4 May 2015. Angular Acceleration.

Purpose

Part 1
to observe the component of angular acceleration by applying different radius, torque, and mass.

Part2
Using the data recorded from part 1, we are going to calculate the moment of inertia and compare it to the true value.

Apparatus:

Angular acceleration apparatus. This apparatus is built of pair of polished disks with a tiny hole that will accept air from the machine, to remove friction and allowing the disks to turn with minimal force. We will use a light hanging mass with a string tied around a torque pulley connected to the disks to provide torque, then measure the angular acceleration experimentally.

Base Theory

• We know that the the equation for torque is: 
• τ = Fd = Iα

• Using the apparatus described, we are going to observe whether this equation really applies to the real life. Analyzing the apparatus used, we can modify the equation and say that the torque F times d is equal to the Tension times radius.

Procedure and Data analysis

for this lab blog, the procedure and data analysis will be described individually under one part section. Since there are two parts of this experiment.


Part 1 (observing the equation of the torque)

Procedure

1. Set up the apparatus so that it can measure the angle as the picture shown.

2. There is 200 marks on the top disk, and this is where the sensor rotary motion come in. We need to set up 200 counts per rotation for the rotation, or else the apparatus going to read 360 counts per rotation as the default set up.
3. To make sure that it gives reasonable data, we also put a motion sensor below the hanging mass and record the data.

Seems reasonable....

4. There are several different aspect that are going to be tested that will be explained in Data Analysis section.
5. To play with the top disk or the combination of the bottom and the top, we need to play with the air hose on the side.
6. We are going to record the data six times with changing properties of angular acc.

Error fix

Our group forgot to set the 200 counts per rotation before we took the data. But, no worries, we can modify the data that we have so that it reads 200 counts per rotation.

What the sensor reads is 360 counts per rotation, so it actually reads: (400/360) = 1.11111 slower than it should be. But, remember that this is in degrees. To convert it to radians we times it to 2pi and get 6.981317 slower. That means we need to times the radians to this number to get true value. Since we can't change the value of the radians, we can change the velocity (which is what we concern about when doing the analysis), which is derived from the radians data and set up the column as follow:

Done! This will fix everything.


Data Analysis

• Now, we have data of angular position, angular velocity, and angular acceleration. The graph of angular acceleration vs time is useless due to the poor timing resolution of the sensors. So to get the angular acceleration we need to plot velocity vs time, and the slope is our answer! (Recall the kinematic equation ωt = ωo + αt)
• But,  we realize that there is some frictional torque happening here (even though we definitely minimize friction using air) as we plot the velocity vs time, one of the slope is higher when the mass is going down, meaning the acceleration is greater, and the acceleration is slower when the hanging mass is going up. (see the graphs)

Expt 1: hanging mass, small pulley, top steel

Expt 2: 2 x hanging mass, small pulley, top steel

Expt 3: 3 x hanging mass, small pulley, top steel

Expt 4: hanging mass, large pulley, top steel

Expt 5: hanging mass, large pulley, top aluminium

Expt 6: hanging mass, large pulley, top steel+bottom aluminium

• Therefore, for the angular acceleration, we need to find the average of these data.
• This the list of experiments that we did:

• Comparing expt 1,2,3 (the tension changing represented the weight of hanging mass)

• Comparing expt 1,4 (the radius changing represented by the size of torque pulley)

• Comparing expt 5,6 (the mass (inertia) changing represented by the type of disk or rotating mass)

Part 2 (comparing the moment of inertia)

Procedure

1. We are going use the data from part 1.
2. We are going to measure the radius and the weight needed to find the inertia.

Data Analysis

• Now, we have data to calculate the experimental value of inertia as follow:

• And we derive the equation to calculate the moment of inertia (handout):

• Now, comparing it to the real value...

• The percent error is

Conclusions

•  the equation τ = Fd = Iα   turns out to be very reasonable.
• We have a very little error in determining the moment of inertia (under 5%), but when we are determining the moment of inertia of the two disks welded together (aluminium plus friction) we have 28.5 % error which is huge. This is probably due to friction in the axis that is not calculated. If we got 1.6 as our angular acceleration, then we got our inertia value, but we got 1.1. So, probably friction is the main reason of this error.

29 April 2015. Ballistic Pendulum.

Purpose
to calculate the initial velocity of the metal ball shot from a gun into a block using the conservation of momentum and conservation of energy.

Apparatus:

A ballistic pendulum model. Picture shown. This model can shoot a bullet from the spring-gun into the block. The arc on the right side will measure the angle in which the block will rise up to.

Procedure

1. We shot the ball from the gun.

2. We measure the angle that is created by the block.

17.2 degrees

3. We also measure the length of the string.

21.7 cm

4. We record all data.

Data Analysis:

• This is the data recorded.

• First, to find the initial velocity formula, we derive it using the conservation theory of momentum and energy.

• Next, we input the data that we have, and we got our initial velocity!

• Now, we are going to calculate the propagated uncertainties of this calculation.

• The initial velocity of the metal ball from the spring-loaded gun turned out to be 5.06 +/- 0.502 m/s. 

Conclusions

• Using the conservation of energy and momentum we were able to calculate the speed of the spring-loaded gun on the metal ball.
• Despite the uncertainty, and other possible factors such as force of the metal rod and friction of the metal ball inside the gun, we found a speed that made sense and found reasonable method of calculating the bullet using the theory of conservation of momentum and kinetic energy

22 April 2015. Collisions in two dimensions.

Purpose
to look at the theory of conservation of momentum and conservation of energy.

Apparatus:

1. A glass table. This will be the base of the collision since it generates less friction that any other type of material that we have.
2. Steel balls and aluminium balls. As the object of collision.
3. Video web-cam, logger pro, and laptop. To capture and analysis the data.

Procedure

1. We set up the glass-table and web cam as the picture shown.

2. We record the weight of each ball used (aluminium and steel).

First and second steel ball in first experiment

steel ball in second experiment

aluminium ball in second experiment

3. Then we record the collision using the laptop and the webcam.
4. Do steel ball vs steel ball first, then steel ball vs aluminum ball.
5. Analysis the data and calculate whether momentum or the kinetic energy is conserved.

Data Analysis:

Steel vs steel

• After we capture the movement of the collision, we analyze it by creating points series in every movement of the two ball, set up the x axis and the y axis, and measuring any necessary apparatus that is going to be used as comparison in the video.

• For example, we also measure the length of the glass table used in this experiment for comparison.

65 meters length

• Then, the dots that we give will give us velocity and position data in x and y direction among other things. For example, below is the position of the ball in x and y direction.

The (0) is the second steel ball that is at rest while waiting for the first ball (5) to collide with it

• To see whether the momentum is conserved, we evaluate it in x and y axis.
• We make new calculated momentum in x-axis and y-axis column for both the first and the second ball using the weight that we already recorded. If the total momentum in x axis (the momentum X first ball plus the momentum x second ball) is constant then momentum in x direction is conserved, so as if the total momentum in y axis is constant then the momentum in that direction is also conserved.
• Here is some the examples of calculated momentum column that we made.

Momentum of the first steel ball in the x-axis

Momentum of the second steel ball in the y-axis

Total momentum of the first and second ball in x direction

• Then, this is the graph of the total momentum of the first and second ball in both x and y directions.

• As you can see, the values are more or less almost constant. That means the momentum is conserved in both x and y direction; momentum is conserved in this collision.
• Next, we need to check whether the kinetic energy is also conserved. Therefore, we created new calculated column of kinetic energy of the first ball and the second ball. Then the TOTAL kinetic energy of both balls.

Kinetic Energy of the first steel ball

Kinetic energy of the second steel ball

Total Kinetic Energy

• This is the graph of the TOTAL kinetic energy with the first and the second KE.

the value of the TOTAL KE is almost constant.

• As we can see, the KE is not as constant as the total momentum. This is because this collision is not really elastic. In non-elastic collision, kinetic energy is not conserved because the energy is also transformed to sound energy and heat. That is why the Kinetic energy after collision is a little bit lower than before collision.

Steel vs aluminium

• We did the same exact procedure as we did in the first experiment. The aluminium ball is the ball waiting for the collision from the steel ball.
• When we do the new calculated column, we need to change the mass of the ball. Here is some example of the new calculated column in the second experiment.

Momentum of the steel ball in y direction

Momentum of the aluminium ball in x direction

Total momentum of both balls in y direction

• Then, this is the graph of the total momentum of the first steel ball and second aluminium ball in both x and y directions.

• As you can see, the values are more or less almost constant. That means the momentum is conserved in both x and y direction; momentum is conserved in this collision.
• Next, we also need to check whether the kinetic energy is also conserved. Therefore, we created new calculated column of kinetic energy of the first ball and the second ball. Then the TOTAL kinetic energy of both balls.

Kinetic Energy of the first ball

Kinetic Energy of the second ball

TOTAL kinetic Energy

• This is the graph of the TOTAL kinetic energy with the first and the second KE.

• As we can see, the KE is not as constant as the total momentum. This is because this collision is not really elastic. In non-elastic collision, kinetic energy is not conserved because the energy is also transformed to sound energy and heat. That is why the Kinetic energy after collision is a little bit lower than before collision.

Conclusions

• Momentum is always conserved in collisions whether the mass is lighter or heavier, the speed is slower or faster as long as there is no external forces acting, such as friction. In this case, friction is there in a very small amount, that is why the momentum is not really conserved (not really constant).
• Kinetic energy is also conserved if only the collision is elastic. In these experiments, collision are only nearly elastic. The kinetic energy is also transformed into heat and sound.
• The errors in this experiment comes from;
• The webcam is a fish-eye like lens; meaning that it is more concave in the middle to capture wider angle. This will messed up the 65 m length of glass table that we use as comparison in the video.
• When we add point series, we did not really click it in the center point of mass. This will alter the velocity and the position of each ball.
• Friction from the glass table, but we ignore that.