4/13 Magnetic Potential Energy Lab

Purpose: To verify that the conservation of energy theorem applies to this system.

Procedure:

FIGURE (1)

For this experiment, we used an air track glider as our cart on a frictionless surface. (FIGURE (1)) We has a magnet attached to one end of the track as well as on the cart.  We wanted to find the relationship between force and separation distance. In order to do this we collected data as we tilted the track at different angles. We gave the cart a little push and recorded velocity and separation of the magnets. We did this five times and then we graphed our data to get a relationship between force and separation distance (FIGURE (2)).

FIGURE (2)

As we can see in the graph in FIGURE (2), the relationship can be modeled by F(r)=.00007975x^(1.937)

In order to obtain the equation for the potential energy of the magnets, we needed to integrate the equation for F(r). When we integrated it we came up with the equation: U(r)=(85.1*10^-6)x^(-.937).

We then graphed the equation for potential energy along with kinetic energy and the total energy. (FIGURE (3))

FIGURE (3)

Conclusion:

The data from the lab pretty well represents the conservation of energy theorem. The graph that we obtain for energy is very similar to our prediction and out total energy line is relatively flat. Errors in this lab could have come from not pushing the cart from the same initial distance each time, not measuring the correct angle (we has some trouble with our measuring apps in the beginning). There also could have been some energy lost to friction still. 

4/8 Conservation of Energy-- Mass-Spring System

Purpose: To prove the Conservation of Energy Theorem in an oscillating mass-spring system.

Procedure:

FIGURE (1)

We started off by setting up the apparatus in FIGURE (1). We hung a spring attached to a force sensor from a rod a few meters off the ground. Under the spring, we set up a force sensor. For this lab, we needed to figure out the spring constant of our spring. To to this we slowly pulled down on a 50 gram mass while the motion sensor was taking data. This gave us a Force v. Time graph and a Stretch v. Time graph, from which we determined our spring constant to be 16.35 N/m. We then hung a 200 gram mass from the spring and recorded the position using the motion detector which gave us a value of .610 m. This was our in initial position and in LoggerPro, we created a new calculated column, "stretched" where we had the equation: stretch= .610-"position", where "position" was the current position the motion sensor was measuring. We then set up calculated columns to measure kinetic energy (.5mv^2), gravitational potential energy (mgh), elastic potential energy (.5kx^2), and a sum of all the energies.

Once we had the columns set up, we attached a 250 gram mass to the spring, pulled the mass down about 10 cm, and then let go as we recorded our data (FIGURE (2)).

FIGURE (2)

Conclusion:
Our total energy line should have been flat, not wavy, since energy was conserved. Possible sources our error could be a miscalculation of the spring constant, not pulling the spring down enough or too much and using a non-uniform spring.

4/6 Work-Kinetic Energy Theorem Activity

Purpose: To prove the work-kinetic energy theorem.

Procedure:

PART ONE

FIGURE (1)

We started this lab by setting up a cart attached to a force sensor with a horizontal unstretched spring (FIGURE (1)). We had the cart on a track that had a motion sensor on the other end.
Once we had the cart all set up we pulled it slowly towards the motion detector  for about 1 m while LoggerPro was graphing (FIGURE(2)).

FIGURE (2)

We integrated the graph for force in order to find the work done at a certain position.

PART TWO

We used the same set up in the second part as we did in the first part. This time, we measured the mass of the cart (m= .503 kg) and we created a new calculated column in LoggerPro that allowed us to calculate the kinetic energy of the cart at any point. For this part of the experiment we puled the cart to a distance of about 1 m from its initial position and then graphed (FIGURE (3)-(5)) as we let go of the cart and the spring pulled it back to its initial position. 

FIGURE (3)

FIGURE (4)

FIGURE (5)

In FIGURES (3)-(5), we analyzed different points of the graph and we got different integration/work, kinetic energy, force, and position values. 

Conclusion:

The values of work and kinetic energy are pretty similar in values. Friction could account for the difference in the values. Other errors could be related to problems with the motion sensor (we were initially having some issues with our motion sensor.)

PART THREE

Procedure:
For the third part of the lab, we watched a movie in which a professor uses a machine to pull back on a large rubber band. The force being exerted on the rubber band is recorder by an analog force transducer onto a graph, which we recreated (FIGURE (6))

FIGURE (6)

The stretched rubber band is then attached to a cart of known mass. Once the cart is released, it passes through two photo-gates a given distance apart. This will give us the distance and time interval between the front of the cart passing through the first photo-gate and then the second. Once we obtained that data we calculated the final speed and kinetic energy:

v = 3.33 m/s

K.E = 23.9 J

Conclusion:

The kinetic energy we calculated from the area under the graph is off by about 7%. The way the professor took that data in the movie was an old way of finding kinetic energy, so we can expect it to be off a little. We also copied the graph that was shown in the video onto our own paper, which leaves room for more error. 

4/1 Centripetal Force with a Motor

Purpose: To use our understanding of centripetal force and motion to predict and test a relationship between Θ and ω. 

Procedure:
We had an apparatus set up that would rotate at a certain speed. (FIGURE (1)) Attached to the top was a string with a rubber stopper at the end. We also had a ring stand with a piece of paper attached to it. The purpose of this smaller apparatus is to measure the height, h, of the rubber stopper when the apparatus is spinning. The other two values that will change each time we do the experiment are θ and r.

FIGURE (1)

 We ran the experiment 6 times and collected h and T for 10 revolutions each run. With that information we could calculate θ (FIGURE (2)) and ω (FIGURE (3)).

FIGURE (2)

FIGURE (3)

We could also calculate angular velocity using theta (FIGURE (4)):

FIGURE (4)

Conclusion:
From looking at it, the values of the predicted angular velocity and the actual angular velocity look pretty close. Another way i check the relationship between the different values was by graphing them (FIGURE (5))

FIGURE (5)

The ideal would be for the slope of the line in FIGURE (5) to be one. This shows us that we were around 37% off. Errors in this lab could come from not timing the period correctly each trial, errors in any initial calculation, or errors in measurement

3/25 Centripetal Acceleration vs. Angular Frequency

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

Procedure:

For this experiment, the professor had a disk set up to rotate. He put an accelerometer on the disk with one axis pointing toward to center of the disk in order to accurately measure acceleration.We measured the radius of the disk to be 13.9 cm. We spun the disk and used LoggerPro to record and graph our data. We used a photo gate and a piece of tape sticking out from the disk to measure the period of the disk. We ran the experiment six different times, giving the disk a different acceleration each time. The data we took is shown in FIGURE (1) & FIGURE (2) below.

FIGURE (1)

FIGURE (2)

 After we obtained our data, we plotted acceleration vs. angular speed^2 (FIGURE (3)) to test the relationship between angular acceleration and angular velocity (α=rω²). To prove that this relationship is true the slope of the graph shold be the radius of the disk (which we measured to be 13.9 cm or .139 m). The slope of the graph in FIGURE (3) is only 7.34 cm (or .0734 m), which means we were off with measurements or calculations by almost 50%.

FIGURE (3)

Conclusion:
The radius I calculated was not very close to the actual value. I could have made a mistake with my calculations. One error could be with the period in either the third or fourth trial. The time for one revolution in these trials does not make sense. The time for one revolution in the fourth trial is larger than in the third trial even though the fourth trial has a larger acceleration. Another source of error could come from starting to measure the period at a different time each trial. 

3/23 Trajectories

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

Procedure:

FIGURE (1)

We started by setting up an incline on our lab table as a place to launch the ball from (FIGURE (1)). We then did a test run to find out the general area of where the ball would land so we could put carbon paper down in order to catch precisely where the ball lands each run.
We did the experiment five times so we would have a range of landing spots. We can take the mean landing spot from that data, which improves our accuracy slightly.
After we obtained all our measurements, we we able to determine that launch speed at the end of the table (FIGURE (2)) by calculating the velocity in the x-direction.

FIGURE (2)

Before we carried out the next part of the experiment we obtained an equation to calculate the distance d as shown in FIGURE (4) (FIGURE (3)).

FIGURE (3)

 The equation we ended up with needed the initial velocity, which we had previously calculated to be 1.42 m/s, and the angle at which the plank was set up, which we later measured to be 48 degrees.

FIGURE (4)

FIGURE (5)

For the next part of the experiment, we set up a plank going from the end of the table to the floor (FIGURE (4)). We did another test run in order to figure out the best place to place the carbon paper on the board and we measured the angle the board made with the ground. We again got multiple values for our distance d (FIGURE (5)) and we found the mean value to be 75.7 cm (.757 m). Earlier, we had calculated the theoretical value of d to be .683 m.






Once we had the experimental and expected value of d, we calculated the uncertainty. In order to do this, we needed our earlier equation for d (FIGURE (3)) to be in terms of x, y, and alpha. (FIGURE (6)).

FIGURE (6)

After we obtained our final equation for d we were able to calculate the uncertainty (FIGURE (7)).

FIGURE (7)

Conclusion:
Our experimental and expected values of d are off by .074 m with an uncertainty of .0045. There are a few possible sources of error in this experiment. One is error in measurement. Besides the obvious sources of error in measurement, such as just reading the wrong number off the ruler, we also could have made a mistake with the carbon paper measurements. For the first part of the experiment we did not realize that there was a certain side of the carbon paper that we were supposed to use. We were surprised by the small difference that the landings had shown each run. We learned this in the second part of the experiment because we were not getting any mark on our carbon paper while it was on the plank. Another source of error may have come from movement by the plank in the second part of the experiment. Even with weights at the end of it, the plank still seemed a little unsturdy. We tried to prevent movement by holding the weights and board in place ourselves. Of course there are always random errors as well.

3/18 Modeling Friction Forces

Purpose: To determine static and kinetic frictional force using different experiments. 

Procedure:

PART ONE: Static Friction

FIGURE (1)

For the first part of the lab, we were determining static friction between a block and the tabletop. We did this by attaching a pulley to the table attaching a string to the block and a cup. We hung the cup off the edge of the table while the block sat on top of the table. We then proceeded to add water to the cup until the block started to move just a little. That meant that the weight of the cup and therefore tension in the string was great enough to overcome the static friction. ( f_static≤µ_sN) By obtaining the mass of the cup we were able to calculate the maximum force one can apply before overcoming static friction (FIGURE (1)). We did this process four times with four different masses and we recorded the mass of the cup with water each time. We then calculated the normal force and frictional force created a Friction v. Normal graph (FIGURE (2)). 

CORRECTION IN FIGURE (2): The mass of the blocks was entered incorrectly. The data should read from top to bottom: 131, 252, 359, 485. 

FIGURE (2)

The slope of this graph gives us the coefficient of static friction: µ_s=.3475 ±.01287

PART TWO: Kinetic Friction

For the second part of the lab, we were determining the kinetic frictional force between the blocks and the tabletop. This time, we had the block sitting on top of the table with a string attached to it. We set up a force probe and tied it to the other end of the string. We then pulled the force probe horizontally while collecting data in LoggerPro (FIGURE (3)). We did this three additional times with larger block masses each time. 

FIGURE (3)

After we finished collecting or data we plotted it on a graph again. (FIGURE (4)) The slope on this graph again gives us the coefficient of static friction: µ_s=.2712 ± .009813. We then know that the coefficient for kinetic friction is anything greater than that. (f_kinetic=µ_kN). 

FIGURE (4)

PART THREE

For the third part of the lab, we placed a block on a horizontal piece of plywood and lifted one end (the one closest to where the block sat) until the block started to slip. We then measure the angle where that occurred. The calculations to find the coefficient of friction are shown in FIGURE (5).

FIGURE (5)

PART FOUR

In the fourth part of the lab we did a similar procedure as part three but we measured speed with a motion detector. From measuring speed, we can calculate the kinetic friction between the surfaces by obtaining acceleration from our graph (FIGURE (6)). We know that the acceleration would be the slope of the velocity graph. Therefore, the acceleration is .7825 m/s^2.

FIGURE (6)

Once we have the acceleration we can use the angle and acceleration to calculate the coefficient of kinetic friction. The steps to find µ_k are in FIGURE (7).

FIGURE (7)

PART 5

In the first part of the fifth part of this lab, we were to derive and equation for acceleration using the coefficient of kinetic friction from part four. The work for that is shown in FIGURE (8):

FIGURE (8)

FIGURE (9)

The second part of the experiment was to calculate the mass that would be required to overcome static friction and move the block, if the unknown mass was attached to a string and to the block and then hooked up to a pulley (FIGURE (9)). After we found the unknown mass we could plug the into our equation for acceleration and get an estimate for acceleration as well. The work for this is shown in FIGURE (10).

FIGURE (10)

When we tried the experiment with our calculated mass (we had to use 40g), the block moved a little bit but it was not a steady movement. We got a steady movement at 50g, which probably means that the precise mass is somewhere between 40g and 50g. We also set up a motion sensor so we could obtain a velocity v. time graph as well as the acceleration. (FIGURE (11))

FIGURE (11)

The graph shows that the acceleration of the block was .573m/s^2. This value is not very to the one we calculated. Errors that may have attributed to this include errors in initial measurements such as the angle measured in part three. This would have led to an error in out calculation of the coefficient of friction and an even bigger error in part 5 calculations. When we were doing the experiment, our motion sensor was also acting very strange. It would take measurements differently each time, and our initial test looked completely different from our final test (shown).

Conclusion:
Overall, the purpose of this lab was fulfilled. We demonstrated everything that we had learned in class about friction and most of our calculations and experimental results were accurate. There were many possible errors in this lab since we did five different experiments. The possible errors may have included: (part (1)) we could have added to much water to the cup. This would obviously lead to a great mass of the cup and therefore a greater frictional force. When we were adding drops to the cup we were not adding drop by drop by drop, because that would take a long time and we did not have that much time in lab; (part (2)) When we were pulling the force sensor we could have pulled it at a slight angle instead of parallel to the table. The force sensor also could have been calculated inaccurately and we would lose accuracy. Also, it it highly doubtful that we were able to pull the blocks at a constant speed and the same speed every time we took new data.; (part (3)) As previously stated, an error in the part could have been the angle we calculated. We could have read the measurement incorrectly or we could have stopped at an angle greater than when the block first started to slip. This would give us a greater frictional value and coefficient of friction.; (part(5)) we could have recreated the 20 degree angle inaccurately which would give the block greater acceleration and therefore a different coefficient of friction.; (part(5)) As previously mentioned, our previous measurements could have given us inaccuracy in this part of the lab. Also, there was the problems with our motion sensor.