6th April 2015: Conservation of Energy Lab

Lab 11: Conservation of energy with a Spring

Purpose: The purpose of this lab is to essentially prove the theory of conservation of energy using a spring and a mass to show the oscillations of motion as a weight is applied to one end of the spring and a force sensor is calculating the energies as the move back and forth.

Spring constant = K

Procedure:  We first began by getting a spring, clamps, a motion sensor, force sensor, and a small mass. We use the clamps to attach a rod to our desk which went up some height and then an additional bar was used to clamp a pole above the spring so that a force sensor could be placed above the object and measure the forces. On the floor was motion sensor that was picking up the displacement of the object and spring as they oscillated. By knowing the total change in distance from the motions sensor and the force that the mass exerted on the spring we came up with a value of 14.86 for the spring constant of our mechanism.

This is a picture of what the mechanism looks like in class with the spring
hanging from the force sensor. 

After we found the spring constant we now need to find the forces of potential, kinetic, and total energies. By using the motion sensor for calculating the displacement and the force sensor to calculate the force, we were able to solve for our energies by inputting calculated columns into logger pro. 

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

Potential Energy mass (PE)= mgh

Spring PE= 1/2kx^2

KE spring= 1/2mv^2

PE gravity spring= mgh

Total Energy= Sum of (initial) KE+PE=PE+KE (final)

By inputing the formulas we came out with several data points that made up several energy curves as follows: 

These are all of the energies that we solved for
by using the different formulas we were given.

By solving for the calculated equations above we got the results from the energy graph in the picture above. With the groups knowledge of conservation of energy, we know that the potential energy of both the mass and the spring should be equivalent. We know that the have a relationship which they mirror each other meaning that if PE of the mass is increasing the PE of the spring should due decreasing and vise versa. In our data we notice that the graphs in fact do correspond with each other and total energy remains constant throughout our graph for the entire experiment. Since we found out the energies and matched those results to our predictions and knowledge of what we know we find that conservation of energy does support the spring and the mass. All in all our experiment was successful in proving conservation of energy within our system.

13th April 2015: Magnetic Potential Energy

Lab 12: Magnetic Potential  Energy with a Cart

Purpose: To prove that there is conservation of energy within the system of an unknown magnet and a glider that is on an air track. By proving an equation for magnetic potential energy then we can prove that the system does have conservation of energy.

Procedure: The materials consisted of a large frictionless air glider and an air track. Attached to the air track was an inverted vacuum which allowed the gilder to be frictionless(almost). We also used books in order to give an angle to the the air glider. On top of those materials we also used magnets which we put at the end of the air glider and the opposite end of the track in order to have a repulsion as the glider glided down the track. We also used logger pro in order to set up a motion sensor that we used to measure the displacement of the glider as it moved closer to the magnet on the opposite side.

These are the materials we used in the experiment the motion sensor is not visible in the picture but is off to the right. 

After we had our track set up and running. We ran several trials which consisted of different angles and each time we moved the angle up we measured the force. Newtons second law of F=(Mass)(acceleration), was used to solve the force of the glider moving on the track, we derived an equation of f=mgsin(ø). We then used logger pro and used the five different forces from the five new angles to create a calculated column, then we compared it to the distance the magnets repulsed. This separation distance vs. force graph was carefully examined, by finding the curve fit of the graph using a power equation, we were able to establish an equation for the magnetic potential force of our contraption. 

Our trial with several books
so that we can increase the angle of our mechanism,

This is the power curve fit to our force vs. separation
equation. Power-fit= y=Ar^B

After we had found the magnetic potential energy equation we made more calculated problems so we could try to prove the conservation of energy. First we calculated a column for the the total energy, which consisted of the two new calculated columns we made for both potential and kinetic energy. We know that the initial potential energy and the final kinetic energy is equal to the total energy of the system. And in our case we are trying to solve for the magnetic potential which we did and to prove it has conservation of energy we had to set our magnetic potential energy equal to the total energy which is potential initial energy plus final kinetic energy. 

This is a graph of all of the calculated columns.
The top is total energy, red is potential, and orange is kinetic.

When examining and analyzing our results we find that there is an error in our data, with the total energy being equivalent to .095 jules approximately, we find that our data is off. For one, it should have the same position as the kinetic energy graph and mirror its every curve. There was most likely an error when we first ran the trial and did the power fit. The most reasonable answer to the error in our graph was the power fit curve which was not perfectly fit to our data because the surface wasn't perfectly frictionless. And the force which we applied to the glider in order for it to come into repulsion contact with the magnet may have been a different applied force every time causing our data to be inaccurate. All in all mathematically we found that energy was in fact conserved, but we could have ran better trials to assure that our data came out to be more sufficient. All in all we had a successful experiment in proving magnetic potential energy does in fact have conservation of energy.

8 April 2015- Work Energy-Theorem Lab

Lab 9: Work Energy-Theorem

Purpose: We examined the proven conjecture of the work-energy theorem that is equal to the kinetic energy graphs area under the curve which is the total work being done.


Procedure/Materials: In order to begin the experiment, we first began by using a metal track that lay across our desk which we placed a cart on and attached a spring to. At one end was the motion sensor which was calibrated to zero, on this same end the cart has a spring attached to the sensor that is unstretched but when we do stretch it the motion sensor will record the data. As soon as we calibrated our motion sensor we began the experiment which involved us pulling the cart back a certain distance. After doing so this distance was measured and then we began recording our data using logger pro. As we released the cart logger pro began measuring data on the force vs. time graph.

This is the force vs. time graph. 

This is a picture of the experiment we did in class to do the force vs. position graph. 

After recording the data we added another column in order to incorporate the calculated kinetic energy of the system deltaKE= .5mv^2. We then used this formula to compare and contrast the value of kinetic force vs time and the total kinetic energy vs time over that same exact time interval, there was some small error. We had to use three separate graphs to show the three different points we tested for each kinetic energy. 

Kinetic energy is .165 and the integral under the force vs time diagram is .152

Area under the curve .344 vs. kinetic energy of .365

area under the curve .453 vs. Kinetic energy .432

These three different intervals are the work energy theorem which is represented by force x time. The area of this integrated curve should be equal tot he kinetic energy with respect to time. As logger pro compared the two graphs we found that this relationship is hared between work energy theorem and kinetic energy, although we had an error. Our data was only of by 1 or 2 percent which could have easily occurred when the motion sensor was measuring data or not zeroing our equipment correctly, leading us to the conclusion that we had a successful lab in learning about the work energy theorem and its relationship with the area under the curve being equal to kinetic energy.

01 April 2015- Centripetal force with a motor

Lab 7: Establishing a Relationship between Angular Speed and Angle of Rotation


Purpose: The purpose of the lab is to find a relationship between angular velocity(w=mv^2) and the angle that the object is rotating at represented by theta (ø).

Procedure/Material: In this experiment, a mechanism was created by attaching a long metal pole to a large tripod which controlled a wooden stick that spun in a circular motion at the very top. This wooden stick ran parallel to the ground. A string was attached to one end of the stick which was sent into a circular motion around and around as it was powered by an electric generator which allowed a constant speed for the conical pendulum to rotate. We did this over again with different wattage to give a large speed which we would calculate theoretically and experimentally to prove our relationship.

Before we start with the experiment and run several trials changing the angular speed, we must first measure the total heigh of the mechanism which is taken to be 2 meters tall. We also needed to know the length of the string because it stays constant at 1.654 meters. There is also a rubber stopper at the end of the string just for safety purposes.

This is the mechanism that we were observing and analyzing.

This is the diagram of what the mechanism looks like
and also what we helped use to solve for the different
variables in order to find a relationship.

Once we got the electric generator to turn on we began watching the top of the mechanism spin allowing the rubber stopper attached to the string to begin spinning, as it reached equilibrium and was stable we began recording the times on our phones and the amount of revolutions (complete circle) done with each different wattage of the generator. In order to find the height from the ground of the spinning rubber stopper, we used a metal rod which had a paper taped at the end. We would higher and lower the height of the metal pole in order to get the best height, we knew how high to put it when the rubber stopper just skimmed the paper. This height was then recorded and represented the distance of h. We repeated this process eight times with eight different heights which we found when the professor began to higher the voltage of the generator allowing the wooden stick with the string and rubber stopper attached to rotate faster. This data will help us to solve mathematically for the angular velocity.

This is the tool used to measure the different heights from the ground of the rubber stopper.

After we worked out the data was used to find the results by using an excel spread sheet. We also used some white boards to show the work that we did too. We first used the first diagram that helped us to solve for the angular velocity (W). By using the force diagram we were able to use trig to find the angle at which the string and rubber stopper swung at (ø) by first finding the distance from the pendulum to the rubber stopper vertically and then using the length of the string to derive a trig equation which we took the cosine inverse of to solve for the angle .

This is the work which solved for the angular velocity that we first started with.

After we solved for theta to find the angular speed we had to convert the answer into radians. We then used the proven physics equation to solve for the angular velocity. We used the physics equation  W= 2(pi)/T and compared it to the equation we derived using our mechanism 

W^2= (gsinø)/cosø(d+lsinø). 

This is the work we used to solve for theta (ø) which is a function we derived.

The equation derived will be compared to a physics law that has been proven which is W(angular speed)= 2(pi)/T. Which is shown above.

Now we plotted an equation for the lows and highs of angular velocity using the derived equation and compared them to the angular velocity of the given W=2(pi)/T. In doing so we found below that the slope of the line is .971 which is the period equation (W= 2(pi)/T).

We found that when we solved for the lows and highs of the equation as well, that the results were almost identical. And the slopes of the lines were almost completely equal. 

This the slope of the high equation meaning that the highest error value it could have had. Which we found the slope to be .9824 

This is the slope of the low equation which is the lowest error value it could have had. We found the slope to be .9696.

The slopes show that we were off by only 2 percent from the from the hypothetical value since the slopes are not completely identical. But as each trial went on the graph shows that the angular velocity did increase. The small error we had could have been from measuring the height of the mechanism as it rotated or when we measured the period which involved several decimals which may have cause the error because of rounding. In conclusion, the experiment was successful in finding a relationship between omega(W-angular speed) and theta(ø-angle). As the angular speed increased the angle also increased showing that the large the angle the higher the angular speed of the object and in this case the rubber stopper. 

25 March 2015: Centripetal Acceleration

Lab 8: Centripetal Acceleration

Purpose: Today in class we used a wheel powered by a small motor which was rotating, known as an accelerometer, which we placed against a a flat disk which then spun in continuous circles which helped us to record the data as a piece of tape passed through a photo gate. By doing so we were able to establish a relationship between angular acceleration and time which lead us to establishing a relationship between centripetal acceleration and angular velocity. Ca=mv^2/r. w=2(pi)/t.

Accelerometer is at the farthest left and the photo gate is the small black object that is attached to the pole under the paper towels.

Procedure: Due to time constraints, professor did the lab for the whole class. He first began by placing the accelerometer next to the disk with the taped photogate. By using the small generator to the far left, (next to the scooter wheel) we were able to power the wheel with different hertz of electricity allowing for the wheel to accelerate at different speeds. By using logger pro the photo gate captures the revolution per second of the wheel spinning. We used this data to record the acceleration, period, and angular acceleration.

Angular acceleration vs. acceleration 

This is the data for acceleration,  rotations, period, and centripetal acceleration. 

Results: As we plot our data in the table we find that our results are true and the radius of the disk spinning is nearly identical to the slope of the line, which we find by comparing our acceleration and omega. We find that the radius is .1386 nearly identical to the spinning wheel, off by only a few hundredths, showing that angular velocity and acceleration do share a relationship.