PHYS4A/ Fall 2014; Periods of Oscillations

Purpose: The goal of the experiment was to calculate the moments of inertia for a triangle and a semi circle. These moments were then used to calculate a theoretical period of oscillation for both objects.

For this experiment, we had to cut a triangle and a semicircle out of a foam and calculate theoretical periods of oscillation under small angles and then relate that to experimental values found with loggerpro. Unfortunately, the triangle data was lost ( I really have no idea where it went) and I only have the semi circle data.

To start the prep for the experiment, we had to find the moment of inertia of the semicircle at the center of mass using calculus. By now, we have done these kinds of calculations many times and you should be able to follow the math below.

Once the moment of inertia at the center of mass was found, we then were able to once again use Newton's 2nd Law to find an angular acceleration of the object. This value was then related to the formula

(angular acceleration) = -(angular velocity)^2 * (displacement)

(alpha) = -(omega^2) * (theta)

Since all we had to look for was the constant before the displacement (theta) and relate that to the equation above, we could figure out an angular velocity value. This value was then used to calculate a period using the equation

(Period) = 2pi/angular velocity

T=2pi/(omega)

 From the calculations above, we were able to find a theoretical period value of 0.889s. We then went over to the setup and use loggerpro with a photogate that was able to record the period of the oscillations. From loggerpro, we got an experimental period of 0.640s. When compared to our theoretical value, we had a 7.72% error. This could be attributed to the pivot not being exactly center or the width of the hanging tape breaking the photogate being too large and recording shorter periods due to a sooner break in the light.

And this brings an end to the PHYS4A Fall 2014 Semester. Not too sure if Prof. Wolf did this on purpose but the final experiment was similar to the very first experiment we did with oscillations in a photogate.  The way we started was the way we ended but more knowledgeable about physics and how the world around us works.

PHYS4A/ Fall 2014; Mass-Spring Oscillations

Purpose: The purpose of this lab was to calculate and observe the changes in period of springs with different spring constants and as hanging mass changed for a specific spring.

For this experiment, the room was split into two groups each with a set of springs in order to lessen the amount of time to do the experiment. Each subgroup was given a spring of which we needed to calculate the spring constant. Using a motion sensor and hanging mass on the spring, we were able to calculate our spring constant by comparing the unstretched length of the spring and the stretched length as the hanging mass was added. We calculated a spring constant of 2.330 for our spring and shared that with the class in the data pool.

Spring Oscillation Set-Up

While the other groups were figuring out their spring constants, we then used several different masses on our spring and measured the period of oscillations. Once the data was recorded, we were able to create the graphs in the form below to look for the relations in the periods of the springs.

Graphs we were asked to analyze

Our Graphs of Period vs hanging mass and Period vs spring constant

 Once all the groups were able to get their constants and periods with a set mass, we were able to create a period vs spring constant graph with data from the class. After that graph was created, we were able to compare that with an experimental period which we calculated in loggerpro and compared that to what the theoretical period turned out to be. As you can see, the two period graphs were fairly close meaning error was not that large. Once this graph was analyzed, we then graphed the multiple hanging masses on our spring and made a similar analysis.

Based on the data, we were able to see that springs with a higher spring constant had a shorter period as the force of the spring resisted stretch. We also noticed that if a heavier mass was put on a spring, the period also got longer. These observations made sense both logically and mathematically using the Period formulas.

PHYS4A/ Fall 2014; Conservation of Linear and Angular Momentum

Purpose: Prof. Wolf does an experiment to show that linear and rotational momentum are both conserved.

In this experiment, Prof Wolf performs an experiment while the rest of us observed and made calculations. For the first part of the experiment, a ball rolls down a ramp and we are asked to calculate the rolling velocity of the ball and the launch velocity of the ball under constant acceleration and through slipping. Calculations below show that with the static friction causing the ball to roll, launch velocity was slightly higher.

Velocity calculations

Angular acceleration was measured by Wolf as we were doing calculations

Once the velocity was calculated, the moment of inertia had to be calculated using the same method as the previous lab with moment of inertia. This was needed so that conservation of momentum could be used in the next part as it would be an inelastic collision and energy would not be conserved. Since Prof. Wolf had already attached a hanging mass to the disks and run the experiment to find angular acceleration, all we had to do was use that and Newton's second law to find the moment of inertia like below.

Moment of inertia of the disks calculations

Prof. Wolf and the lab set-up

Prof. Wolf then ran the experiment to make better use of time as we waited for groups to get their values. The ball was released from the top of the ramp and the ball was launched into a metal trap on the spinning disk which then started to spin due to rotational momentum being transferred into the disk.

Calculations for omega

 Once the experiment was over, using conservation of momentum, we were able to calculate a theoretical omega value once the collision occurs. This omega value was then compared to the graph below where the position in radii was recorded vs time. The slope of this graph was the angular velocity of the spinning disk and the value of 1.775 rad/s was compared to our calculated value of 1.95 rad/s.

The values were fairly close and shows that momentum was conserved throughout the experiment since we were able to use the conservation to calculate what we needed for the final goal.

PHYS4A/ Fall 2014; Angular Momentum

Purpose: Using angular momentum and the concept of conservation of momentum, we will predict how high a ruler will swing after an inelastic collision with a piece of putty.

For this experiment, we split the problem into two parts in order to show that momentum is conserved in a rotational collision. To do this, we had set up a ruler and putty scenario where the ruler would start at an angle and be released from rest so that it would hit a piece of putty resulting in an inelastic collision. The goal would then be to calculate the momentum and show that the momentum was conserved throughout the process.

Ruler and putty set-up

The above set-up was created with a camera set across from the ruler centered on the location that the putty ended up at the peak of the rotation. Once we were set-up, the ruler was lifted and released from rest to hit the putty and pick it up. The camera was used to record a video of the motion and loggerpro was used to analyze the data. 

LoggerPro's view of the collision and plotting the points frame by frame of the path of the putty.

When analyzing the video, we had to put a point on the path of the putty, frame by frame. Once all of the points were plotted, we were able to see a fairly smooth curve along a circle. Once the video was finished analyzing, we calculated a theoretical final height for the putty by splitting the experiment into two intervals. For the first interval, we used conservation of energy starting at the initial point at rest to just before the ruler hit the putty.

Total Energy = Potential Energy = Rotational Kinetic Energy

mgh = (1/2)Iw^2

By using conservation of energy we were able to calculate the angular velocity just before the ruler hits the putty. This angular velocity was then used to calculate for the angular velocity just after the collision using conservation of rotational momentum. Initially, the rule was moving at some angular velocity. Then, once it makes contact, its mass changes and it has a new angular velocity. Since the collision was inelastic, energy was not conserved but momentum was. This new omega was then used in the next part of the calculations to finally come up with final height.

All the messy work in our calculations (which Prof. Wolf called beautiful)

Final part of the calculations where we find the height.

The last part to find the height was by using conservation of energy once again. Using the angular velocity of the moment right after collision as the initial speed, we were able to compare that to when the ruler and putty stopped at the end of the path. By using conservation of energy again, this time starting with rotational kinetic energy and ending with gravitational potential, we were able to calculate the height of the putty to be 0.303m.

Knowing this height, we went back to loggerpro and set an axis at the origin of the collision, then using the analysis data it was able to give us a final height. Since both the calculations and loggerpro were both using the same axis, the values could be directly compared to each other. From loggerpro, we got a value of 0.2325m where the theoretical was 0.303m. The experimental value being lower does make sense due to frictional forces such as air drag or friction at the pivot. Therefore, we were able to prove that momentum is conserved in a rotational collision.

PHYS4A/ Fall 2014; Moment of Inertia Pt. 2 (Triangle)

Purpose: To find the moment of inertia of a solid triangle that is revolving about its center on a spinning disk apparatus.

For this experiment, we will be using the same apparatus that we used in the angular acceleration experiment done, about two labs ago. First, we calculated the theoretical moment of inertia for a triangle using calculus.

Deriving the moment of inertia for a triangle about its center.

To find the moment of inertia about its center, we first had to find the moment of inertia about the end of the triangle. The parallel axis theorem was then used to move the axis of rotation to the middle of the triangle. Once the moment of inertia was found at the center of mass, the experiment was done on the air ring to obtain values that can be compared to the theoretcial value.

The triangle was then mounted on the spinning air disk and the same process used to find the angular acceleration was used. However, this time, moment of inertia of the disk was found. These moments were recorded on the board seen above and compared to the theoretical value also found above. The experiment was done twice with the long side up and the long side down to show that the moment of inertia at the center of mass should not be different. This was also proven as the moment of inertia was very close to each other.

The same method as the angular acceleration lab was used but this time, to calculate the experimental moment of inertia

The moment of inertia when compared to the theoretical was very off due to it being a magnitude off. This may be due to the way the measurements were taken as the expression for the theoretical was correct.

PHYS4A/ Fall 2014; Moment of Inertia Pt. 1

Purpose: This experiment was done in order to find the moment of inertia of a spinning disk and to use that info in order to predict how long it would take for the cart to reach the bottom of the track.

For this experiment, a metal wheel was set up on a chair. A cart on rollers was set at an angle on a frictionless track and laid against the chair. For the experiment, we needed to calculate the moment of inertia for the spinning disk apparatus. This required the moment of inertia for not only the disk, but the two ends as well.

Calculations for the moment of inertia of the disk, two ends, and dimensions of the apparatus.

Once we found the moment of inertia, we were able to use Newton's Second Law to calculate the to find the translational acceleration of the disk.

Once the moment of inertia was calculated, acceleration was calculated

Once translational acceleration was calculated, kinematics were  used to determine the time that it
would take the cart to reach the bottom of the ramp which we calculated to be 8.27s. The experiment was then actually done in order to measure the time it took to go down the ramp by timing it with a stopwatch. 

Kim and Ivan timing the fall of the cart

It was recorded that it took 10 seconds to reach the bottom. This value was compared to the theoretical value of 8.27 s with a percent error of   17.3%. This could be due to the calculations not including frictional torque which would have given us a larger time needed. Thus the calculations do conceptually match the experimental data.

PHYS4A/ Fall 2014; Angular Acceleration

Purpose: This experiment was done to calculate the angular acceleration of disks of different masses and see how it affects the angular acceleration of an object.

For this experiment,  a hanging mass is attached to a pulley and a rotating disk. 

Experimental Set-Up

The bottom disk is a solid steel disk which has air flowing over its surface. The top disk is either an aluminum disk or a steel disk. The two disks can be made to either spin independently or together by allowing air to flow between them or directing the air through the holes. By using these techniques and understanding that torque in this case is equal to the radius multiplied by the force (tension).

Torque = r* T 

The set-up with the hanging mass

By changing different aspects of the experiment, we are able to see whether acceleration increases or decreases. Running the experiment with different parameters and recording all of the info, it is apparent that as the hanging mass increases, the angular acceleration increases. When the diameter of the torque pulley changes angular acceleration increases proportionally. The same is seen if the mass of the disks increase.

The angular acceleration  graphs were found using logger pro and taking linear fits of the angular velocity over time graphs like the one below. When looking at the acceleration, due to the orientation of the set-up, negative acceleration was the acceleration as the hanging mass moved up and the positive acceleration is when the hanging mass moves down.

Angular velocity vs Time for Trial 1

In the above graph, velocity is either increasing or decreasing at a steady rate meaning that angular acceleration is constant. This was done for each graph with the acceleration as the hanging mass went up, went down, and the two accelerations averaged to fill in the table above. The same thing was done for all six runs with similar graphs. The average acceleration is used because by taking the average, you are removing systematic error due to friction, making the results more accurate. The observations taken earlier support the equation given where

Torque = (radius)*(Tension) = (Moment of Inertia)*(angular acceleration)

or 

Tr = r*T = I(alpha)

This matches with the above observations where the radius of the of the pulley either got bigger or smaller and when the hanging mass changed which changed the tension in the string. The mass of the disks were changed to get different I's or moments of inertia. As masses increased, the moment also gets larger about the center of mass. Dividing torque by a larger I gives a smaller angular acceleration which was observed. This showed our experiment supports the theory and the experiment was a success.

PHYS4A/ Fall 2014; Collisions in Two-Dimensions

Purpose: To create 2-D elastic collisions between two metal balls. Conservation of energy is used to determine the loss from rolling friction.

Two steel balls sitting on top of a glass surface

Setup with the glass surface and a camera above looking down on the glass. 

Paige setting up the balls for collision and making sure they are in view. 

In this experiment, we are making more collisions to practice how to calculate different things using real world examples . In this case, we are creating 2-D collisions between two metal balls and calculating the changes in momentum after collision. The first collision uses an aluminum ball which collides with a steel ball of more mass. The second collision uses two steel balls which and momentum was again calculated. To set up the collision, one ball was set in middle of the surface and the other ball was set with an initial velocity towards the first ball. The camera connected to logger pro was able to record the paths of the ball using the video analysis software. This analysis was used to create postition vs time graphs. The linear fit of each graph was performed with the slope being equal to the velocity of the ball.

Trial 1 with aluminum ball and steel ball

Trial 2 with two steel balls

After obtaining all the velocity values, energy was calculated for all of the runs. 

Energy after first collision 

Energy after second collision

Since these are not perfectly elastic collision, you can see that energy is not conserved. However, momentum is conserved.

PHYS4A/ Fall 2014; Impulse Momentum Energy

 Purpose: For this experiment, we are trying to determine an object's impulse and show that impulse is equal to momentum.

Based on the impulse-momentum theorem, the impulse of an object is equal to the change in momentum of the same object. Momentum is the the idea that it takes a force in order to stop a moving object and is similar to the idea of inertia which is a resistance to change. When an object has a large momentum, it is moving very fast or has a very large mass meaning that it would take a great force in order to slow it down to a stop.

Momentum = p = m*v

Impulse is the quantity of an applied force over a period of time. This can be derived from Newton's Second Law where....

F = m*a = m*v/t

Which becomes

F*t = m*v

Impulse = J = F*t = m*v = Momentum

Based on the derivation above, the impulse is equal to the change in momentum of an object which is what we will be trying to prove in the experiment below. But first, meet Bob, the clay model who will be assisting us today in the experiment. 

Bob and his dog Spot. No animals were harmed in the execution of this experiment.

For the experiment, we were creating different types of collisions to measure the changes in momentum. This was done by creating inelastic and elastic collisions between two carts on a frictionless track. These are collisions where the objects either stick together or bounce off each other after the collision. In order to simulate an inelastic collision, two carts were gathered with one set up on its side with a plastic bumper sticking out. A secondary cart (blue in the picture) was also set on the track with a force sensor which had a rubber stopper at the end in order to measure force from the impact and to protect the equipment from any unnecessary damage. Two runs were done, first with no mass and then with a 500g mass added to it for the second run.

Cart prepared for second run with 500g mass added on top.

Force vs Time for Trial 1

Velocity vs Time for Trial 1

Once the inelastic collisions were done, Logger pro was used to analyze the data. When looking at the force vs time graph, it is apparent that there was suddenly a force within a small amount of time. The integral of this curve can be taken and the resulting value is impulse. This value was then compared to the velocity graph in order to see if impulse is equal to the change in momentum. As seen in the velocity vs time graph, velocity suddenly drops at the same time that the impulse occurred. This change signifies the impact that occurred and is the relation used to prove the theory mentioned earlier. Since the velocity dropped, that means momentum also dropped which is what also occurred to the impulse.

Set-up for the inelastic collision.

For the next part, an inelastic collision was set up by replacing the rubber stopper on the force sensor with a metal nail. A wooden post was set up at one end of the track and Bob was placed on the post acting as a clay ball which would create an inelastic collision. The idea for this part of the experiment is that the cart would stick to the clay to drop the velocity to 0. Energy would no longer be conserved in this situation, however momentum still would be, The same method of data collection and analysis was used to see the relation in an inelastic collision.

Bob being stuck by the nail. His eyes were covered so he wasn't scared.

Force vs Time for Trial 3(Inelastic)

Velocity vs Time for Trial 3 (Inelastic)

As you can see in the graphs, just like before the force vs time graph and the velocity vs time graph both followed a common trend of decrease at the same time. This proves that they have some correlation and due to the the nature of the experiment, there is some error so the graphs are not perfect. However, this shows that in all cases, impulse is directly proportional to the change in momentum of an object.

PHYS4A/ Fall 2014; Unknown Magnet Energy

Purpose: For this experiment we are using a magnet in conjunction with a slider on an air track to determine an equation for the magnetic potential energy.

To set up the experiment, we used an air track attached to a reverse vacuum thus creating an airflow through many small holes throughout the track. By having air flow, we were able to set a metal slider on top so that it could move along the track without friction. By moving the slider towards a magnet on one end of the air track, we were able to see a repelling force when the slider and magnet were close enough. This was due to a magnet also set on the slider of an opposite polarity. In order to get the most accurate results, the track was made to be level so that there was no sliding due to gravity. And the motion sensor was set up on the end opposite the magnet to measure displacement of the slider.

Running the experiment with a small force on the slider.

Elevating the track using wooden blocks and books

Once the track was level, we measured the initial height of the track in order to establish an origin. Once this height was recorded, the track was elevated to a small angle so that the slider would now move due to gravity. Using Newton's Second Law, we were able to create a force equation of mgsin(theta)=ma. By creating a force column and measuring the distance between the two magnets at the end, we were able to plot a force vs separation graph. By taking a power fit of this graph, an equation y=Ar^B is used. This equation was equivalent to a magnetic potential energy function.

The slider at the end of the track in front of the magnet.

 Using this equation, we could calculate the magnetic potential energy. By using this to create a calculated column for energy done, we could create several other columns for energy. By calculating total energy before the slider moves, we were able to equate that to final energy after each run. Kinetic energy was (1/2)mv^2 which could be calculated from data from loggerpro. Using the inital potential energy and kinetic energy which was equal to total energy of the system and equating that to final magnetic potential energy, we were able to see if we had calculated the correct potential energy and show that energy is conserved. The graphs shown do have some fluctuation but that can be due to unaccounted friction that may have occurred. Total energy was around 0.012J.

Bottom graph showing the measured energy levels according to the formulas we input. Top graph shows a total energy which is mostly constant.

PHYS4A/ Fall 2014; Conservation of Energy

Purpose: To experimentally show conservation of energy by attaching a mass to a metal spring and force sensor and allowing it to oscillate over a motion sensor. The motion sensor then measures the changes in distance to calculate kinetic, potential, and total energy of the system.

Experiment: To start the experiment, we attached a metal spring to a force sensor that was pointing down and zeroed. We calculated the difference between the spring length stretched and at rest. By knowing the total change in distance and the force exerted by the spring, the spring was calculated to have a spring constant of 20.86.

Experimental set-up of spring on force sensor as the mass is oscillating over the motion sensor.

Graph of force vs position where the slope of the graph is the equivalent to the spring constant. Due to the spring constant being equal to Force/ Displacement. 

Once the spring constant was calculated, the mass was set to oscillate over the motion sensor in a vertical direction. By recording the change in distance, energy for the spring and mass could be calculated using various equations. 

Kinetic Energy (Mass) = (1/2)mv^2

Potential Energy (Mass) = mgh

Elastic Potential Energy (Spring) = (1/2)kx^2

Elastic Kinetic Energy (Spring) = (1/2)mv^2

Gravitational Potential Energy (Spring) = mgh

Total Energy = Sum of Energies calculated above

By inputting these formulas into loggerpro and making newly calculated columns, we were able to get several energy curves for analysis.

Graphs of the different types of energy for the calculated columns.

As seen above, the graphs oscillate according to they type of energy it corresponds with. Based on conservation of energy and the potential energy of the mass and the elastic potential energy of the spring should oscillate with relation to each other. By knowing that the total energy should be constant throughout the system, if potential energy of the mass goes up, elastic potential should go down. This is apparent in the graphs above as you see the graphs oscillating and canceling out any changes that are done to total energy. Overall, total energy is constant as seen in the first graph with very little oscillation change thus supporting the theory of conservation of energy. 

PHYS4A/ Fall 2014; Work-Energy Theorem

Purpose: We looked at the work-energy theorem which equates the area under a kinetic energy graph to total work being done.

To set up the experiment, a cart was attached to a metal spring and placed on a metal track. The cart was positioned to be at rest with the spring at a neutral length, where it is at rest and neither stretched or unstretched. This was the zero position that was used to calibrate a motion sensor attached to the end of the track. Once calibrated and setting a positive axis on loggerpro, the cart was pulled back a small length to measure a positive distance. The cart was then released and loggerpro was used to measure the force vs time and plot it in a graph.

Cart and mass at rest while spring is unstretched

Cart and mass after moved a distance x so that spring is stretched and containing potential energy.

Once the data was collected, a new calculated column was created for kinetic energy using the formula K=(1/2)mv^2. Kinetic force vs time and kinetic energy vs time were both graphed. The integral of kinetic force vs time was equivalent to the total kinetic energy over the same interval of time.

As seen in the figure above, the total area under the graph is the integral of kinetic force. The integral is equal to the calculated kinetic energy for the region.

The three points are three different intervals that correlate with kinetic energy intervals with a % error of less than 1%.