20-May-2015:Predicting the final height of a clay-stick combination using Conservation of Energy/Conservation of angular momentum theorems

     The purpose of this experiment was to predict how high a clay-stick combination would rise after a collision and to compare actual results with the prediction. This would be achieved by using conservation of energy and conservation of angular momentum relationships.

The setup for the experiment consisted of a meter stick that was pivoted at one end, or close to it.

The meter stick would be released from a horizontal position and swings down. When it reaches the bottom of its swing it collides inelastically with a piece of clay. This was achieved by wrapping the clay with tape and wrapping the bottom of the meter stick with tape.

The clay-stick combination continue to swing with each other to some final position. Below is the overall setup.

A camera was setup at the opposite end of the table to capture video for Logger Pro's video analysis. another meter stick was setup just behind the piece of clay to give Logger Pro a distance to reference. 

The mass of the meter stick and the piece of clay were both measured using a balance. 

     Before the experiment began, the predictions were made. The motion of this experiment was divided up into three parts. Energy was used when the meter stick is released from a horizontal position to the moment right before it collides with the clay. Conservation of angular momentum was used during the inelastic collision with the stick and clay. And finally, Energy was used again when the clay-stick combination move together right after the collision to when they reach their highest point.

Part 1: Energy

     For the first part, the meter stick started with gravitational potential energy and ended with kinetic energy as well as gravitational potential energy. GPE was set to equal zero at the point where the meter stick was horizontal. So, the initial GPE was zero. GPE was calculated from the center of mass which is at the 50 cm mark since we assumed a stick of uniform dimensions. This means the distance from the pivot to its center of mass was 49 cm (represented by "y" in the calculations) since the pivot was located at the 1 cm mark. The parallel axis theorem was used for the inertia of the stick since the pivot is neither at the center of mass nor directly at the end. Angular velocity was solved for so that it may be used for the next part.

Part 2: Angular Momentum

     Now that the angular velocity before the collision is known, conservation of angular momentum could be used. This was used to calculate the angular velocity after the collision. The inertia of the system after the collision is the sum of the inertia of the stick and the clay. The inertia of the clay was treated as a point mass and calculated by mR^2, with R being the distance from the pivot to the end of the meter stick.

Part 3: Energy

     Knowing the angular velocity after the collision allowed for the calculation of final height for the clay-stick combination. The clay and stick both started off with GPE and KE, and ended with GPE. Below were the final calculations needed for the height prediction.

A prediction of 0.3827 m was found after plugging in all numbers.

     Now the experiment was run and the video was analyzed. The origin was set to zero where the piece of clay rest before the collision. The video was run and a data point was placed where the clay was at it highest. This gave an x and y-coordinate, but we were only interested in the y-coordinate. That y-coordinate was the final height we compared with the prediction.

     The predicted height was 0.3827 meters and the experimental height was 0.3795 meters. A percent error of 0.84%. This really good result shows that conservation of energy/conservation of angular momentum is a good model to describe the motion of a swinging mass, and it also tells us that the two theorems can be used together to achieve results. Errors in the experiment could have been due to assumptions of no air resistance, assumptions of uniform dimensions on the meter stick, and error in the video analysis.

13-May-2015: Finding the moment of inertia of a uniform triangle

     The purpose of this experiment was to determine the moment of inertia of a uniform right triangle about its center of mass for two perpendicular orientations of the triangle.
     This experiment compares the theoretical and experimental inertia's of the triangle. First, the theoretical calculations were made. To find the moment of inertia about its center of mass, the parallel axis theorem was used. This made the calculations easier since it is easier to calculate moment of inertia about one end of a triangle and we already know where the center of mass is located on a triangle. Below is the derivation of the moment of inertia of the triangle about its center of mass.

     The setup for the experimental part had a triangle mounted on a disk. The disk "floats" on air making it nearly frictionless. A string it attached to a torque pulley, which is also mounted on the disk. A hanging mass is attached to the opposite end of the string and hung over another pulley. A picture of the setup is below. The hanging mass starts at the top and is released. It moves up and down as the disk and triangle spin.

     To find the experimental value, The inertia of the disk plus the triangle holder needed to be compared to the inertia of the disk plus the triangle. Taking the difference of the two values would give the inertia of the triangle. The equation I=mgr/(alpha)-mr^2 was used for the experimental portion. Alpha, angular acceleration, was found using the average of the up and down values of the slope of the angular velocity graph that was produced using Logger Pro. 

     Below is the graph of angular velocity of when the triangle was not mounted on the disk. In this graph, three slopes in both the up and down motion were used. The average was taken of those up and down slopes, and then the average of those two numbers were taken. 

     The triangle's dimensions were measured using a caliper and its mass was measured with a balance. The dimensions and mass of the disk and the torque pulley were also measured with the same tools. They were needed to calculate the inertia. 

     After the angular acceleration was found, simply plugging in the numbers into the equation I=mgr/(alpha)-mr^2 was the next step. The inertia of the disk was added to the inertia of the disk and holder using the angular acceleration measured in each of their trials. Then, the inertia of the disk was added to the inertia of when the triangle was mounted. The difference of the two final values was taken. 

     Using this experimental approach yielded a value for the inertia of the triangle (when it was upright) to be 2.482*10^-4 kg*m^2. Using the equation for the moment of inertia of a triangle about its center of mass, which was derived earlier, and the dimensions of the triangle found with the calipers, gave a theoretical value of 2.442*10^-4 kg*m^2. This resulted in a percent error of 1.6%. A really good result! The exact same approach was used for a second trial, but this time the triangle was placed sideways with its longer side as the base. That trial resulted in better data with an experimental value of 5.587*10^-4 kg*m^2 and a theoretical value of 5.634*10^-4 kg*m^2. A percent error of 0.83%. 

     The experiment was a success and the derived equation for the triangle's moment of inertia about its center of mass was accurate. The data shows that when the triangle is sideways its inertia value is larger. This means that it is harder to rotate as its base gets longer. That agrees with what our intuition would tell us. Sources of error included assumptions that the spinning disk was acting completely frictionless and errors and uncertainties in using calipers to measure dimensions.

11-May-2015: Moment of inertia and frictional torque

     The purpose of this experiment was to predict the time it would take for a cart to travel some distance down a ramp by applying Newton's 2nd Law and torque/inertia relationships.
     The apparatus used for this experiment is shown below. It was a symmetrical mass with a large center disk and smaller cylinders on each side of it. The entire thing was able to spin about its center.

One end of a string was tied to one of the smaller cylinders and wrapped around it. Connected to the other end of the string was a cart. The cart was placed on a ramp, which rested on a counter making some angle with the horizontal. Below is a picture of the whole setup. The cart was released and the time it took to travel 1 meter was recorded. This was repeated a few times to get a rough number for the time.

The actual experiment was short and sweet but the theoretical calculations took a little more time. So, lets take a look at those because that is what we need to compare the experimental value with.

     First, the moment of inertia of the whole apparatus was needed. Since it was made up of cylinders, the equation (1/2)MR^2 was used. The inertia of the individual parts were added to find the total inertia. In order to find the mass and radius, though, measurements were made with vernier calipers and the mass was derived using a volume relationship. Below were the steps taken. 

     

     The final step was to derive an equation for the time it would take the cart to travel down the ramp. Newton's laws were applied to the cart and torque relationships were applied to the apparatus. Manipulating the equations gave an expression for linear acceleration. Once linear acceleration was found, playing with kinematic equations allowed time to be solved for in terms of distance traveled and acceleration. Below is the derivation.

     The theoretical calculations produced a time of 9.21 seconds for the cart to travel 1 meter down the ramp. The experiment yielded a consistent time of about 9.70 seconds. A percent error of 5.3%. This error could have been caused friction from the cart and ramp or the string not being parallel to the ramp. 

04-May-2015: Effects of mass and diameter variations on the angular acceleration of a spinning disk

     The purpose of this experiment was to determine what factors affect angular acceleration and to find a theoretical and experimental moment of inertia, and compare them, for the object in question.
     The object in question is a disk and pulley system. A disk lays flat with a torque pulley attached on top of it, and a string connected from the torque pulley over another pulley. The other side of the string has a hanging mass connected to it. A diagram is shown below.

The apparatus works by pushing air through it so the disks "float" on the air, making them nearly frictionless, similar to how air-hockey tables work. 

     Before the experiment began, measurements were made with calipers for the mass and diameter of each disk (there were 3, two steel disks and one aluminum), each torque pulley (a small and large one), and the mass of the hanging mass.

     Logger Pro was used to record the spin of the disks. On the side of the disks are 200 marks that a sensor reads and transmits to Logger Pro. This enables Logger Pro to produce graphs of angular position, angular velocity, and angular acceleration vs time as the hanging mass moves up and down. The angular acceleration vs time graph was useless due to the poor timing resolution of the sensors so it was not used. 

     Below is an example of one of the angular velocity vs time graphs used to find angular acceleration. Linear fits were used to find the slope, which was angular acceleration. Note that the positive slope was when the hanging mass was moving down and the negative slope was when it moved up. Each experiment gave similar graphs.

The average angular acceleration of the absolute values of the slopes from above was used. This was because of frictional complications in the system. As the mass moved down, torque from the string sped the disks up while frictional torque from the disks slowed them down. This means that angular acceleration as the mass descends is less than the ideal angular acceleration where there is no friction. Also, as the mass moves up, torque from the string slows the disks down and frictional torque also slows the disks down. This means that angular acceleration as the mass ascends is greater than the ideal angular acceleration where there is no friction. So, to compensate for this, the average angular acceleration was used.

     There were three factors that were changed over the course of the experiment. The hanging mass was varied, the radius of the torque pulley was varied, and the mass of the disks were varied. Below is the raw data table of the masses used and the angular acceleration numbers recorded by Logger Pro. The top of the picture has an explanation of what was varied with each experiment.

     The data in the "average angular acceleration" column tells us a lot. The data shows heavier hanging masses result in faster angular acceleration. In this experiment, angular acceleration went up by 0.6 rad/s^2 for every 25 grams of hanging mass added. The data shows the larger torque pulley gives a faster angular acceleration. And finally, the heavier the disks, the slower the angular acceleration. 

     

27-April-2015: Ballistic Pendulum

     The purpose of this experiment was to find the initial velocity of a ball by using the conservation of energy/momentum theories. This experiment was done by shooting a ball with some initial velocity into a container. The container had a hole in it so the ball would stay inside. The apparatus is shown below. The container is held up with string so it can move side to side.

     To the right of the container are tick marks for degrees. Also next to the container is a light rod that gets pushed out some distance by the ball and container. The rod is stiff enough that it does not fall back down to its original position. Doing this allowed for a reading of the angle the container made with the vertical at its furthest point after the ball collides with it. Below is a picture of the rod after the experiment and the angle tick marks.

     Since it was assumed momentum and energy were conserved, equations for both were produced. The experiment was split into two parts in order to make the equations work. The first part focused on the initial momentum of the ball and the final momentum of both the ball and container. The initial momentum of the container was neglected because it was initially at rest, therefore its velocity was zero and its momentum was zero. The equation for the conserved momentum was: 

     The second part of the experiment focused on conservation of energy after the collision. The ball and container were treated as one system. The collision gave the system some kinetic energy. Since the system moved to the right and up, the kinetic energy turned into gravitational potential energy and the system at its peak had zero kinetic energy. The equation for the conservation of energy was:

     The final velocity in the change of momentum equation is the same as the initial velocity in the conservation of energy equation. So, re-writing the momentum equation in terms of the ball's initial velocity and plugging that into the energy equation allowed for the solving of the ball's initial velocity. Below is the algebra needed to get to that final equation.

     We have the equation for the initial velocity of the ball being fired, but the work is not done yet. There was uncertainty in the experiment that needed to be accounted for. Measurements were made for the mass of the ball, the mass of the container, the length of the string holding the container up, and the angle the container made with the vertical at its highest point. So, a number for uncertainty was needed to give a range for the speed of the ball, and this was found with the equation for uncertainty. 

     The equation for uncertainty is quite a mouthful to describe in words, so below is the equation written out and each partial derivative used. In the top of the picture is the equation, in the blue are the partial derivatives, in the red are the measurements made, and in the bottom right in black are the uncertainties in those measurements.

     After letting the calculator figure out the rest, it returns a number for the initial velocity of the ball to be

 v=1.522 m/s (+-) .1479 m/s 

     This experiment was not just about getting a final answer to this problem. There is nothing to compare the answer to unless cameras and video analysis are involved, although the speed might make sense considering the ball is not shooting out of an actual gun. This experiment was also about how to approach problems differently. People are used to seeing a problem as a whole, but this problem required it to be broken down into two parts and use different theorems together.

     

22-April-2015: Collision in two dimensions

     The purpose of this experiment was to determine if momentum and energy are conserved in a two-dimensional collision.
     The setup consisted of two small, steel balls placed on a level table, one initially stationary and the other given some initial velocity. The ball with an initial velocity collides with the stationary one and they move away at some angle between them after the collision.
     The collision was recorded with a camera that was mounted above the table. Logger Pro's video analysis was used to capture and analyze the collision. Below are the camera settings used in Logger Pro.

     Data points were used to track each ball's position in the video. The origin was set to where the two balls collide, and also oriented so that the initial moving mass is traveling along the x-axis only. Subsequent position vs time graphs in the x and y-axis were produced.

Logger Pro's video analysis: Blue dots track the ball that had an initial velocity. Red dots track the ball that was initially stationary.

Below is the graph of the dots in the video analysis above. The blue and red dots are from the ball with initial velocity while the green and burgundy dots are from the stationary ball.

     To figure out if energy and momentum was conserved, equations were needed. The equations used were:

Note that there is no initial momentum or energy for the second ball because it is at rest.

     The balls had an x and y component when they moved so the velocities in each equation were substituted with the sum of the squares of the x and y velocities. The x and y velocities were found by analyzing the graph of the dots above. Only one example will be shown since all four trails of dots were analyzed the same way.
     Below is the position graph of the ball with an initial velocity. A linear fit was done before and after the zero position. This produced a slope for each linear fit which was the initial and final velocities in the x-direction for the first mass.

     After the velocities were found, simply plugging in the numbers and solving for each side of the equation was the next step. There were no units of length specified in this particular setup but the overall numbers should give the general idea of what was happening. The initial momentum and final momentum were calculated to be 12.40 units and 14.46 units, respectively. A difference of 2.06 units. Close, but this shows momentum was not completely conserved due to assumptions and errors. For now though, this data tells us that momentum was conserved in a larger frame of reference. Now, take a look at the energy data.

     The equation shown earlier for energy was used, and plugging in the numbers gave initial and final energies of 1150.54 units and 790.70 units, respectively. A difference of about 360 units. That is much more than the difference seen in the momentum part. Intuition tells us this difference is too big to consider energy to be conserved. 

     The data suggests that momentum can be conserved in a two-dimensional collision but that does not mean energy was conserved. Energy was lost during the collision to sound and heat. The glass table was assumed to be frictionless and therefore the calculations would not be perfect. The experiment was repeated again with a marble of different mass than the stationary steel ball. Although the results were better, the conclusion was the same. Momentum was conserved but energy was not.

15-April-2015: Impulse Momentum Activity

     The purpose of this experiment was to test and confirm the impulse-momentum theorem. The idea was to calculate a cart's momentum and compare it with impulse by integrating a force vs time graph created by colliding the cart with something. This was because the impulse-momentum theorem says that impulse is equal to the change in momentum of an object There were 3 parts to this experiment.

Part 1

     The first part of the experiment consisted of a moving cart colliding with a stationary cart. The stationary cart had a spring like component sticking out from it that the moving cart would collide into. The idea is that in a perfectly elastic collision, the moving cart would crash into the stationary cart and bounce back with the same magnitude of momentum.
     The setup had a stationary cart being held in place with a clamp at one end of a level track.

 A motion sensor was placed at the opposite end of the track. A second cart was placed on the track. A force sensor was taped onto the cart, the force probe facing the stationary cart. An index card was placed on the back of the cart so the motion sensor could detect the cart's position easier.

The mass of the cart was weighed using a digital balance. Its mass was 692 grams. This was needed for later calculations. Now the experiment could begin.
     The motion sensor was activated and the cart was given a push toward the stationary cart. The moving cart collided with the springy part of the stationary cart and then bounced back toward the motion sensor. The motion sensor produced a velocity graph and the force sensor produced a force vs time graph. Below are the graphs.

Top graph: velocity vs time
Bottom graph: force vs time

     By taking the average velocity before and after the collision, the change in momentum was able to be calculated. Taking the integral of the force vs time graph gave a value for impulse. Below was the relationship that was being used to compare the data and sample calculations for this first experiment. Each experiment after this used the same calculations so they will not be shown.

     The calculated change in momentum was 0.7541 N*s and the value given from the force vs time graph for the impulse was 0.7481 N*s. These very close numbers, off by 0.006, suggest the impulse-momentum theory holds true for this experiment.

Part 2

     The same set up was used for Part 2, except 400 grams of mass was added to the cart. This experiment produced very similar graphs to Part 1, shown below.

Using the same methods from part 1, the calculated change in momentum was 1.298 N*s and the impulse obtained from integrating the force vs time graph was 1.248 N*s. These two numbers are further apart than the ones in part 1, but they are still very close. This suggests that the impulse-momentum theorem holds true for more massive objects in a similar experiment.

Part 3

     For part 3, the impulse-momentum theorem was examined in an inelastic collision. The same cart with the added 400 grams of mass was used but this time a nail was put in place of the rubber stopper on the force probe, and some clay was put in place of the stationary cart. 

     It was predicted that the impulse would be smaller than the impulse of the previous two parts since the final velocity would be zero. It was also predicted that the impulse and momentum would be nearly equal as they were in the previous two parts.

     The cart was given a push and it collides with the clay and stops. Below are the graphs that were produced. The top graph is the velocity graph and the bottom is the force vs time graph. The same techniques were used to analyze these graphs as in the first two parts.

     The calculated change in momentum was 0.4175 N*s and the impulse from the graph was 0.4275 N*s. Again, this experiment resulted in numbers very close to each other, only off by 0.01. 

     The data for all three experiments was very promising. All three showed a change in momentum that was very close to the impulse obtained from the area under a force vs time curve. Although not perfect, these results suggest the impulse-momentum theorem to be true. Reasons for the imperfection could be attributed to friction between the cart and track, since it was assumed it was a frictionless surface. Also, in the first two parts, the stationary cart's spring bumper was not perfect and could not send the moving cart back at the same speed in which it came.