For the triangle with axis of rotation on the apex.

For the Half circle with axis of rotation at its midpoint

Then we used the Torque = I alpha equation to solve for angular velocity.

Summary:

First we set up the apparatus like this. 

The tape is added to keep the paperclip attached to the triangle. The strip of notecard is used so that the photo gate kept track of the triangle's period as it oscillates back and forth.

Unfortunately, there was no picture of the semi circle with the axis of rotation at the midpoint, but it was set up similarly to the triangle.

Then set up a graph by bring up the physical pendulum file and ran the program. 

The period of the triangle with axis of rotation at its apex is 0.8627 seconds

The period of the semi circle with axis of rotation at its midpoint, which is 0.6949 seconds

Then compare the experimental value to the predicted value.

The percent error for the calculations are

When comparing the experimental period to the calculated period, we found that they were both very similar to each other as indicated by the very low percent difference from each other (3.41% for the semi circle and 0.7546% for the triangle). It was safe to assume that the mass wouldn't affect the period very heavily since mass does not affect the period of the triangle according to Newton's second law. The masking tape would have affected the period more since the masking tape affects the dimensions of the shapes, so the period of the shapes may have been greater than its true value.

Some external factors that may have affect our values from the experiment include air resistance, since air resistance slows down the shape's angular velocity as it moves back and forth, and friction between the hook and the paper clip, which would have also slowed down the shape's angular velocity. 

05-31-2017: Lab 19: Conservation of Energy/ Conservation of Angular Momentum

Lab 19: Conservation of Energy/ Conservation of Angular Momentum

Kevin Nguyen

Lab Partners: Kevin Tran, Jose Rodriguez

Date of Lab Performed: 05-31-2017

Purpose: The purpose of this lab is to use the concept of angular momentum of the meter stick and clay and conservation of energy and conservation of angular momentum to find how high the meter stick and clay has risen. 

Theory:

Picture of the apparatus

In this experiment, we used the conservation of energy/ conservation of momentum to solve for the final height the clay has risen after the meter stick hits it.

We used conservation of energy first to find the angular velocity right before the meter stick collides with the clay

Then use the concept of angular momentum to find the angular velocity of the system right after the collision.

Then use the conservation of energy in order to find the maximum height the meter stick plus clay rises.

Summary:

Picture of the apparatus

First, we measured the mass of the clay (and the 3 metal pieces that acted as a stand for the clay) and meter stick since they were useful for our predicted height calculations. We set up the apparatus like above. Note that the axis of rotation is at the 10 cm mark, so the moment arm is 90cm long. 

Then record a high speed video of the ruler swinging, colliding with the clay, and rising to its maximum height. After that, put the video on logger pro, set the origin at the clay, which we considered as a point mass, and set the distance between the axis of rotation and the clay to be 0.9 m. 

Ignore the blue dots

After that, jump to the point of the video where the ruler is at its maximum height. Then, using the video tool, measure the distance between the x-axis and the maximum height the ruler and clay went. 

The experimental result was 0.276 m

Then compare the experimental results to the calculated results

Data Table

Calculated result

The height calculated was 0.253 m

The percent difference between the two values is 8.696%

Conclusion

There is a relatively large difference between the calculated result (.253m) and the experimental result (.276m). There are several reasons for the relatively large difference between the two values.

First, since we did not take into account the air resistance on the meter stick, the angular velocity may have been slower than what it should have been. A slower angular momentum would have resulted in a lower height.

Second, the friction on the axis causes the ruler to rotate slower and therefore have a slower angular velocity, causing the ruler to travel less than what it should have traveled. 

05-24-2017: Lab 18: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque
Kevin Nguyen
Lab Partners: Jose Rodriguez and Kevin Tran
Date of Lab performed: 05-24-2017

Statement: The purpose of this lab is to find the moment of inertia and frictional torque of the apparatus so that we can solve for the time it takes for the cart (that was attached to the small cylinder) to travel down a sloped surface.

Theory: To find the moment of inertia of the entire apparatus, we needed to find the individual moments of inertias of the the disk and the two side cylinders attached to it. To find those moments of inertias, we found the mass of the entire apparatus (given on the side of the disk), the diameters, and the thickness of the disk and the two cylinders. After getting those measurements, we found the individual masses of the two side cylinders and the disk by using the formula

Mass 1 and Mass 3 represents the mass of the side cylinders. Volumes 1 and 3 represents the volume of the side cylinders and volume 2 represents the volume of the disk

Mass two represents the mass of the disk

After finding the individual masses of the two cylinders and the disk, we used the formula

M represents mass, R represents the distance from the center of the axis of rotation.

                                         
To find the moment of inertia of each part.

After finding the individual moments of inertia of the disk and two cylinders, we added them together to find the total moment of inertia of the entire apparatus.

Then we had to find the frictional torque of the apparatus.

To find the frictional torque, we used the concepts

Torque equals to the inertia of the system times the angular acceleration.

The change in angular velocity over change in time equals to angular acceleration.

Then, for the second part of the experiment, we set up the apparatus like this.

We were tasked to find the time experimentally and theoretically.

First, we timed how long it took for the cart to travel down the slope (1 meter).

                                                    
Second, we found the time theoretically using these equations.

We then compared the experimental time to the theoretical time to see if they are close to each other.

Summary:
We grabbed one of the apparatus that was available on the table.

We then measured the diameter of the disks and two cylinders. After taking necessary measurements, we measured the frictional torque on the apparatus.

We put a tape on the disk in order to easily indicate the rotational speed of the disk. We then recorded a high speed video of the disk spinning until it stops. To find the angular deceleration of the disk, we put the video on the logger pro program, set up a angular velocity vs time graph, and plotted the position of the tape each 2 frames until the disk stopped rotating. We then used the slope of the angular velocity vs time graph to find our angular deceleration, which allowed us to find the value for frictional torque.

                                                          

The angular deceleration we got was -0.1049 rad/sec^2

Then, we set up our apparatus like below

             

The apparatus is connected to a 500 gram cart and the cart is on a track that is inclined 49 degrees. We measured how long it took for the 500 gram cart to travel one meter down the sloped track (from rest). We ran the experiment two more times (for a total of three times) in order to get an accurate average time.

Measured Data:

Experimental Time

Dimensions of the apparatus

Calculated Results:

The time that we calculated was 6.67 seconds

                                                     
Conclusion

The time that we calculated through theoretical means was very accurate since the calculated time was only off by 0.05 seconds compared to the average experimental time.

Although the results agreed with one another, there are sources of uncertainty that may have produced error in the results.

First, we assumed that there was no friction on the track. By making that assumption, the value of acceleration of the cart would have been slightly lower than what it should have been.

Second, air resistance on the cart was not considered, so the cart could have also been slower than what we've calculated.

22-May-2017: Lab 17: Physics 4A Lab

Lab 17: Physics 4A Lab
Kevin Nguyen
Lab Partners: Kevin Tran, Jose Rodriguez
Date of Lab Performed: 22-May-2017

Statement: The purpose of this lab is to find the moment of inertia of the right triangular thin plate around its center of mass, for both of its perpendicular orientations.

Theory/ Introduction:
For this lab, we used two approaches to find the moment of inertia of the triangular masses.

For the first approach, we used the idea that we can subtract the inertia of both the triangle and disk from the inertia of the disk in order to find the inertia of the triangle.

Each moment of inertia is defined by this equation.

In this equation, "mg" represents the weight of the hanging mass (in Newtons), "r" represents the radius of the torque pulley (in meters), and alpha average represents the sum of the absolute value of the angular acceleration of an ascending and descending hanging mass.

In the second approach to find the moment of inertia for the right triangular thin plate around its center of mass, we used the parallel axis theorem, which states that

Since it is easier to find the moment of inertia of one vertical end of the triangle rather than the inertia around the center mass of the triangle, we do this

Summary of the apparatus:

First, we used the apparatus already set up in order to find the inertia of the top disk since it is the only one that is spinning in the experiment. 

We measured the mass of the hanging mass, the radius of the torque pulley, and angular acceleration ascending and descending since those measurements are crucial for solving for the inertia of the disk and the inertia of the disk and triangle. We took measurements of the angular acceleration by running the logger program. We found those values by using an angular velocity vs time graph and finding the slope of the positive line and negative line.

Graph of the disk alone
Angular acceleration up = 5.775 rad/sec^2
Angular acceleration down = -6.494 rad/sec^2

After finding the inertia of the disk, we placed the thin triangular plate in two orientations shown below and found the moment of inertia for both of them.

The longer side of the triangle is the base

Graph where the longer side is the base
Angular acceleration up = 3.698 rad/sec^2
Angular acceleration down = -4.142 rad/sec^2

The shorter side of the triangle is the base

Graph where shorter side is the base
Angular acceleration up = 4.789 rad/sec^2
Angular acceleration down = -5.357 rad/sec^2 

After taking necessary measurements, we plugged the values into our equations.

For the moment of inertia of the shorter base, we got the value 2.059 x 10^-4 kgm^2. For the moment of inertia of the longer base, we got the value 4.169 x 10^-4 kgm^2.

For the second approach, we first need to derive an equation for the inertia of a right triangle around the vertical end. Before calculations, we measured the dimensions of the triangular plate since we need those values to calculate the moment of inertia of the perpendicularly oriented triangles. The derivation can be found here. 

After finding that the moment of inertia of the oriented triangles is (1/18)MB^2, we use this equation to solve for the inertia of the perpendicularly oriented triangles.

The moment of inertia of the triangle with the short base is 2.528 x 10^-4 kgm^2 and the moment of inertia of the triangle with the long base is 5.840 x 10^-4 kgm^2. 

Conclusion

In the end, the values of inertia for the triangles did not match each other although they both do agree that the triangle with the longer end as the base had a larger moment of inertia. The moment of inertia for the triangle with the short base found in the first and second approach differ by nearly 20%. The moment of inertia for the triangle with the long base found in the first and second approach differ by 33%. This was found by using the equation 

Equation for percent difference

There are multiple sources of uncertainty that may have gave us a large percent difference between the results. 

First, there is air resistance acting on the hanging mass as it ascends and descends, which could have lowered the angular acceleration of the apparatus. 

Second, there is also air resistance acting on the triangle as the apparatus spun, which also lowered the angular acceleration of the system,

Third, since we assumed that there was no friction in this experiment, the values of angular acceleration may have been lower than what it should have been. 

29-May-2017: Lab Angular Acceleration part 2

Lab: Angular Acceleration part 2
Kevin Nguyen
Lab Partners: Kevin Tran, Jose Rodriguez
Date performed: 17 May 2017

Statement: The purpose of this lab is to determine the moment of inertia of each disks (steel bottom, steel top, aluminum top, steel bottom and top) using data collected from part 1 of the lab.

Theory/ Introduction:

Free body diagrams

In this lab, our task was to find the moment of inertia for each disks. The problem that was present was that since there is some frictional torque in the system, the angular acceleration of the system when the hanging mass is going down wasn't the same as when the hanging mass is going up.

In order to determine the inertia of the disks, we first indicated counter-clockwise as the positive direction, the string goes down when the disk is spinning counter clock wise and vice versa, and that frictional torque is going in the clockwise direction. We used Newton's second law to predict that

(1) Torque of string (down) - Torque of friction = Inertia of disk x angular acceleration (down)

where the "angular acceleration (down)" is positive. When the hanging mass is rising, the tension of the torque of the string goes in a clock wise direction along with frictional torque, giving the equation

(2) -torque of string (up) - torque of friction = Inertia of disk x angular acceleration (up)

where the "angular acceleration (up)" is negative. Since we know that torque of the string equals to the tension force of the string times the radius of the torque pulley and that tension force is unknown, we used Newton's second law to find the tension of the string. We need two equations for tension of the string: one for descending hanging mass and one for ascending hanging mass. 

Descending

Weight of hanging mass - tension (down) = net mass x acceleration (down)

Since we know that acceleration = angular acceleration (down) x radius of torque pulley

Weight of hanging mass - tension (down) = net mass x angular acceleration (down) x radius of torque pulley

Then we multiply radius of torque pulley to both sides since we see that we can get tension (down) x radius of torque pulley = Torque of string (down)

(3) Tension (down) x radius of torque pulley = Torque of string (down) = (weight of hanging mass x radius of torque pulley) - (net mass x angular acceleration (down) x radius of torque pulley^2)

Ascending

Weight of hanging mass - tension (up) = net mass x acceleration (up) = net mass x angular acceleration x radius of torque pulley

(4) -tension (up) x radius of torque pulley = -torque of string (up) = -weight of hanging mass x radius of torque pulley - (net mass x angular acceleration (up) x radius of torque pulley^2)

Then we plugged in (3) into (1) and (4) into (2), which will give:

(5) (weight of hanging mass x radius of torque pulley) - (net mass x angular acceleration (down) x radius of torque pulley^2) - torque of friction = Inertia of disk x angular acceleration (down)

(6) -(weight of hanging mass x radius of torque pulley) - (net mass x angular acceleration (up) x radius of torque pulley^2) - frictional torque = Inertia of disk x angular acceleration (up)

Since we don't know frictional torque, we cancelled it out by subtracting (6) from (5), which will give 

Inertia of disk = (net mass x gravity constant x radius of torque pulley) / ((| angular acceleration down|  - | angular acceleration up| )/2) - (net mass x radius of torque pulley^2)

which is simplified to 

(7) Inertia of disk = (net mass x gravity constant x radius of torque pulley) / (| average angular acceleration| ) - (net mass x radius of torque pulley ^2)

If we wanted frictional torque, we can add equations (5) and (6), giving

(8) frictional torque = (net mass x radius of torque pulley^2) (| angular acceleration up|  -|  angular acceleration down| ) /2

Summary of apparatus:

The set up is the exact same as the one from angular acceleration part 1 lab blog. 

List of measured data

List of calculated results

Also includes calculations

Conclusion

From the calculated results of experiments #1 to #3, it seems that the calculated inertias of the disks are very close to each other (5.235 x 10^-3 kgm^2 vs 5.394 x 10^-3 kgm^2 vs 5.350 x 10^-3 kgm^2), so these experiments are accurate in terms of the angular accelerations given from the apparatus. 

From experiment #4, it seemed that doubling the torque pulley radius slightly increased the inertia of the disk from #3 to #4 from 5.350 x 10^-3 kgm^2 to 5.438 x 10^-3 kgm^2. 

From experiment #5, it seem that using a top disk that is nearly 2 to 3 times lighter than the steel disk decreased the moment of inertia of the disk approximately 3 times from the moment of inertia from experiment #4 from 5.438 x 10^-3 kgm^2 to 1.887 x 10^-3 kgm^2. 

In experiment 6, it seemed that having both steel disk spin nearly doubled the moment of inertia from experiment #4 from 5.438 x 10^-3 kgm^2 to 9.396 x 10^-3 kgm^2.

There may be sources of uncertainty that may have not allowed us to get the true values of inertia of the disks. 

First, there may be some air resistance acting on the hanging mass as it descends and ascends, which may lower the overall value of angular accelerations attained from the logger pro graphs (from Angular acceleration part 1 lab blog). 

Second, the tools used to measure the angular acceleration of the system (logger pro, motion detector of apparatus, etc) may not be that accurate (for example, the angular acceleration is off by a couple decimal places). As a result, the angular acceleration may be larger or smaller than the true value.