Lab 13 Magnetic Potential Energy

Purpose: The purpose of this laboratory is to verify that conservation of energy applies to a system with a glider being pushed by a magnetic force.

The apparatus that we used consisted of an air track connected to an air blower on one end and a glider on the track with magnets on the front and back and a motion sensor on the end of the track.
We use the apparatus to measure the distance between the magnet on one end of the track and the magnet on the glider.

Procedure:

The first thing we did was find an equation for the magnetic potential energy. Generally for a system that has a non-constant potential energy like this one the potential energy U is caused by an interaction force F and is given by the relationship U(r) = - integral from infinity to r of F(r)dr, where r is the separation distance. We assume the force is 0 when r = infinity. so what we have to do is find an equation for F(r) .

But first we had to find out a way to measure the distance "r" which was the distance between the magnet on the track and the magnet on the glider. We did this using the motion sensor to measure the distance between the aluminum plate on the glider and the motion sensor and subtracting that distance to the distance r between the magnet on the glider and magnet on the track. This gave a distance that we would always subtract from the distance the motion sensor read. Allowing us to find "r" at any position for the glider. 

To find an equation for F(r) we had to collect data for the Force and distance of r at different angles so we elevated the track at many different angles and measured the angles and the distance r.

To get the force from the data collected we figured that the force pulling the glider to the end of the cart was a horizontal force parallel to the cart and the only force that could be was the horizontal force of friction, since the track was frictionless, which was mgsin(theta). So to calculate the force we used the measured angle and the measured mass of the cart which was .346kg. And we already talked about how to get r at the top so using this information we plotted a Force vs distance r graph and used the power law fit F = Ar^n to get a trendline for the graph.

Next we had to verify the conservation of energy is true. So we made columns that we needed to calculate the magnetic potential energy and the kinetic energy.

We plotted the graphs for kinetic energy and magnetic potential energy to see if energy was conserved.

The fact that the graphs are opposite tells us that the energy is conserved because when the kinetic energy goes to zero it turns to magnetic energy. Ideally the magnetic potential energy peak is supposed to be higher to show that the kinetic energy lost completely transferred to magnetic potential energy, but there was error as there is in all cases. Maybe the collision between the magnets was not perfect or the track was not completely frictionless because of the air molecules its going through .

Lab 12 Conservation of energy- mass-spring system

Purpose: The purpose of this laboratory is to examine the energy in a vertically oscillating mass-spring system, where the spring has a non-negligible mass.

The apparatus that was used in this laboratory is a mass spring system consisting of a rod clamped to the table with another rod clamped horizontally at the top of the vertical rod. Attached to the horizontal rod is a force sensor. Hanging from the force sensor we have a spring and a 50g hanging mass. Under the hanging mass we have our motion sensor. The purpose of the apparatus is to help us measure the force and position of the hanging mass and spring.








The first thing we did was find the spring constant "K" for the spring. To do this we pulled the hanging mass and let the spring oscillate and we recorded data for force and position of the spring.
We plotted the data in a Force vs. Position graph and added a linear fit. The slope of the linear fit gave us the spring constant "K" which turned out to be 9.1N/m.

 Figuring out the value for our spring constant "K" we were able to find and equation for the elastic potential energy of our spring which is :

EPE = 1/2K(stretch)^2

 

Where the "stretch" is the distance of the unstretched position minus the distance of the position of the spring at any given time.

 

Next we had to derive an equation for the gravitational potential energy of the spring itself. To do this we:

 

1. Choose a representative piece dm of the spring and wrote an expression for it in terms of dy. the expression is dm = M/(H-yend)dy where M is the mass of the whole spring and H is the distance from the ground to the top of the spring. and yend is the distance from the bottom of the spring to the ground.

 

2. We wrote and expression for the GPE of the little piece of mass "dm" which was GPE = Mgy/(H-yend)dy where M is the mass of the whole spring g is acceleration of gravity y is the distance of dm to the floor, H is the distance of the top of the spring to the floor and yend is the distance from the bottom of the spring to the floor.

 

3. lastly we summed the gravitational potential energies of all the dm pieces in the spring and our expression for the gravitational energy of the spring turned out to be M/2gH + M/2gyend. which can also be written as Mg(H+yend)/2.

 



 Next we had to derive an expression for the kinetic energy of the spring itself. To do so we followed the same three steps above only we changed our origin to the top of the spring and our positive direction was down.

1. Our expression for the small piece of the spring dm is dm = M/Ldy where L is the Length of the spring and M is the mass of the whole spring.

2.Our expression for the kinetic energy of the small piece of the spring is 1/2(M/L)dy(y/L)^2(vbottom)^2. where M is the mass of the whole spring, L is the length of the spring, y is the y is the distance of dm from the origin and v bottom is the velocity of the bottom of the spring.

3. Our expression for the kinetic energy of the spring is KE = 1/6M(vbottom)^2. where M is the mass of the whole spring and vbottom is the velocity of the bottom of the spring.

 Other energy equations that we need and were given to us in this lab are:

Kinetic energy mass = 1/2(Mhanging + 1/3Mspring)v^2 where v is the velocity of the hanging mass.

Gravitational potential energy in mass = (Mhanging + 1/2Mspring)gy

What we did next was make calculated columns for the KEmass, EPEspring, GPEmass, KEspring, and GPEspring and the energy sum of all the energies, and we plotted the graphs in logger pro.

 The black line on the graph represents the sum off all the energies while all the other graphs represent the individual energy by themselves. The fact that the black line is close to being a straight horizontal line shows that energy is conserved. Although the ideal result is to have a perfectly horizontal straight line for the sum of the energies there are error factors that affected the results.
for example I believe we were having trouble with the distance that the motion sensor was calculating, also when we forgot to write down the mass of the spring so we had to guess which spring we had the following day and we measured the mass of that spring. It could have been incorrect which could have led to error as well.

Lab 11 Work-Kinetic energy theorom

Purpose: The purpose of this laboratory is to learn how to calculate work using a force vs. distance graph. Also we will see how the work don on the cart by the spring compares to its change in kinetic energy. Finally, Using the kinetic energy theorem the work will be determine from a graph and the kinetic energy will also be calculated and both with will be compared with each other to see the relationship.

The apparatus that we used measures the force of the spring as the cart is being pulled and plots the data into logger pro.
Procedure:
1. Calibrate the force probe with a force of 4.9N applied.
2. Set up the ramp, cart , motion detector, force probe, and spring as shown in the diagram. you may wish to use cart "stops" or something else to support the spring so that it can be horizontal and unstretched.
3. Be sure that the motion detector sees he cart over the whole distance of interest- from the position where the spring is just unstretched to the position where it is stretched about .5 m .
4. Open the experiment file called L11E2-2(stretched spring) to display the force vs. position axes .
5. Zero the force probe and the motion detector with the spring supported loosely and unstretched. Verify that the motion detector is set to "Reverse Direction", so that toward the detector is the positive direction. Then begin graphing force vs. position as the cart is moved slowly towards the motion detector until the spring is stretched about .5 m . (keep your hand out of the way of the motion detector).
6. Sketch your graph.
a. determine the spring constant of your spring explain how you did that.
b. use the integration routine in the software to find the work done in stretching the spring.

here is a sketch of our graph of force vs. time.

Using the integration routine in logger pro we were able to find that the work is .6561 J.

Expt 2. Kinetic Energy and the Work Kinetic Energy Principle.

We use the same apparatus for this part of the experiment.

Procedure:
1. Measure the mass of the cart. The mass of our cart was .573 kg.
2. Under Data- new calculated column, enter a formula that would allow you to calculate the kinetic energy of the cart at any point.
3. Be sure that the motion detector sees the cart over the whole distance of interest- form the position where the spring is stretched about .5 m to the position where it is just about unstretched.
4. Make sure that the x-axis of your graph is "position". Zero the force probe with the with the spring hanging loosely. Then pull the cart along the track so that the spring is stretched about .5m from the unstretched position.
5. Begin graphing, and release the cart, allowing the spring to pull it back at least to the unstretched position. when you get a good set of graphs, save them.

Note that the top graph displays the force applied by the spring on the cart vs. position. It is possible to find the work done by finding the area under the curve using the integration routine in the software. The kinetic energy of the cart can be found directly form the bottom graph for any position of the cart.

6. Find the change in kinetic energy of the cart after it is released form the initial postion (where the kinetic energy is zero) to several different final positions. Use the analysis feature in logger pro. Also find the work done by the spring up to that position. Record these values of work and change in kinetic energy in a table determine from your graph the position of the cart where it is released and record it in the table.

 Work and kinetic energy was found for many positions in the graph and it indicates that the work and kinetic energy is the same value only the work is negative because the cart is initially moving and at the end it is at rest.

The work - energy principle states that W = 1/2mvf^2 - 1/2mvi^2

Expt. 3 Work- KE theorem

Procedure:
On the laptop is a movie file entitled Work KE theorem cart and machine for Phys 1 .mp4. In the video, 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.
The stretched rubber band is then attached to a cart of known mass. The cart, once released, passes through two photogates a given distance apart. By knowing the distance and the time interval between the front of the cart passing through the first photogate and then the second photogate, you can calculate the final speed and thus the final kinetic energy of the cart.

At a convenient place in the movie, stop the movie, make a careful sketch the force vs. position graph, and determine the work done by the machine in stretching the rubber band in it.

Use the data in the last few frames of the video to get mcart , change in distance of photogate and change in time of photogate and to determine the final KE of the cart attached the machine. Compare the results to what you would expect from your Fore vs. Postion graph. Comment on uncertainties in your results.

The Force distance graph that we drew from the video is .

We calculated the work to be 25.675J.

And the calculated Kinetic energy turned out to be 23.9 J.

The calculations are close, but there could have been error from the graph because we technically approximated some of the values on the graph because they could not be 100 exact and some points landed between two points on the y axis and not exactly on a value. But the method gives a good approximation to correct amount of work done.

Lab 9 Centripetal force with a motor

The purpose of this laboratory is to come up with a relationship between omega (w) and theta which is the angle that the string makes in the following diagram between L and the vertical.

The apparatus we used is an electric motor mounted on a tripod. A long shaft is connected to the motor . At the end of the shaft a horizontal rod is connected to it. At the end of the horizontal rod there is a long string of length L. Attached to the end of the string there is a rubber stopper. to measure height each we used a ring stand with a horizontal piece of paper sticking out of it.

The purpose of this instrument is to be able to use it to collect data for the period to find omega for different speeds of the motor. And to see the relationship that omega has on the size of the angle.

The first thing we did for this experiment was measure the H, R, and L. These measurements were H= 2 m, L= 1.645 m, R= .98 m.

We collected data for h and the time of ten revolutions for a certain speed:

Trial 1:
h = .455 m
t10 = 39.55 sec

Trial 2:
h= .578 m
t10= 34.6 sec

Trial 3:
h= .832 m
t10= 28.8 sec

Trial 4:
h= 1.025 m
t10 = 25.6 sec

Trial 5
h= 1.26 m
t10 = 21.4 sec

Trial 6
h= 1.485 m
t10= 17.5 sec

Next we had to find a relationship between theta and omega to do so we created a free body diagram to find the net force equations.

Although we have found a relationship between theta and omega we do not know what theta is so we are going to find out what we use to plug into theta with the given data that we measured.

we can use the following diagram of part of the apparatus to find out what theta is using trigonometry.

We know the measured length of L, and we can find one of the sides of the right triangle by subtracting the height of h that we measured with the height of H that was also measured. Knowing this we can derive an equation for theta as follows :

Plugging theta into the first equation we found and simplifying gives us the following expression:

Another way to find omega is by using the period T using the following expression:

Using these to equations we plugged them into logger pro and created a graph out of them the y axis was omega using the measurement of the height h. And the x axis was the omega found by using the period.

The graph tells us that the relationship that we derived for theta and omega is a good model . This is because our graph was a fairly straight line with a slope that is really close to one. The slope being close to one says that you get the same values using either equation to find omega. But like always our model is not perfect because there is a little bit of error that could have came from the fact that we had to guess when the weight at the end of the string actually crossed the paper on the ring stand so that we could start the time. Also taking the time for only ten revolutions is a small amount. To get a more accurate measure for the period it would have been better to take the time of 100 revolutions.

Lab 8 Demonstration Centripetal acceleration vs. angular frequency

Purpose: To determine the relationship between centripetal acceleration and angular speed.

In this experiment, we used a heavy rotating disk with an accelerometer attached to it. We used this device to measure how long it takes for the disk to make a certain number of rotations at different range of speeds.

Angular speed is the change in angle over time and is called omega . Centripetal acceleration is referred to the force acting on an object that's trajectory is a circle and is directed towards the center of its circular path.

To find angular speed we use the following formula :

 

 

Where t is the time for one rotation.

In this experiment the professor measured the acceleration and time of ten rotations of the disk and the whole class used the same data. We put the data into logger pro and inputted formulas into logger pro to calculate the omega and omega^2 values and got the following.

Then we made an acceleration vs. omega^2 graph and added a linear trendline.

The equation of the line given from logger pro is the same as :

Where ac is centripetal acceleration and w^2 is omega^2 and the slope of the trendline is r . This equation shows the relation ship between centripetal acceleration and angular speed to be as the angular speed becomes greater the centripetal acceleration will also become greater.

The graph shows that our radius is .1371 meters and when we measured the radius in class it turned out to be .139 meters. The radius in our graph was really close to the measured value in class. There could have been error as always because the equipment we were using was not the most expensive and was many years old . The accelerometer might not have been perfect at measuring the exact point of the rotation that is why we used ten rotations instead of timing just one.

Lab 5 Trajectories

Purpose: To use your understanding of projectile motion to predict the impact point of a ball on an inclined board.

 
 

The apparatus is used to examine the trajectory of a metal ball and the carbon paper is used to mark were the metal ball lands on the white paper.




















Procedure:

1. Set up the ramp system like in the picture above.
2. Mark the point from which the ball will be launched on the ramp with some tape so that the ball will be released from the same point for each trial.
3. launch the ball and check to see where it lands so that you can tape the blank piece of paper with the carbon paper on top in the same spot where the ball landed.
4. launch the ball 5 more times from the same place as was marked on the ramp and verify that it lands in the same place each time.
5. Measure the height of the bottom of the ball when it launches off the ramp. And measure the distance from the table's edge that it lands.
6. From your measurements, determine the launch speed of the ball .

7. Attaching an inclined board, like the one pictured on the image to the right the ball will now hit the board a distance d when it is launched. Derive an expression that will allow you to determine the distance of d given that you have the initial velocity V0 and the angle .

8. Do a trial run to see where the ball is going to land on the board. With tape attach the white paper and carbon paper on the spot where the metal ball landed on the board. Measure the angle of elevation of the board and perform five more trials from the same spot.

9. Determine the experimental value of d and report your experimental value with uncertainty

10. Compare experimental and theoretical values for d. comment on sources of uncertainty or error in the experiment.

 

 

 

We measured the distance from the edge of the ramp to where the ball landed in the x direction. And we measured the height of the edge of the ramp to the ground in the y direction. We got the measurements to be :

 



 

distance in y direction:     .94 meters

 

distance in x direction:     .68 meters

 

Using the distances measured we were able to calculate the initial velocity of the ball when it launches by using the method to solve a projectile motion.

 



 



Splitting the projection of the ball into an x and y direction we used the kinematics equation in the image above to solve for time of flight in the y direction. Solving for the time we were able to solve for the initial velocity in the x direction and we calculated the initial velocity to be 1.55 m/s.

 

For the second part of the experiment we attached a ramp to the table and find the distance d where the ball is going to land. We derived two equations to solve for the distance d:

 

 



We calculated the initial velocity in the first part of the lab and we measured the angle of the board using an app that measures angles on our phone. So we have two equations and two unknowns the unknowns being d and t. Since the time t is the same for both x and y we solved each equation for t and made the equations equal to each other then we solved for d and got an expression that solves for d.

 

 

Calculated initial velocity :  1.55 m/s

 

Measured angle of board : 48 degrees

 

g = 9.8 m/s^2

 

 

The distance we calculated where the ball is supposed to land on the board is .814 meters. So we launched the ball 5 times and got a range between .841m and .869m.

To see if our calculated value is acceptable we have to use the method of propagated error.

 

 





To be able to calculate the propagated error we are going to have to derive an expression that involves x and y and d together. So we used the kinematics equations for x and y and we solved for t again and made the equations equal to each other. Then we solved for initial velocity so that we could plug it into our equation to get a new equation that has d, x and y.

 

 







Now that we have the appropriate expression we took the partial derivatives and calculated the range of our uncertainty in the calculation.

 

 

 

the uncertainty that we calculated was  +/- .0155 .

 

As you can see our calculated value for the distance with the uncertainty is not within the range of the measured values.

 

Calculated d and uncertainty : .8143 +/- .0155

 

Measured range: .841m - .869m

 

The reason that the calculated value might not be in the range of the measured value was that we could have been measuring the distance with the meter stick from the wrong point. There was space between the edge of the ramp and the edge of the ramp . Another factor that could have caused error was that the board was not taped to the floor and table so it could have moved and the board was also not flat the wood was bent a little.