13-April-2015: Magnetic potential energy

     The purpose of this experiment was to verify that energy was conserved for a magnetic system. In this experiment, a cart has a magnet on one end of it and approaches another stationary magnet. When the cart gets close to the magnet, the energy goes all into magnetic potential energy, which repels the cart back the way it came. The problem is that there is no equation for magnetic potential energy (MPE) like there is for gravitational or elastic potential energy. So, one needed to be found.
     There exists a relationship between force and potential energy, and it is that potential energy is the negative integral of force. For instance, we know that the force of a spring is defined as F=-kx. If the negative integral is taken you get (1/2)kx^2, which we know to be the equation for elastic potential energy. The same approach was taken to find the MPE. First, lets look at the setup.
     The setup was an air track and a glider cart. The track worked by blowing air through it so that the air comes out of tiny holes, lifting the cart, and essentially making the system frictionless.

setup of glider cart on air track

     The glider cart had a piece of square metal attached to the top so a motion sensor could detect it. It also had a magnet attached to one end (the small circle with tape over it).

     At the very end of the air track was another magnet with the same polarity. A motion sensor was set up on the same side as this magnet (shown below).

     The idea was to come up with some force function that could be integrated to give potential energy. One way to find the force of the magnets was to lift up the end of the track that did not have the magnet and let gravity push the cart to approach the magnet. The cart will not touch the magnet since the two magnets are repelling each other, but the cart will eventually come to rest at an equilibrium point some distance, r, from the stationary magnet. At that point the only forces acting on the cart are gravity, which pushes the cart toward the magnet, and the force of the magnet pushing back on the cart. Since the cart is in equilibrium, the two forces equal each other.
     By tilting the cart at various angles, a set of distances between the magnets were recorded as well as the force from them. The force was calculated by the equation mgsin(theta) that could be found from drawing a free-body diagram of the tilted cart and track. The Force and distance (r) data was put into Logger Pro and graphed. Below is that graph.

The graph curved much like an exponential graph, so a power fit was done. This means the equation for the force of the magnets was going to be

The power fit provided constants for A and n. They can be seen in the data set below.

Taking the negative integral of the force function gave a MPE function of

     A way needed to be found to measure the speed of the cart and the distance between the magnets. The motion sensor was used for this step. The cart was placed so that the distance between the two magnets was 10 cm. The motion sensor was activated and the distance, d, from the motion sensor to the piece of metal on the cart was recorded. That distance, d, minus the 10 cm would be used to calculate r in the MPE function above. So, r was defined as the position, x, of the cart at any given time minus (d-0.1m), so r=x-(d-0.1). Note that (d-0.1) is a constant. Since the motion sensor can measure the distance, it can measure velocity to find the kinetic energy (KE) of the cart. The mass of the cart was found with an electronic balance.
     All equations were found to figure out if energy is conserved. All that was left was to put the equations in new calculated columns and then to do a trial. The columns in Logger Pro are below.

TE is the total energy of the system.

     The motion sensor was activated and the cart was given a gentle push toward the stationary magnet. The cart got close to the magnet, momentarily stopped so that kinetic energy was zero, and was then repelled back by the magnet. You can see from the graph below that as kinetic energy approaches zero, magnetic potential energy goes up to its highest point.

     Adding these two energies gives a line for Total Energy, which was straight across. It was not perfectly straight due to errors and uncertainty but it was close enough to tell us that energy was indeed conserved for the magnetic system. Sources for uncertainty included the measurements made for the angle at which the track was lifted, the measurements with a ruler for the distance between magnets, and the subsequent calculations for force of the magnets. The air track was also not perfectly frictionless, which was an assumption made for the experiment.

08-April-2015: Conservation of energy using a mass-spring system

     The purpose of this experiment was to determine if energy was conserved in an oscillating mass-spring system. Normally when approaching an energy problem with a spring, the mass of the spring is ignored. For this experiment, however, the mass of the spring will not be ignored.
     Before starting the lab, an equation for gravitational potential energy (GPE) of the spring needed to be found. Since the spring was hanging, the GPE of the spring varied with its y-position above the ground. The length of the spring (y) was defined as the height from the ground to the very top of the spring minus the y-position of the bottom of the spring. So, y=H-y0. A small piece of the mass of the spring was represented by dm, and a small length of the spring was represented by dy. A relationship between mass and length was found to be

dm/M=dy/y 

Rearranging this to solve for dm gave an equation of 

dm=(M/(H-y0))*dy

 GPE is defined as mass*gravity*height, so plugging in dm for mass gives an equation of 

Integrating the above equation from the end of the spring to the very top sums up the GPE of all points on the spring. So, after integrating, an equation that can be used for the GPE of the spring is

Since the first part of that equation is a constant, it was ignored for this experiment. Therefore, the final equation used was

The exact same process was used to find the sum of the kinetic energy (KE) of all the points in the spring, since they spring is not moving uniformly. The final equation used for KE of the spring was

Vend represents the velocity of the bottom of the spring.

     After the equations were derived, the setup started. A force sensor was attached to a horizontal rod, some height above ground. The spring was attached to the force sensor and allowed to hang vertically. A motion sensor was placed on the floor, directly underneath the spring. A hanging mass, with a small piece of paper on the bottom of it, was attached to the bottom of the spring. The piece of paper was there so that the motion sensor was able to identify something, since the spring was thin.

Setup of spring, force sensor, and hanging mass. Motion sensor is out of frame.

     The first step was to determine the spring constant associated with this particular spring. The forcer sensor and motion sensor were setup with logger pro to provide force vs time data and stretch vs time data. This was so a graph of force vs stretch could be plotted, so that finding the slope of that graph would be the spring constant. The collect data button was pressed and the hanging mass was slowly pushed down toward the motion sensor. Here are the resultant graphs.

A linear fit of the force vs position graph showed the slope to be 8.057 N/m, the spring constant!

     The next step was to find an equation that can be used to calculate the stretch or compression of the spring while oscillating. This was needed to calculate the elastic potential energy (EPE) of the spring. To do this, the force sensor was disconnected and just the motion sensor was used. The hanging mass was held on the spring so that it was unstretched. The motion sensor was activated and the position relative to it was recorded. The stretch or compression was found by the unstretched position minus the position at any given time. This formula was input in a new calculated column in Logger Pro.
     All three energies were now able to be calculated. To recap, the energies involved for the whole system are: EPE, GPE of the spring, KE of the spring, GPE of the hanging mass, and KE of the hanging mass. It was time to do the final experiment. First, new calculated columns were made for each energy listed above. The equations derived in the beginning were used in their respective energy's calculated columns. Below are the columns in Logger Pro. The last column on the right is Total Energy of the System. That column is calculated by adding all the other energy columns. It was predicted that the Total Energy of the System would remain constant, as energy would be conserved.

The experiment could now start. A lot of work for an experiment that lasted no more than 10 seconds!
     The hanging mass was pulled down about 10 cm. The motion sensor was activated so data could begin collecting, and the mass was let go. The motion sensor produced velocity and position graphs which were needed to calculate the energies. After the experiment ended, a graph of Total KE/GPE/EPE and the sum of those energies vs position/time were produced. Below are the graphs.

   
     The first graph shows roughly straight lines for each total energy. This tells us that the initial energies were roughly equal to the final energies, so energy was somewhat conserved. The second graph shows the oscillation. The energies are opposing each other, producing a not-so straight line for the total energy of the system(the pink line). The total energy of the system varied between 0.9 and 1.0, not a huge variation but enough to know there was error somewhere in the experiment. Upon closer look at the position data, it showed that the hanging mass was too light, causing it to oscillate past its equilibrium point and therefore sending EPE to zero.

06-April-2015: Work-Kinetic Energy Theorem

     The purpose of this experiment was to find a relationship between work done by a spring on a cart and kinetic energy of a cart. There were three parts to this experiment. The first part involved work done by a non-constant spring force. The second involved the work-kinetic energy principle. The third part involved the work-kinetic energy theorem.

Part 1

     The first part of the experiment required the measure of the work done when a spring was stretched a certain distance and the finding of the spring constant. First, a cart, spring, force sensor, and motion sensor were setup. The cart was placed on a level ramp, with a spring attached to one end of the cart. The other end of the spring was attached to a force sensor. The motion sensor was placed at the opposite end of the track. Logger Pro was opened so that the data could be recorded. The force sensor was calibrated and zeroed. The trial began by slowly, but steadily, moving the cart down the track towards the motion sensor. This would stretch the spring, which would pull the force sensor. The motion sensor and force sensor provided a force vs position graph.

setup of the cart, spring, and force sensor

Motion sensor at opposite end of the track

Here is the resultant graph of force vs position.

Since it is known that force of a spring is F=kx, the slope of the best fit line of the graph is going to be k, the spring constant. The spring constant was found to be 2.785 N/m. 

Since work done is force*distance, the area under the curve of the graph is equal to the work. Using the integration feature of Logger Pro gave a number of 0.4519 N*m. 

So, from this part you can see it is possible to find the work done using a graph.

Part 2

     The second part of the experiment asked to plot a graph of force vs position and a graph of kinetic energy vs position, and to then analyze the graphs to see any relationship between them. The setup was exactly the same as in part 1. In Logger Pro, a new calculated column was added to the table for the calculation of kinetic energy (KE). The formula input was KE=(1/2)mv^2, where m was the mass of the cart and v its velocity. The mass was found using an electronic balance. 

     For this part, instead of starting the cart next to the force sensor where the spring would be in a relaxed position, the cart started off near the motion sensor so that the spring was stretched and the force sensor had an initial force. When the cart was released, the spring pulled the cart all the way back. The motion sensor recorded the cart's velocity so that the kinetic energy was able to be calculated. 

     Here is the resultant graph. The purple points represent the force vs position graph and the orange points represent the kinetic energy vs position graph.

     The work the spring does on the cart can be found the same way as before, by using the integral feature of Logger Pro to find the area under the curve. The kinetic energy is simply the point on the graph at a certain position. If the work done up to a certain position is compared with the kinetic energy at that position, it shows the two are equal. For example, here is a portion of the graph which compares the two:

     It shows the integral to be 0.3044 N*m and the KE of the cart at that point to be 0.298. The two numbers are very close to each other. This analysis was done at several points to ensure the relationship is consistent, which it was. So, this shows that the work done by the spring on the cart is equal the the change in kinetic energy. 

Part 3

     This final part involved watching a video of a professor performing an experiment. The professor uses a machine to pull back on a large rubber band. The force being exerted on the rubber band is recorded by an analog force transducer onto a graph. A cart of unknown mass is then attached to the rubber band. The cart is released and passes through two photogates a certain distance apart. The photogates were used to calculate the final speed of the cart, which would allow for the calculation of the final kinetic energy of the cart. A rough sketch of the graph produced by the professor in the video looks like:

     From the first two parts of this experiment, it is known the work can be found by calculating the area under the curve of a force vs position graph. So, for this graph it can be done by breaking the graph up into shaped we know how to find the area of and then adding up the areas. The first section is a triangle, the second is a rectangle, and the last two are trapezoids. The areas of those shapes are easy to calculate, and so they were. The final number for the total work was 25.675 J.

     The video provided data for the mass of the cart, the distance between the photogates, and the time when the cart passes through the photogates. With that data the velocity of the cart was found to be 3.33 m/s and the mass was 4.3 kg. So, the final kinetic energy of the cart was calculated to be 23.88 J. We expected a value close to 25.6 from the above graph, and we got pretty close. The difference between the numbers might have been caused by the sketch of the graph. Since the actual graph was not perfectly straight, but instead was quite squiggly, the actual area under it would be different. Also, the experiment the professor performed was all done by hand with no help of computers. The accuracy depended on the consistency of the person performing the experiment. Thank science for computers! 

01-April-2015: Centripetal force with a motor

     The purpose of this experiment was to come up with a relationship between angular speed and the angle with the vertical at which an object spins. The setup for this experiment used the apparatus shown below.
 

     The apparatus had an electric motor mounted on a tripod. A vertical shaft coming out of the motor. A horizontal rod attached to the top of the vertical rod. A string tied to the end of the horizontal rod and a rubber stopper at the end of the string.
     As the motor spins at higher speeds, the mass (rubber stopper) revolves around the vertical shaft at a larger radius and the angle it makes with the vertical increases. Below is the diagram for measurements made on the apparatus.

     The height from the ground to the horizontal rod (H) was measured with a meter stick. As was the distance from the vertical rod to the end of the horizontal rod (R), and the length of the string (L). To find the height from the ground to the mass (h), a piece of paper was put in place under the revolving mass. As the mass revolved, the paper was slowly risen up until the mass hits the top of the paper. The height from the ground to the top of the paper was then measured with a meter stick for each trial. The distance from the edge of the horizontal rod to the mass (r) was found by Lsin(theta), and the angle was found by arccos((H-h)/L).

     Finding a relationship between the vertical angle, theta, and angular speed, omega,  called for equations to solve for the variables. The equation for the angle was found with simple trigonometry to be arccos((H-h)/L). To get an equation for angular speed, omega, a free body diagram for the mass was used. Below is the free body diagram for the mass in motion.

Finding the sum of forces in the x and y directions gives

     Now that the sum of forces were found, the variable omega (angular speed) can be solved for. After some manipulations, an equation for omega gave

and since r=R+Lsin(theta), the end equation is

     So, the relationship between the angle and angular speed was found.

     After the equations were solved for, the trials began. The motor was turned on and the mass began to revolve. The period for the revolution was found by using the stopwatch feature on a smartphone. Each time the mass passed by a point of reference, the time was recorded. That was done for a total of 10 times for each trial. Below is the data recorded and the subsequent calculations for the angle and angular speed. Note that calculating the angular speed required the angle to be converted to radians.

     Next, the actual measured angular speed needed to be found. This was found by the formula 

     The period, T, for each trial is located in table above, in the column "T for one revolution."

The measured angular speed and the predicted angular speed were put into a table in Logger Pro (shown below).

The data was then graphed and a linear fit was done.

     This graph shows something very important. It shows that there is indeed a relationship between angular speed and the angle at which an object revolves, and that relationship was shown in the equation from earlier:

 

     If there were no relationship between them, the graph would not produce a straight line. Below is the data for the linear fit of the above graph.

     The slope is 1.042, very close to 1. Of course, since there were uncertainties in all measurements made with the meter stick, the slope was never going to be exactly one. There were also some factors that could have negatively affected the data. When the mass was revolving, the entire apparatus was rocking back and forth. Also, there was human error in trying to record the period of the revolution. It would be extremely difficult to accurately record the period with the human eye and a stopwatch.

25-Mar-2015: Centripetal Acceleration vs. Angular Frequency

     The purpose of this experiment was to determine the relationship between centripetal acceleration and angular speed. The formula a=rw^2 was used as a reference point for this experiment. It was hypothesized that a graph of acceleration vs angular speed would produce a slope, which would represent the radius from the object in motion to the center of its circular path.
     The demonstration was done using a flat, heavy rotating disk. A scooter wheel, powered by a motor, was placed next to the disk. The wheel was touching the side of the disk so that when it was powered up, the wheel rotated the disk. An Accelerometer was taped to the top of the disk. A piece of tape was sticking outward from the accelerometer so that it passes through a photogate sensor at the same place once every revolution.

Setup of the demonstration. Photogate sensor to the left, disk in the middle, and scooter wheel to the right.

   
     The motor was powered up and set the wheel into motion, which set the disk into motion. The first trial was done with the motor running at 4.8 volts. Logger Pro was used to make recordings of each time the accelerometer passed through the photogate. It was also setup to the accelerometer itself, providing graphs of acceleration vs time. This procedure was repeated five more times for a total of six trials, with each trial having a larger voltage going through the motor.
     Here is an example of data collected from the photogate sensor. The table on the left shows the time when the accelerometer passes through the photogate. Under the "State" column, when a 1 shows up is the time that was being looked at. The first "1" was the starting point. Ten "1's" were counted after that to find the time after ten rotations. This data was used for calculations later.

Table and graph given by the photogate sensor.

     Here are the graphs the accelerometer produced. For each trial (shown by the voltage used), a statistical analysis of the graph was done. That produced a mean value of the graph. That mean value was used as the acceleration for that trial. This data was saved for later use. 

Graphs produced by accelerometer.

     Once all data was collected, a graph of acceleration vs angular speed needed to be produced. The collected data was input into a table in Logger Pro. Below are the first three columns. 

     Next, angular speed needed to be found for each trial and input into a new column. The formula used for angular speed, represented by the greek symbol "omega", was

     To get the time for one rotation, the time after 10 rotations was subtracted by the start time, and then divided by ten. The calculation for angular speed was done for each trial. The last calculation was to square the angular speed. This was because a graph of acceleration vs angular speed squared was needed to find the slope, which would be the radius of where the accelerometer was located on the disk. 

     The radius was measured by with a ruler to be 13 cm. The first and last column were plotted in a graph and a linear fit was done.

     From the linear fit data above, the slope shows 0.1366, which translates to 13.66 cm. The experiment outcome matched the hypothesis. The measured radius and the experimental radius did have some difference in it. This could have been caused by uncertainties or errors in the lab. It was assumed that there was no friction on the spinning disk. There were also uncertainties in how accurate the photogate sensor was in reading the tape going through it. Would the data have been more accurate if the tape going through the sensor was thinner? For now, a rough approximation is good enough to show that, yes, there is indeed a relationship between centripetal acceleration and angular speed, and it is the radius of the circular path.

23-Mar-2015: Predicting impact points of trajectories.

     The purpose of this experiment was to use the understanding of projectile motion to predict the impact of a ball on an inclined board. The setup for this was quite simple. Two ramps were used, one lying flat and the other attached at an inclined angle so a ball could roll down. The flat ramp was set up flush with the edge of a table.

setup of ramps

     Once the ramps were setup, a steel ball was released from the top of the inclined ramp. It rolled down and flew off. The point at where the ball hit the floor was noticed, and regular printing paper, with a piece of carbon paper over it, were put on that spot. This was done so when the ball hits the carbon paper it makes a mark on the regular printing paper. The ball was launched five times to ensure it hits roughly the same spot each time.

     The measurements of the height of the ramp and the distance from it where the ball lands were needed for future calculations. A meter stick was used to measure both. The height of the ramp was measured to be 0.874 meters and the distance from the ramp where the ball hit the floor was 0.734 meters. It might be hard to see in the picture (below and to the right), but there are a small cluster of dark spots where the ball hit the paper.

Measurement of the height of the ramp from the floor.

Measurement of the distance from the ramp where the ball hits the floor.

  

     The next step was to calculate the launch speed of the ball using the measurements taken above. The equation delta x=V0x*t was used since the ramp it was launching off of was flat. First, the time it was in the air needed to be calculated. This was done by using the formula

delta y=V0y*t+(1/2)gt 

and assuming constant acceleration in the y-direction. Since the initial velocity in the y-direction is zero, the formula can be rearranged and solved for t, so that

t=[(2*delta y)/g]^(1/2)

Delta y was a measured quantity, so after plugging in the numbers, the time was .422 seconds. That number was plugged into the formula delta x=V0x*t, which was solved for V0x so that

V0x=(delta x)/t

Delta x was a measured quantity and t was solved for in the previous step. Plugging in the numbers gave an initial x-velocity of 1.74 m/s.

     The next step was to place a wood board at the edge of the table and incline it so that when the ball is launched, the ball will hit the wood board instead of the ground. Before starting the trial, a prediction was made as to where the ball would hit the wood board. This prediction was done by using the measured quantities from earlier and using trigonometry. Below is the picture of the setup of the inclined board.      

     As you can see, the bottom of the inclined board was positioned where the ball would strike the ground. This allowed for the finding of the angle between the board and the ground using 

tan(theta)=y/x

in which y and x were the measured quantities from earlier. The angle was calculated to be 46.3 degrees. Finding the angle then allowed for the prediction of the distance the ball would hit on the board. First, note that y=dsin(theta) and x=dcos(theta). The kinematic equations 

y=(1/2)gt^2

x=V0t 

were used to make the prediction. Substituting dsin(theta) and dcos(theta) for y and x, respectively, gives new equations of 

dsin(theta)=(1/2)gt^2

dcos(theta)=V0t 

The equations were solved for t^2 and set equal to each other, so that

(2dsin(theta))/g=(d^2cos^2(theta))/V0^2

d was then solved for giving a final equation of 

d=(2V0^2sin(theta))/(gcos(theta))

     After plugging in the numbers, the ball was predicted to hit the board 0.936 meters down. 

     After the prediction was made, it was time to start the trial. The piece of printing paper, with carbon paper over it, was positioned around the predicted impact point. The ball was then launched five times from the same spot to make sure it hits roughly the same point on the board. After all trials were completed, the point where the ball hit the board was measured using a meter stick. The distance was recorded to be .890 meters. 

     The final numbers of a predicted measurement of 0.936 m and an actual measurement of 0.890 were not terribly close. But, remembering that there were measurements made, and therefore uncertainties in those measurements, gave some hope that the predicted measurement was within some range of the uncertainty. The total uncertainty was calculated using the usual propagated uncertainty formula, where the partial derivative is taken with respect to some variable, and that partial derivative is multiplied by the uncertainty in the measurement of the variable. The distance that was solved for earlier, however, only had one variable in it, theta. So, some algebraic manipulation was needed to get the x and y variables in the formula for distance. Using 

y=(1/2)gt^2

t was solved for so that t=(2y/g)^(1/2). Then, using 

x=V0t

V0 was solved for so that V0=x/t. At that point, t was plugged into this equation to give 

V0=x/(2y/g)^(1/2)

This V0 was plugged into the equation 

 d=(2V0^2sin(theta))/(gcos(theta))

and simplified to a final equation of 

d=(x^2/y)sec(theta)tan(theta)

     Now that the three measurements with uncertainty are in the formula for distance, the partial derivatives were taken. All numbers were plugged in to solve for propagated uncertainty, which was (plus or minus) 0.0477 m. It worked! The propagated uncertainty gave a predicted range of 0.883 m to 0.9837 m, in which the actual impact measurement was within. There were ways in which this experiment could have had greater errors, such as, not releasing the ball from the same height each time or errors in reading measurements. There were also assumptions made, such as no air resistance when the ball was in its trajectory, and no friction when the ball was rolling down the ramps.