Kevin Tran
Lab partners: Kevin Nguyen, Jose Rodriguez
June 2, 2017

Conservation of Energy/Conservation of angular momentum

Purpose: To determine the height of the clay/meterstick when it rises experimentally and comparing it theoretically.

Introduction: We will be setting an apparatus on a table, we will attach the end of a 1 meter stick on it. Then, we will be having a small piece of clay floor. We dropped the ruler from its horizontal height and let it gain speed. It will then collide with the clay and they will stick and reach a maximum height. To do this, we attached tape to the end of the meter stick and also taped the clay, which will allow them to stick together when collided.

Below is what the setup looks like.

I would like to note that:

Meter stick: 0.088 kg

Clay: 0.03 kg

The pivot is at 0.1 meter.

Our next step was to video the before and after collision. We set up loggers pro to allow us to find the maximum horizontal and vertical distance that the clay/ meter stick travels. As a result, loggers pro gave us a height of 0.246 meters and a horizontal distance of 0.712 meters.

Below is the moment of inertia of the stick.

Below is the work of gravitational energy to angular velocity(omega). As a result, the angular velocity is 5.48 rad/s

Below is the theoretical work after the collision of the meter stick and the clay, which we solved for angular velocity(omega). As a result, we have an angular velocity of 2.626 rad/s

Below is the work to solve for our predicted height. As a result, we have a height of 0.21 meters. 

 Below is our work for uncertainty.

Conclusion: We were able to use conservation of energy and angular momentum to find our the height of the meter stick and clay when it is going up. As a result, calculated result was different to the experimental result by 0.03 meters, which gives us a 14% uncertainty.

The cases for uncertainty could be:

-Air resistance, which could cause energy lost.

-The collision of the meter stick and clay may lose energy in the experiment.

Kevin Tran
Lab partners: Kevin Nguyen, Jose Rodriguez
May 27, 2017

Finding the moment of inertia of a uniform triangle

Purpose: To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

Introduction: We mount the triangle on a holder and disk. The upper disk floats on a cushion of air. A string is wrapped around a pulley on top of and attached to the disk and goes over a freely-rotating frictionless pulley to the hanging mass.

Below is the parallel axis theorem, which states that:

First, I like to note about the measurements within this experiment.

Top shell disk

-Diameter: 0.126 m

-mass: 1.356 kg

Bottom shell disk: 

-Diameter: 0.126 m

-mass: 1.343 kg

Aluminum disk

-Diameter: 0.126m

-mass: 0.467

Smaller pulley

-Diameter: 0.0248m

-mass: 0.0110kg

Larger puller

-Diameter:0.0498m

-mass: 0.036kg

Hanging mass

-mass: 0.025kg

Below is one of the three set ups that we compared.

With Loggers Pro, we were able to calculate the angular acceleration of the direction in the downwards and the upwards direction. The rising slope represents the downwards direction and the descending slope represents the upwards direction.

-The downwards angular acceleration for the system without a triangular piece is 5.775 rad/s^2

-The upwards angular acceleration for the system without a triangular piece is -6.994 rad/s^2
The average angular acceleration is 6.38 rad/s^2

Below is the calculated Inertia for the system without a triangle.

Next, below is the system with the triangle point pointing up.

Below are the calculation to solve for the center mass of the triangle









































Below is the inertia when the triangle is rotated around the edge and around the center of mass.





Below are the results of angular acceleration for the downwards and the upwards direction. 

-The downwards angular acceleration for the system without a triangular piece is 4.789 rad/s^2

-The upwards angular acceleration for the system without a triangular piece is -5.357 rad/s^2
-The average angular acceleration is 5.073 rad/s^2

Below is the calculated inertia for the system with the triangle tip facing up.

Next, below is the set up with the triangle turned 90 degrees.

Below are the results of angular acceleration for the downwards and the upwards direction. 

-The downwards angular acceleration for the system without a triangular piece is 3.698 rad/s^2

-The upwards angular acceleration for the system without a triangular piece is -4.142 rad/s^2
-The average angular acceleration is 3.92 rad/s^2

Below is the calculated inertia for the system with the triangle rotated 90 degrees.

We were able to find the difference of moment of inertia with no triangle with the triangle in each position. The moment of inertia of the triangle in a vertical position minus moment of inertia of the system with no triangle was 2.56 x 10^-4.

0.0012 - 0.000944= 2.56x10^-4 kg*m^2

We did the same for the triangular plate in the horizontal position. As a result, we found that its moment of inertia is 6.06 x 10^-4 kg*m^2

0.00155- 0.000944= 6.06 x 10^-4 kg*m^2

Conclusion: 

By finding the difference between the moment of inertia of the rotating system and then the moment of inertia of a triangular plate in two different positions, we were able to find the moments of inertia of the triangular plate itself. The values of these numbers represent the amount that it takes for the triangle(s) to spin around a fixed axis. The data allowed us to come to a conclusion that it takes a higher value for the triangle to spin around its center of mass in the horizontal position. The types of errors that could of affected our results are the air resistance of when the system is spinning and the mass of the triangles may not have been completely in uniform density.

Kevin Tran
Lab Partners: Kevin Nguyen, Jose Rodriguez
May 26, 2017

Moment of Inertia and Frictional Torque

Purpose: To determine the moment of inertia of a spinning object and the frictional torque in the system and then test the friction torque by comparing the time it takes to descend a cart with the experimental value and theoretical value.

Introduction: In this experiment, we used the rotating part of the apparatus to determine its moment of inertia. We performed two parts for this experiment.

Below is the apparatus that we used to performed this lab. In the picture below, there are 3 rotating parts in the system: 2 small pulleys and 1 large pulley

Small pulley( cylindrical shape ):

-Diameter: 0.031 meters

-Height: 0.05 meters

Large pulley( cylindrical shape ): 

-Diameter: 0.1997 meters

-Thickness: 0.016 meters
















The mass of the whole system is 4.802 kg. The number in the picture below gives us the total mass of the system.

Below is the side view of the apparatus.

Note:

We can't exactly assume the entire system as one mass. So, we have to break it into 3 parts. We find the inertia of the individual pulleys and summed up the inertias in order to find the total inertia of the system.

First, I like to note that below are the volumes of the 3 pulleys.
V1/V3: the two cylinders on both sides
V2: The cylinder in between 1 and 3.













In order to find the individual inertias of each of the cylinders, we have to find the mass of each of them. Below is the formula to calculate for an individual mass. Our "M total" is 4.802 kg.

Below are the calculated results for the masses.
m1/m3: the cylinders on both sides.
m2: the cylinder in between 1 and 3.



Now with the masses, we were able to find the total inertia of the system.

Moment of inertia of the cylinder: 

  

Our next step is to find the frictional torque of the system. So, we spun the wheel to calculate the angular acceleration as it's slowing down within the 10 second time range. We jotted down points on the video to allow us to find the positions at multiple time frames, which will eventually give us a chart of a omega/time chart. As a result, we have an angular acceleration of -0.105 rad/s^2 




Since we know that Torque= (Total inertia) x (Angular acceleration), we were able to find the frictional torque of the system.

Friction torque= (0.02 kgm^2) x (-0.105 rad/s^2) 

Friction torque= -0.0021 N*m

Our second part of the lab was to attach a 547 gram dynamic cart to the apparatus. The cart will be rolling down an inclined track for a distance of 1 meter. The inclined track was angled at approximately 49 degrees.






































My lab partners and I timed the cart as it rolls down 1 meter. We performed a total of three trials. As a result, we got:

Trial 1: 6.67 seconds
Trial 2: 6.58 seconds
Trial 3: 6.6 seconds

On average, the time was 6.62 seconds.

Below is the calculated work to compare the results of both the calculated time and the stopwatch time.

























As a result, our calculated results was 6.74 seconds.

In comparison to the 6.62 seconds and the 6.74 seconds, the differences are off by 0.12 seconds off but relatively close. By calculating, we have a 1.78% error in the timing.

Conclusion: 
-By comparing to the theoretical time 6.74 seconds, there is a 1.78% error, but we could say that they are really close to the true value.
-There are some uncertainties in this experiment:
     -Uncertainty from the radii measured.
     -Uncertainty from the angle measured.
     -Uncertainty from timing the time the cart takes to roll down the distance of 1 meter.
- Overall, we could say that the method we use to find the frictional torque is accurate.

Kevin Tran
Lab Partners: Kevin Nguyen, Jose Rodriguez
20 May, 2017

Angular Acceleration: Part 2

Purpose: To find moment of inertia with the given results from part 1, which are the hanging masses, radius of the torque pulley, and the average of the absolute values of the angular accelerations when the hanging mass is going up and down when it going down.

Introduction: I determine the experiment values for the moments of inertia of the disks with the data from part 1.

Below is the experimental values for the moments of inertia of the disks.

Conclusion: In comparison to all the EXPTs, the moment of inertia changes for each case due to the conditions: rotating mass, radius of the torque pulley, and angular acceleration.

Kevin Tran
Lab Partners: Kevin Nguyen, Jose Rodriguez
20 May, 2017

Angular Acceleration Part 1

Purpose: To find a measured value for the moment of inertia by applying torque to an object that can rotate and measure the angular acceleration, which gives us torque and acceleration data.

Introduction: We used a device that allows us to apply a know torque onto an object that can rotate. By allowing that to happen, we are able to measure the angular acceleration, which will eventually be used to measure values for the moment of inertia. The device we used is a system that requires air in order to activate. (picture below). In this experiment there are two disk stacked on top of each other that helped us get our results.

Before I can continue, I would like to note about the measurements and masses of the following equipment:
-the diameter and mass of the top steel disk: 1,357g, 126.5mm +/- 0.1mm, +/- 1g
-the diameter and mass of the bottom steel disk: 1,348g, 126.3mm  +/- 0.1mm, +/- 1g
-the diameter and mass of the top aluminum disk: 466g, 126.4mm  +/- 0.1mm, +/- 1g
-the diameter and mass of the smaller torque pulley: 9.99g, 25mm  +/- 0.1mm, +/- 1g
-the diameter and mass of the torque pulley: 36.3g, 48.9mm  +/- 0.1mm, +/- 1g
-the mass of the hanging mass supplied with the apparatus: 24.58g +/-1g

We plugged the power supply into the Pasco rotational sensor. We connected a cable to the Lab Pro at Dig/Sonic 1, so the computer is reading the top disk. We opened Logger Pros and went to our devices and chose "Rotary Motion" to give us a two graphs, which are the Angle(rad)/Time(sec) graph, and Velocity(rad/sec)/Time(sec) graph. In this case, we excluded the Angle(rad)/Time(sec) graph and used the Velocity(rad/sec)/Time(sec) graph. The reason for the exclusion is because of the poor timing resolution of the sensors. Then, we set up the equation in the sensor settings to 200 counters per rotation. 

Next, we turned on the compressed air so that the disks are rotating separately. With the string wrapped around the torque pulley and hanging mass at its highest point, we started collecting data with our loggers pro. 

Below are the data collection. First I would like to note that:

- Expt #1, 2, and 3: Effect of changing the hanging mass.

- Expt #1 and 4: Effect of changing the radius and which the hanging mass exerts a torque.

- Expt 4, 5 and 6: Effect of changing the rotating mass.

Below are the 6 graphs that helped my lab partners and I get our results. How we found angular acceleration for the downwards and upwards direction is by the slope in the graphs. The rising slope gives us the angular acceleration when it is going downwards and the descending slope gives us the angular acceleration when it is going upwards. How we found the average angular acceleration of each EXPT is by adding both the angular acceleration in the upwards and the downwards direction and divided by two.

EXPT 1:

EXPT 2:

EXPT 3:

EXPT 4:

EXPT 5:

EXPT 6:

Conclusion: In comparison to the first 3 EXPT, increasing the hanging mass only will increase the angular acceleration in the downwards and the upwards direction. In EXPT 1, the average angular is 1.148 rad/s^2. On the other hand, EXPT 2 is 2.24 rad/s^2, and EXPT 3 is 3.387 rad/s^2. To sum up, the change of mass in the hanging mass will increase angular acceleration. Now to compare EXPT 4 and EXPT 5. Next, changing the top disk will change the angular acceleration. When applying the steel disk on the top, the average acceleration is 2.192 rad/s^2. In comparison to the EXPT 5 with an aluminum disk on the top, the average angular acceleration is 6.186 rad/s^2. Note that the mass of steel disk is 1,357 g and the aluminum disk is 466 g. The change of rotating mass in the top disk tremendously affects the angular acceleration in the upwards and the downwards direction. In addition, changing the radius of the torque pulley does affect the angular acceleration. An error that might affect these results is the possible friction between the top and bottom disk when accelerating.

Kevin Tran
Lab Partners: Kevin Nguyen, Jose Rodriguez
May 1st, 2017

Ballistic Pendulum

Purpose: To determine the firing speed of a ball from a spring-loaded gun.

Introduction: We have a ball with a mass of 7.64 grams that undergoes an inelastic collisions with a nylon block with a mass of 77.6 grams, which is being supported with vertical strings. After the ball collides with the nylon block, both will rise through some angle, which loses kinetic energy and gains potential energy. When the system reaches its maximum height, the kinetic energy will be zero. 

Below is the model that we used to determine our angles. We pulled on the cannon to give it some initial potential energy. Once we released the string, the cannon will uncompress and launch a ball into the nylon block. We repeated this process 5 times to get 5 different angles.

Data Collected:

Mass of ball: 0.00764 kg

Mass of Nylon block: 0.0776 kg

Trial 1- 20.5 degrees +/- 0.5 degrees

Trial 2- 20.5 degrees +/- 0.5 degrees

Trial 3- 19.5 degrees +/- 0.5 degrees

Trial 4- 19.5 degrees +/- 0.5 degrees

Trial 5- 20.5 degrees +/- 0.5 degrees

We will use 20.5 degrees +/- 0.5 degrees as an average of all 5 angles.

The first left half represents conservation of linear momentum in the initial and final stage. The right half represents conservation of energy. We set up an equation for momentum when the ball inelastically collides with the nylon block. We have two unknowns, the initial velocity (what we're looking for) and final velocity. In order to find final velocity, we go to our conservation of energy and create and expression for "final velocity(Vf). The final velocity of the momentum expression is equal to the initial velocity of kinetic energy; therefore, we can connect both equations together.


Since we have an equation to solve for initial velocity(firing speed), we find our values and plug them in.

We measured the length of the string of 0.205 m.

In order to find our "Hcm", we can calculate by H= L-Lcos(angle).

H = 0.205m - (0.205m)(cos20.5 degrees)= 0.013 m




When we plugged in our values into the equation that we created, we get a firing speed of 5.63 m/s.

Next, began the verification process. We sticked a piece of carbon paper onto another piece of paper on the ground, which is where we expected the ball to land.

First, we measured the height from the ground to the center of the ball. As a result, the height is 0.976 m, and the height from the ground to the tip of the table is 0.895 m.

Next, we need to find the distance from the table to the spot where the ball landed on the carbon paper. In order to find the origin in where to start measuring, we used a string in order to get the best starting point to measure. 

Then, we used a two meter stick and an additional ruler to measure where the ball landed. We launched the ball three times and took the average distance that the ball lands. As a result, the distance from the table to the land is 2.144 m

Now, with the following data, we can use kinematics to solve for our velocity. In this case, we used two kinematics formulas to solve for this. (work shown below).


In comparison to the two velocities: 5.63 m/s and 4.8 m/s, the velocities are not the same but are close.

Conclusion: Overall, the types of errors that could occur within this lab is possibly air resistance when the ball is in mid air. Also, there could be friction within the nylon block that results us with different velocities when we performed the verification process. This lab allowed us to combined the conservation of linear momentum and conservation of energy to help us find our initial velocity/launch speed of the ball.

Kevin Tran
Lab partners: Kevin Nguyen, Jose Rodriguez
29 April, 2017

Collisions in two dimensions

Purpose: To determine if momentum and energy are conserved by looking at a two dimensional collision.

Introduction: We performed two different experiments: collisions in two dimensional with two ball of the same weight and different weights. We performed these experiments on a leveled glass table. We set one ball in the middle of the leveled glass table at rest and we collided it with another ball with some initial velocity.

Below is the set up we used in order to perform this experiment. We have a level glassed table with an apparatus to record the collision. We used our phones as the recorder. We made our phone capture 60 frames per second to get a more accurate data.

Below is a picture of the results of the collisions between two balls.

This is the process after we video capture the collisions of two balls in 60 frames per second.

First, we started off with the collisions of two balls with a similar mass. To be exact, we used two marbles with a mass of 0.004 kg

Below is the results of both marbles in a two dimensional graph. In order to get this graph, students need to

1. Options-> Movie Options

2. Override frame rate to 60 to 120fps(frames per second))

3. Advance the movie 2 or 4 frames after adding a new point.

This graph is a position/time graph. In the y-axis, we have two x's and two y's since we have the positioning of both marbles.

For the video, we used two different colors for each ball in order to determine its position in a certain time frame, which will result in the creation of our position/time graph.

As a result from this graph, we got the following velocities from the two balls from the slopes.

Intial:

Vy= 418.5 cm/s

Final:

Vx= -138 cm/s

Vy= 284 cm/s

Vx2= 136.2 cm/s

VY2= 108.5 cm/s

With the velocities, we can determine if momentum and energy are conserved by using the formula Pintial= Pfinal. 

As a result, both the initial and the final momentum are relatively the same. So, momentum and energy are conserved.

Next, we performed the 2nd experiment when two balls of different masses result in a collision. To be exact, we will be using a marble of 4g and a metal ball of 30g for this case. The marble will be placed at rest in the middle of the leveled glass table while we apply some velocity to the metal ball in the direction of the marble ball. 

The graph is a position/time and it is set up similar to the 1st experiment. We have time(x-axis) and the positions of both balls in the x and y direction in the y-axis.

As a result from this graph, we got the following velocities from the two balls from the slopes.

Intial: 

Vy= 288.6 cm/s

Final:

Vx= -35.54 cm/s
Vy= 305.7 cm/s
Vx2= 140.1 cm/s
Vy2= 402.9 cm/s

With the velocities, we can determine if momentum and energy are conserved by using the formula Pintial= Pfinal. 

As a result, the initial and final momentum in the x direction are not the same; therefore momentum and energy is not conserved, but the initial and final momentum in the y direction are relatively the same.

Conclusion: We were able to determine if momentum and energy is conserved by performing this lab. We realized that momentum and energy is conserved when we collided two balls with a similar mass in comparison to the collision with two different masses.