October 17th, 2016: LAB 15: Collisions in 2-D

Michell Kuang
Lab Partners: Charles, Anthony
Date performed: 10/17/16

In this lab, we examined two-dimensional collisions and determined whether the conservation of energy and momentum principles applied. For our experiment, we did a collision with two balls of the same mass, and a collision with two balls of different masses. Our setup included a leveled glass table on which we were able to place two balls stationary. Directly above that table, we used a ring stand and an apparatus to hold our phone so that we could record the collisions in slow motion. These videos will later be transferred over to a computer so that we can analyze data on Logger Pro.

 

Setup of glass table on which collisions were performed 

Before starting the experiment, we took measurements of the length of the table and the masses of the balls that we were using for our collisions. Our table length was 0.653m (this will later be used as a scale on Logger Pro). We had three balls: two steel balls of roughly the same mass (m1=0.067kg, m2=0.0667kg) and a glass ball of different mass m3=0.197kg. With this data, we were able to start the experiment and perform the collisions. For our experiment, we wanted to make sure that after the collision, the balls rolled off at some decent angle from one another. Once we recorded videos for both of those, we transferred them onto a computer and into Logger Pro.

In Logger Pro, after setting up the scale of the table, we clicked on the position of the balls as they moved in the video per 4 frames. This produced a graph of x and y positions of the balls vs. time.

 

Logger Pro view:  Same mass collision, graph of x and y positions vs. time

We then created calculated columns for x and y positions of the center of mass of the system and x and y velocities of the center of mass of the system. Additional graphs of these vs. time were produced.

 

Logger Pro view: Same mass collision, columns of data and video with edits on right

 

 Logger Pro view: Same mass collision, graph of x and y positions of cm of system vs. time

Logger Pro view: Same mass collision, graph of x and y velocities of cm of system vs. time

The same steps were repeated for the different mass collision.

 

Logger Pro view: different mass collision, graph of x and y positions vs. time

 

Logger Pro view: different mass collision, columns of data and video with edits on right

 

Logger Pro view: different mass collision, graph of x and y positions of cm of system vs. time

 

Logger Pro view:  different mass collision, graph of x and y velocities of cm of system vs. time

October 12th, 2016: LAB (): Ballistic Pendulum

Michell Kuang
Lab Partners: Charles, Anthony
Date performed: 10/12/16

Note: This lab was an addition to the original lab manual, which is why there is no number.

In this lab, we try to determine the firing speed (launch speed) of a ball from a spring-loaded gun. To do this, we used an apparatus that allowed us to shoot a ball into a nylon block that is supported by four strings. Once the ball is "absorbed" into the block, it hits a piece of metal that we use as our angle indicator, and that shows the angle at which the ball+block system rise.

Setup of "ballistic pendulum" apparatus

First, we measure the mass of the ball and the block. The ball we used was 0.0073kg +/- 0.0001kg and our block was 0.0806kg +/- 0.0001kg. We also measured the length L from the top where the strings were attached to the center of the nylon block. Our L was 0.225m +/- 0.002m. The "cannon" had three different notches that we could pull it back to fire the ball, and we used the first notch. After recording all this data, we fired the ball and repeated this four times to get an average.

Angles to which the block rose for each run; average angle for all runs

Now that we had this average angle, we used it to calculate our h, which is the height that the ball+block system rose. Using that and all of our measured data from earlier, we were able to determine a firing speed of 5.0568m/s. We calculated the uncertainty for this launch speed as well. Finally, we used this launch speed to find our theoretical (guessed) value of the distance that it would go if we launched the ball horizontally into the air.

 

Calculation of theoretical launch speed and comparison of theoretical distance vs. actual distance

 

Calculations for uncertainty in our launch speed 

Both our guessed and measured values came out to be 2.3m, meaning the actual, experimental launch speed is about the same as our calculated one. This shows that our experiment was a success!

October 10th, 2016: LAB 13: Magnetic PE Lab

Michell Kuang
Lab Partners: Charles, Anthony
Date performed: 10/10/16

In this lab, our goal was to verify that energy is conserved. Our setup consisted of a frictionless glider with a magnet on an air track. On either ends of the air track are magnets to which the cart's magnet opposes (because they're the same polarity). This, along with the air from the track, keeps the glider in motion back and forth across the track. With this setup, we tested and verified that conservation of energy applied to the system.

 

Setup of air track and glider with aluminum reflector attached onto glider

To start, we had to tackle the problem of not having an equation for magnetic potential energy. Since this is so, we had to find an equation for it, and to do that, we had to find the interaction force F(r).

First, we weighed our glider (0.352kg) and leveled the air track. Then, we found our value for r, the separation between the glider, more specifically the aluminum reflector that sits on top of the glider (attached so that the motion sensor can better detect the motion of the glider), and the magnet at the end of the air track. To do this, we used a digital vernier caliper to measure the distance between the magnet at the end of the track and the magnet on the glider. Then, we used a ruler to measure the distance between the motion sensor and the square piece. We then subtracted the first value from the second, and got the value 0.168m. This number, however, is not our r. Our separation r is the position of the glider minus this value (since the glider will be moving positions).

Once we determined how to find our r values, we collected data while tilting the track up at different angles. We used an iPhone app to measure the angle.

 

Data of track at different angles and the corresponding r values

Now using Logger Pro, we plotted a graph of F vs. r, assuming a relationship in the form of a power law: F=Ar^n. On Logger Pro, our A and n variables will be shown as A and B, respectively.

 

Logger Pro view: Graph of F vs. r with curve fit; values of A and n shown in data box

Our A came out to be 0.0004724 and our n was -2.004. Using this, we were able to find our equation for magnetic potential energy by negating and taking the integral of this function:

F=0.0004724r^-2.004

Once we had that equation, we were able to verify the conservation of energy. To do this, we set the motion sensor to record 30 measurements per second. Then, starting at the end of the track, we gave the cart a push to get it moving. Using our function for r that we derived earlier, we created a new calculated column to calculate the separation at various different positions. 

 

Logger Pro view: Graphs of Position vs. Time and Velocity vs. Time

These graphs show the point at which the glider began to go backwards once it was pushed back by the same-polarity magnet. We then created three more calculated columns, one for KE, one for Umag, and one for the total energy, Etotal. We also plotted a graph for all three of those vs. time.

 

Logger Pro view: Graph of KE, Umag, and Etotal vs.Time

As you can see, as potential energy increases, kinetic energy decreases. This makes sense if energy is actually conserved. Looking at the total energy, it is also consistent with the conservation of energy principle.

October 5th, 2016: LAB 12: Conservation of Energy in a Mass-Spring System

Michell Kuang
Lab Partners: Charles, Anthony
Date performed: 10/05/16

In this lab, our goal is to show that the energy in an oscillating mass-spring system is conserved. Our setup includes a vertically-oscillating spring with a mass attached to it. On the floor below the mass and spring is a motion sensor that detects the motion of the mass as it moves up and down.

 

Setup of mass-spring system with motion sensor on floor

Before we could actually start the physical part of the lab, we had to do some calculations to show that the gravitational potential energy (GPE) of the spring is equal to mg(H+y)/2, H being the height from the very top of the system (extended arm from which the spring and mass hang) to right above the motion sensor, and y being the height from the bottom of the mass (when supported with hand so that the spring is NOT stretched) to right above the motion sensor. This calculus was done by the professor on the board, and as a class our job was to listen and understand to the point that we could do those calculations on our own.

The same was repeated for the equation for kinetic energy of the spring. Using the same concept and selecting a little "representative" piece of the spring, the professor showed us that KE is equal to 1/2(1/3mspring)v^2.

After that, we were able to start on the experiment. First, we had to take measurements for the mass of the spring and the mass hanger. For our lab, instead of using a mass+mass hanger, we just used a 1kg hanging mass. The mass of our spring was 0.065kg. Using the motion sensor and Logger Pro, we measured our y and confirmed that value using a meter stick. We measured 1.36m. Then, we released the hanging mass from the support of our hand and let it hang at rest (not oscillating), and measured that height, which came out to be roughly 0.718m. This is our equilibrium position. Finally, to put our system into motion, we pulled the spring slightly (about 10cm) so that it started oscillating. Then we pressed start on Logger Pro to begin collecting data.

 

Logger Pro view: Graphs of position and velocity vs. time; calculated column for stretch of spring

The result was a sinusoidal graph for our position vs. time graph, which was expected from the oscillating system because it moves up and down. For our velocity vs. time graph, we also got a sinusoidal graph because the speed at which the spring falls downward is faster than the speed at which it recoils back up. This is due to the constant weight of the mass pulling the spring downward. Since the system is oscillating, this pattern repeats. We moved on to create three more calculated columns, one for each energy (kinetic, gravitational potential, and elastic potential). For each of these energies, we produced two graphs, one of the energy vs. position and another of the energy vs. velocity. Each of our graphs produced fits correctly with what we should expect.

 

Logger Pro view: Graphs of KE vs. position and KE vs. velocity

 

Logger Pro view: Graphs of GPE vs. position and GPE vs. velocity

 

Logger Pro view: Graphs of EPE vs. position and EPE vs. velocity

For our KE vs. position graph, it is a parabola facing downwards because the kinetic energy reaches its maximum at the equilibrium position, which we measured earlier in the lab to be about 0.718m. The graph shows that the top of the parabola is at about 0.7m, so our data is accurate here. For our KE vs. velocity graph, since kinetic energy is directly related to the velocity squared, when the value of velocity is greater, so is the kinetic energy. When the velocity is 0, kinetic energy is also 0.

For our GPE vs. position graph, the GPE increased as the position increased. This is because the GPE of the spring is directly related to the position (y) of the mass above the ground (in our case, the motion sensor), as we showed earlier in the lab. For our GPE vs. velocity graph, it is a repeating ellipse, which makes sense for the same reason as earlier. Because the GPE is directly related to the position, it is bigger when the value of position is bigger. Although this graph doesn't show position, we know that there are two points at which the velocity is 0: at the top of its oscillation and at the bottom of its oscillation. In other words, although the mass+spring system is constantly in oscillation (motion), it stops briefly when it bounces back up (right before it falls again), and again when it is pulled all the way down (right before it bounces back up again). Looking back at the graph, you can see that at velocity=0, GPE is at its maximum AND its minimum, which means that it was at its highest and lowest position, proving that this graph is correct.

Finally, for our EPE vs. position graph, the EPE decreased as the position increased. This is because the system's energy is initially stored in the spring as EPE when it is at rest (mass hanging and spring stretching down). As the mass+spring move up and position increases, energy is converted into kinetic energy, thus decreasing the elastic potential energy. For our EPE vs. velocity graph, the reason for the ellipse is similar to the last explanation for our GPE vs. velocity graph. Instead of being directly related to position, however, the elastic potential energy is directly related to the stretch of the spring.

Moving on to the last part of the lab, we created a calculated column called Esum, which is the sum of the KE, GPE, and EPE. To finish off, we produced two graphs: one of Esum, KE, GPE, and EPE vs. position, and another of Esum, KE, GPE, and EPE vs. time.

 

Logger Pro view: Graphs of Esum, KE, GPE, and EPE vs. position, and Esum, KE, GPE, and EPE vs. time

October 5th, 2016: LAB 11: Work-KE Theorem Activity

Michell Kuang
Lab Partners: Charles, Anthony
Date performed: 10/05/16

Our goal in this experiment is to understand how work and kinetic energy relate to help explain the work-kinetic energy theorem. To do this, we measured the work done while stretching a spring to a specific distance (about 0.6m) on a track. On the track we have a cart with which we pulled the spring. A white square is attached to the cart magnetically so that the force sensor can more easily detect the motion. In addition, we placed a book under the spring so that we could lay it out flat and make sure that it was unstretched.

Setup of cart and track with which we stretched spring; force sensor on one end

We first had to calibrate the force sensor to 4.9N. Then, we had to zero the force sensor so we could set an initial "x=0" position. With the force sensor connected to a laptop, we collected data on Logger Pro. This produced a graph of force vs. position. To determine the spring constant of our spring, we did a linear fit on the resultant graph, and our spring constant came out to be 29.39 N/m. To find the work done, we used the integration routine function on Logger Pro. Our work done was 2.755 Nm.

Logger Pro view: Graph of force vs. position

For the second part of the experiment, we did the same thing again by pulling the cart along the track, but this time we created a column on Logger Pro to calculate the kinetic energy of the cart. We first had to measure the mass of the cart, which was 0.714kg. Then, using the same spring (so we could use the same spring constant that we calculated in the first part), we again pulled the cart a distance of 0.6m. This time, we made a graph of both force vs. position and kinetic energy vs. position.

Logger Pro view: Graphs of force vs. position and kinetic energy vs. position

Automatically, the kinetic energy vs. position graph should have a data box that tells us the kinetic energy at any point on the graph. Using this, we found the kinetic energy for a few different positions, and then found the corresponding work done at that same position. Once again, we used the integration routine function to determine that. We picked four points, and recorded this data onto an excel sheet.

Excel sheet of work and kinetic energy at certain positions

If done right, the experiment should produce values for work and kinetic energy that are almost the same. Our values are fairly similar. One definite source of error in our lab was that we didn't read the manual carefully, and ended up switching our springs because our first one was difficult to work with. As you can see in my lab's third image, we got a different spring constant doing the experiment the second time around (the slope from the linear fit). Another reason for the discrepancy in our values may be that we didn't calibrate or zero the force sensor correctly. 

All in all, our lab was more or less successful. One can see that work and kinetic energy are definitely related. Since energy cannot be created nor destroyed, the work done on an object converts energy from one kind to another. Initially, all the energy in the spring is elastic potential energy. Then, when we pull the spring, kinetic energy increases (is gained) an amount equal to the work done.

October 3rd, 2016: LAB 9: Centripetal Force with a Motor

Michell Kuang
Lab Partners: Charles, Anthony
Date Performed: 10/03/16

In this experiment, we examined centripetal force by putting an object in circular motion using this apparatus:

Centripetal force apparatus with electric motor

There is a motor attached to the apparatus that puts the object in motion. By turning up the power of the motor, we increased the speed at which the object spun, which also increased the radius and angle. Our goal is to find a relationship between the angle theta and angular velocity.

To do this, we based our model off the equation:

ω^2=(gtanθ)/0.70+1.855sinθ

How we derived this equation is easier to understand if one looks at this diagram of the apparatus: 

Lab manual sheet with added labels and numerical data

As can be seen at the top of that sheet, the length L of the string can be separated into its x- and y- components, Lsinθ and Lcosθ, respectively. Lsinθ is going to be our R2, and Lcosθ is our H-h. Using this information, we can derive the equation.

Derivation of equation

Moving onto the experimental part of the lab, we took data for 6 runs at which the apparatus swung at 6 different speeds. We used a meter stick to measure our values for H and R. Our values for ω were obtained by timing how long it took for the object to make 10 rotations. For our h values, we put a piece of paper on a ring stand and slowly raised it until the object in motion hit it. Then, we moved that ring stand away from the spinning object, and measured the height h with a meter stick.

Data of 6 runs

After collecting all of this data, we put it into excel so we could calculate our theoretical values of ω and compare them with our experimental ones. 

Excel sheet of data

Our theoretical values turned out to be very close to the experimental values, with the % error being no more than 0.09%. Discrepancies in these values could possibly be from inaccurate readings of measurements such as H or from inexact timings of the time it took for the object to make 10 rotations. Although we can't really make our timing of the rotations more accurate, we can change our value of H on excel and see how that affects our data.

Excel sheet of data with modified H

Now, our  % error is no more than 0.06%! Regardless of this change, however, our initial values were accurate enough for us to confidently say that theta is, in fact, related to angular velocity as it is described in our equation.

September 28th, 2016: LAB 8: Centripetal Acceleration vs. Angular Frequency

Michell Kuang

Lab partners: Charles, Anthony

Date performed: 09/28/16

For this lab, a wooden rotating disk was set up in the back of the room for us to observe. To determine a relationship between centripetal acceleration (force) and angular speed, we ran and compared the experiment at varied radii, voltages, and masses. Each run ran for a total of 10 rotations (1 rotation for each time the piece of tape on rim of disk passed through the photogate). The attached force sensor relayed data onto Logger Pro, from which we got our values for force.

Setup of rotating disk with force sensor at center and mass attached

Looking at the equation for centripetal force:

 Fcentripetal = mrω^2

You can see that centripetal force is directly related to the mass, radius, and angular velocity of the object rotating. This means that if any of the factors on the right increase, then the centripetal force on the left would consequently increase as well. Additionally, if any of those factors decreased, then the force would also decrease.

First, we ran the experiment a couple times with varied radii. The mass was set at 200g and the voltage at 6.1 volts. We started with a radius of 19cm, and increased with random increments all the way up to 54cm. Here is an example of what the data looked like on logger pro:

Logger Pro view: Recorded data with 200g of mass, 6.1 volts, and 28.5cm radius

We then repeated this process with varied voltages, and finally varied masses. In our lab, a mass was attached by a string to the force sensor that was adorned at the center of the rotating disk.

Excel sheet of data

As expected, the force increased with each increase in radius, voltage, and mass. This experiment proved our guess that the force of centripetal acceleration is directly related to the mass, radius, and angular speed of a rotating object.