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. 

15-May-2017 Lab 16: Angular Acceleration Part 1

Lab 16: Angular acceleration Part 1

Kevin Nguyen

Lab Partners: Jose Rodriguez, Kevin Tran

Date Lab was performed: 15-May-2017

Purpose: This lab is designed to observe the effects of changing the hanging mass, torque pulley size, and weight of disk on the angular acceleration as the mass moves up and down. 

Theory/ Introduction: In this lab, we are trying to measure the effects of changing hanging mass, torque pulleys, and weight of the spinning disk on angular acceleration. On the apparatus, we know that there is a known torque acting on the rotating disk(s). This torque is represented by

where I represents inertia of the system and the alpha symbol represents the angular acceleration of the system.  Since torque is equal to 

we ran experiments to find the effects of doubling and tripling force, doubling the distance from the axis of rotation, doubling inertia, and dividing the value of inertia by 3 has on the angular acceleration.

Summary: First we used one of the apparatuses that were already set up.

We attached the cable coming out of the apparatus into the logger pro device, which is plugged into the computer.

When setting up the computer, we used rotary motion and set the equation of the sensor settings to 200 counters per rotation. 

We adjusted the hose clamp to test the bottom disk and top disk spinning before taking measurements. 

We measured the masses and diameters of the top steel disk, bottom steel disk, top aluminum disk smaller and larger torque pulley, and the mass of the hanging mass attached to the apparatus.

We then set up the apparatus and logger pro so that when the experiment starts, logger pro records the angular velocity of the spinning disk(s) when it is turning clockwise and counterclockwise. 

After doing so, we ran 6 separate experiments.

In the first experiment, we used only the hanging mass that came with the apparatus, the small torque pulley, and only the top steel disk was spinning. 

In the second experiment, we doubled the mass of the hanging mass, used the small torque pulley, and only the top steel disk was spinning.

In the third experiment, we tripled the mass of the hanging mass, used the small torque pulley, and only the top steel disk was spinning.

In the fourth experiment, we used the original hanging mass, used the large torque pulley, and only the top steel disk was spinning.

In the fifth experiment, we used the original hanging mass, used the large torque pulley, and only the top aluminum disk was spinning. 

In the sixth experiment, we used the original hanging mass, large torque pulley, and the top and bottom steel disk were spinning.

In each of these experiment, we recorded a angular velocity vs time graph.

Measured Data:

Average angular acceleration of 1st experiment

Average angular acceleration =  | angular acceleration down |  +  | angular acceleration up |  /2

                                                   =   | 1.061 rad/sec ^2 |  +  | -1.235 rad/ sec^2 | 

                                                   = 1.148 rad/second ^2

Calculated results:

Graph of 6th experiment

Graph of 4th experiment

Graph for 5th experiment

Graph of second Experiment

graph for 3rd experiment

Graph of 1st experiment

Explanation of Graphs:

The reason why we needed to graph an angular velocity vs time graph is because we are able to get the angular acceleration down and up by taking the slope of the line when the slope is positive (line is going upwards) and when the slope is negative (line is going downwards).

Conclusion:

For the conclusion, I'm using the first experiment data as the base to compare experiment's 2, 3 and 4 data to. The first experiment used a 24.6 gram hanging mass, top steel disk spinning, and small torque pulley. The 4th experiment is used as a base to compare experiment's 5 and 6 data to. Experiment 4 uses a 24.6 gram hanging mass, larger torque pulley, and top steel disk spinning.

In the second experiment, it seems that nearly doubling the hanging mass from the first experiment (from 24.6g to 49.6g) causes the average angular acceleration to approximately double from 1.148 rad/s^2 to 2.240 rad/sec^2.

In the third experiment, tripling the hanging mass from the first experiment from 24.6 g to 74.6 grams seems to triple the average angular acceleration from 1.148 rad/s^2 to 3.387 rad/s^2.

In the fourth experiment, it seems that using original hanging mass and the larger torque pulley (the larger torque pulley has 5.0 cm diameter, which is double the diameter of the smaller torque pulley) doubles the angular acceleration from 1.148 rad/s^2 to 2.192 rad/s^2.

In the fifth experiment, with the original hanging mass and larger torque pulley, using a lighter aluminum top disk that is 3 times lighter than the steel disk (466 grams vs 1362 grams) increased the angular acceleration 3 times the angular acceleration from the fourth experiment from 2.192 rad/s^2 to 6.1855 rad/s^2.

In the sixth experiment, with the large torque pulley (which doubled the torque value) and original hanging mass, we doubled the inertia of the system by having both steel disk (top and bottom) rotate. Having both disk rotate lead to making the angular acceleration nearly halved the angular acceleration from the fourth experiment (2.192 rad/s^2 vs 1.2745 rad/s^2)

Sources of uncertainty

One source of uncertainty is the frictional torque between the top and bottom disk. Since the disks aren't entirely frictionless, the frictional torque will lower the values of angular acceleration from what angular acceleration would be without friction.

Another source of uncertainty is that there is some air resistance acting on the hanging mass as it goes down and up, which can also decrease the angular acceleration of the system.

The pulley that the string lies on shown below

may also have friction acting on the string as the hanging mass goes up and down, causing the string to sometimes slip. The string slipping may cause the angular acceleration of the system to decrease.

26-Apr-2017: LAB Ballistic Pendulum

LAB: Ballistic Pendulum
Kevin Nguyen
Lab Partners: Jose Rodriguez, Kevin Tran
Date of lab performed: 26 April, 2017

Purpose of Lab: The purpose of this lab is to determine the firing speed of a ball from a spring-loaded gun.

Theory: In this lab, we are trying to find the firing speed of the ball. To find the firing speed of the ball, we must use the conservation of momentum and conservation of energy. There are two parts to this experiment. First, conservation of momentum occurs when the ball undergoes an inelastic collision with the hanging block. The ball's initial momentum is converted to the block's (plus ball) final momentum. Second, the conservation of energy occurs when the ball's initial kinetic energy is completely converted to gravitational potential energy at the block's maximum's height. Using the conservation of energy, the initial speed of the block plus ball can be found. The initial speed of the block plus ball will then be used in the equation for conservation of momentum in order to find the firing speed of the ball.

In the second part of the lab, we shot the ball outwards (on the floor) in order to find the velocity of the ball in the x-direction. In order to find the velocity (in the x-direction) of the ball, we needed to measure and record the horizontal and vertical displacement. After recording those two measurements, we would use 2-d kinematics in order to find the horizontal velocity of the ball.

Summary:

We first obtained this apparatus and made necessary adjustments to the strings so that the block's compartment lined up with the barrel of the gun. We then leveled the base of the apparatus, the length of the strings, and measured the mass of the ball and block. After we made the measurements, we put the ball in the barrel and pulled back and locked the spring into position. We put the angle indicator to zero degrees. We fired the ball into the block and recorded the maximum angle the block rose. We repeated this for five times and then took the average angle of the five trials.

Next, we set up the apparatus just like above except we put the block on top and pointed the barrel towards open space like below.

We made sure the apparatus is leveled before shooting the ball. To record the ball's horizontal distance, we shot the ball first to estimate where it landed. We then placed notebook paper and carbon paper so its position can be measured. To measure the vertical distance, we recorded the distance from the ground to the middle of the cannon barrel. We then did three trials so that we can get an average horizontal distance the ball traveled.

Table of measured data:

Calculated results:

Conclusions: 

The values of launch speed that we got from those two different measurements were different from each other by 9.28%. Their difference may have came from several factors. We did not take into account air resistance when doing kinematics to solve for launch speed, so the true launch speed may have been higher than what we originally got. We also didn't take into account how the speed of the ball plus barrel was slowed down by the angle measurer.