15-April-2015: Magnetic Potential Energy Lab

Lab 12: Magnetic Potential Energy

Purpose: To verify that the conservation of energy exists within a magnetic system.

Apparatus:  For this experiment,we used a glider on an air track. The glider had a magnet attached to it and another magnet was fixed to one end of the track, we positioned a motion sensor at this point.

Air track with vacuum.

 Procedure:  In order to prove the conservation of energy in this system, we must calculate the KE of the glider and compare it to the magnetic PE between two repelling magnets. Since an equation for PE between two magnets does not exist, we must find one. We achieved this by conducting several runs, where we raised the track to different angles. From here, the glider would travel along the track and stop at distance away from the fixed magnet. This is due to the magnet attached to the glider. The the glider would stop due to the repelling forces between the magnets. At this point, we measured the angle of elevation (theta) and the separation distance between the magnets. We gathered a sufficient set of data so that we could derive an expression for the PE between two magnets. The rest of the lab was fairly simple. We leveled the track and calculated the KE of the glider with the aid of the motion sensor. Then we determined if energy was conserved by graphing the KE of the glider, our expression of PE between the magnetics, and the sum of the energies. If energy is conserved, the sum of the energies should be constant.

Run 1

Finding PE Magnetic 

In order to find PE Magnetic, we must derive an expression for the force acting on the glider based on the separation distance between the magnets F(r).

 

First, we found the forces acting on the glider by drawing a FBD.

• Fx (along slope): F=mgsin(theta)
• Fy: N=mgcos(theta)

Note: We assume the track is frictionless.

 

As mentioned above, several runs were conducted at different angles. We collected theta and separation distance(r) for each run.

 

Mass of cart (measured) = .347 kg

Theta was measured with phones.

For F, theta was converted into radians.

 

Run 1

• Theta = 3 +/- 0.1 degrees
• r = .036 m
• F =  0.178 N

Run 2

• Theta = 6.8 +/- 0.1 degrees
• r = .018 m
• F =  0.403 N

Run 3

• Theta = 11.6 +/- 0.1 degrees
• r = .036 m
• F =  0.684 N

Run 4

• Theta = 14.7 +/- 0.1 degrees
• r = .011 m
• F =  0.863 N

Run 5

• Theta = 19.0 +/- 0.1 degrees
• r = .0085 m
• F = 1.107 N

Run 6

• Theta =25.0 +/- 0.1 degrees
• r = .0065 m
• F = 1.437 N
   

F was calculated in LoggerPro, under a calculated column using F=mgsin(theta)

 

From here, we found an expression for F(r) by graphing  Force vs r on LoggerPro.

• x-axis: Force (N) = mgsin(theta)
• y-axis: r = separation distance

We plotted the points and took a power fit of the graph to find F(r). 

 

As you can see, our function is:  F(r) = .004344r^-1.157

Finally, we can find a function for the PE between the magnets [U(r)] simply by taking the integral of F(r)*dr.

• U(r) = 0.02766879r^-0.157

 

Now, we can finally verify the conservation of energy within the system.

First, we leveled the track and then placed a motion sensor near the fixed magnet at the end of the track. Then we ran a test run to determine the relationship between the distance the motion sensor reads and the separation distance between the magnets.
Next, we recorded and calculated the following on LoggerPro:

• Time
• Position of cart
• Velocity of cart
• Separation Distance: "Position"- 0.25
• KE of cart: (1/2)m*v^2         m=.347 kg (measured)
• Umag: U(r) = 0.02766879r^-0.157    where r = separation distance
• Total Energy: KE + Umag

We predict that the energy graphs will mirror each other. This means that KE will decrease to zero and then increase to its original value; while Umag should be nonexistent until KE = 0. Therefore, if we graph Total Energy, it should look like a straight line.

Results:

Position and Velocity graphs

KE=Orange, Umag=Red,Total Energy=Blue

As you can see, our graphs are slightly off. The error seems to originate from Umag, since KE is constant and reaches zero. However, we still proved that energy is conserved within the system.

Conclusion:
In this lab, we verified the conservation of energy within the system to a certain degree. First, we determined a function for the forces acting on the cart based on the separation distance of the magnets. Then, we calculated the PE, U(r), within the magnets and the KE of the cart. We compared the graphs of U and KE over time. Finally, we calculated the Total Energy of the system and determined whether energy was conserved. Unfortunately, our U graph was slightly off from the predicted result. This is due to the uncertainty in the power fit of the F vs. r graph. Furthermore, there is also uncertainty in our measurements of magnet separation. Lastly, there could also be a systematic error since we assumed that the air track was frictionless.



13-April-2015: Conservation of Energy Lab

Lab 11: Conservation of Energy--Mass-Spring System

Purpose: To show the conservation of energy in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.

Setup:

 As seen above, a force sensor is set so that a spring can hang from it without interference from the table. A motion sensor is set on the floor to measure the spring's movement.

As seen above, a spring is hung from the force sensor. H and y will be used to determine the spring's unstretched position and the GPE of the spring.

 

Pre-Lab: As a class, we identified the types of energy that will be present during the experiment and we derived equations for each of them. We will need this equations in order to determine the total energy of the system.

• KE(hanging mass) : (1/2)m*v^2
• GPE (hanging mass):  Position (measured by motion sensor)*m*g
• EPE: (1/2)k* stretch^2
• GPE (spring): (Position*Mspring*g)/2
• KE (spring): (Mspring*v^2)/6

How we determined GPE Spring:

First, we chose a representative piece (dm) of the spring. Next, we wrote an expression for the GPE of that piece. Then, we summed the GPE of all the pieces of the spring from y to H. Finally, we solved the integral to find our equation for the GPE Spring.

 

The same process was used to find the KE Spring:

 

 

Essentially, our job for this lab is to calculate all of the energies shown above. Once we achieve this, we then graph the sum of all the energies and based on this graph,we determine if energy is conserved.

 

Procedure: In the first part of the lab, we found the spring constant of the spring.

• Mass of spring (measured) = 64 +/-.1 g
• Unstretched position of spring (measured)= 48.5+/- .1 cm

Determining the Spring Constant

 

First, we calibrated the force sensor with a 1 kg mass and reversed the direction of the motion sensor. We then hung the spring on the force sensor, and zeroed it. Next, we attached a 50g mass to the spring and pulled on it, while collecting data. We did this to verify that the sensors were setup properly and that LoggerPro was able to plot the data.

 

Next, we followed the same steps and recorded data. We collected Force vs. Time and Stretch vs. Time data for the oscillating spring with an additional 50g mass. These graphs are needed in order to calculate the spring constant of the spring.

Graph 1:

• x-axis: Force (provided by force sensor)
• y-axis: Time

Graph 2:

• x-axis: Stretch (Position measured by motion sensor - Unstretched position)
• y-axis: Time

 

We then graphed Force vs Stretch to determine the spring constant of the spring. The spring constant would be the slope of the Force vs Stretch graph. 

• x-axis: Force (provided by force sensor)
• y-axis: Stretch (Position measured by motion sensor - Unstretched position)
• Calculated Spring Constant: 14.86 N/m

In the second part of the lab, we added mass to the spring and recorded data while it was oscillating. Then, we calculated all of the energies involved and determined if energy was conserved.

Conservation of Energy

Energies involved: 

• KE(hanging mass) : (1/2)m*v^2
• GPE (hanging mass):  Position (measured by motion sensor)*m*g
• EPE: (1/2)k* stretch^2
• GPE (spring): (Position*Mspring*g)/2
• KE (spring): (Mspring*v^2)/6

Next, we added 250 grams to the spring. The hanging mass is now 300g. Then, we followed the same procedure; we pulled on the spring and recorded data. The equations shown above were applied to calculated columns on LoggerPro and were then plotted. We also created a new column to include the sum of KE, GPE, EPE.

• Note: When KE=0 and EPE=0, all energy is found in GPE

 

  From Left to Right: Time, Force, Position, Velocity, Acceleration, Stretch, KE, EPE, GPE, KE Spring, GPE Spring, Total Energy

The graphs for all the energies can be seen below:

 

Finally, we ran the experiment one last time. This time we only graphed Total Energy vs. Time

 

Bottom Graph: Energy Total vs. Time

As you can see the graph has a constant oscillation. Ultimately, this means that energy is conserved since no energy was lost or gained.

 

Conclusions:

In this lab, we proved the conservation of energy in a mass-spring system. First, we determined the spring constant of the spring by graphing Force vs. Stretch. Then, we calculated the various energies of the hanging mass and the spring. Finally, we graphed the energies and noticed small patterns. The main one being that the sum of the energies has a constant oscillation. This proves that energy is conserved within the system, since no energy was lost or gained.

 

Uncertainties: This experiment heavily relies upon the use of force and motion sensors. This means that our values our affected by the precision of the instruments. Furthermore, human error also comes into play when measuring the unstretched position of the spring. In this case, the uncertainty is +/-.1 cm. This could have an effect on the value of the spring constant. Human error can also be at fault, when analyzing the data on LoggerPro. Ultimately, the concept behind the lab is sound and the uncertainties were negligible to the point that it did not drastically affect the final result. 

6-April-2015: Work-Kinetic Energy Theorem Lab

Lab 10: Work-Kinetic Energy Theorem Activity

Purpose: To understand and apply the Work-KE Theorem and to determine the work done on an object from a Force vs. Position graph.

Experiment 1: Work Done by a Non-constant Spring Force

Setup:

Above, you can see a track on a horizontal surface. On one end of the track, there is a motion sensor. On the other end, we positioned a force sensor and it is held up by a rod and a C-clamp. A spring is attached to the force sensor and to the end of the cart.

In this portion of the lab, we measured the work done on a stretched spring through a measured distance. First, we collected data for the force applied by a stretched spring vs. the distance the spring is stretched and plotted the results.

Procedure:
First, we calibrated the force sensor with an applied force of 4.9 N. Then, we did a test run to see if the motion sensor recorded the cart's position. Once this was achieved, we zeroed the force sensor and motion detector. We also reversed the direction of the motion detector, so that toward the sensor is the positive direction.
Now, for the experiment. We began graphing force vs. position, as the cart was moved slowly towards the motion detector. Logger Pro then recorded the force applied and the cart's position.

Data:
The data was plotted on a Force vs. Position graph.

• x-axis: Force Applied (N)
• y-axis: Position (m)

 

The picture above shows two Force graphs because we mistakenly collected data for 2 runs. For the purpose of the lab, only one graph was analyzed.

 

We then found the work done in stretching the spring by taking an integral of the graph.

• Work Done (Area under graph) = 0.1520 Joules

We were also able to determine the spring constant from the graph. This was achieved by finding the slope of the graph.

• Slope of Force vs. Position graph: Spring Constant = 6.969 N/m  

 

 

Experiment 2: KE and the Work-KE Principle

 

Setup: Same as in Experiment 1.

 

In part 2 of the lab, we will examine the work done by the spring and the change in KE of the cart.

 

Procedure: 

Procedure is basically the same as in Part 1, but a few changes were made. First, we added a Kinetic Energy graph by creating a new calculated column that would calculate the KE of the cart at any point.

• Measured mass of cart = .504 kg
• KE=1/2(m*v^2)

 The second, data was now taken after releasing the cart. We pulled the cart back and began graphing after we released the cart.

Data:

 

Red Graph: Position vs. Time
Green Graph: KE vs. Time

Like in Part 1, we found the work done by the spring by finding the area of the graph between two positions . We also noted the change in KE at this point. Next, we found the change in KE at different positions and compared it to the work done at that point.

• At x = .195 m : KE=.453 J and W=.432 J
• At x = .259 m : KE=.365 J and W=.344 J
• At x = .364 m : KE=.165 J and W=.152 J

Conclusions:
The work done on the cart by the spring and the change in kinetic energy are essentially the same. Our values do not match because the force sensor was not set to zero. As you can see, our Force graph does not start at zero. In conclusion, the work done on an object is conserved into Kinetic Energy. This can be seen in the experiment. As the work done on the spring system was conserved and then interpreted as the change in Kinetic Energy of the cart.

Experiment 3: Work-KE Theorem
This portion of the lab involved watching a video clip. In the clip, a professor uses a machine to pull back on a large rubber band. The force being exerted on the rubber 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 cart's final speed and the final KE of the cart.

Essentially, we are finding the area under the Force vs. Position of the rubber band and the KE of the cart. Based on the Work-KE Theorem, the answers should match.

In order to find the area under, we broke it down into 4 different shapes: triangle, rectangle, and two trapezoids.

• Area of Triangle: (1/2)b*h = 9.18
• Area of Rectangle: b*h = 7.48
• Area of Trapezoid: (1/2)(b1+b2)*h = 2
• Area of Trapezoid: (1/2)(b1+b2)*h = 7.935
• Total Area under graph = 26.595 J

Then, we were given the following information:

• m cart= 4.3 kg
• t= .045 sec
• change in position (x)= 15 cm
• v= x/t
• KE=(1/2)m*v^2

Velocity = .15/.045 = 3.33 m/s
KE = (1/2)(4.3)(3.33)^2 = 23.9 J

As you can see the result for Work and KE are off. KE = 23.9 J and Work = 26.595 J. This can be the result of the uncertainties of the experiment. Essentially, the uncertainties of this experiment come from the precision of the instruments used. The experiment depends upon the accuracy of the analog force transducer and the photogate. The concept of how we solved for work and KE is correct, but due to imprecise readings our answers are off.

01-April-2015: Centripetal Force with a Motor

Lab 9: Centripetal Force with a Motor Lab

Purpose: To determine a relationship between Angle of Rotation (theta) and Angular Speed (w).

Apparatus: 

Here we see, an electric motor mounted on a surveying tripod. There is a long shaft going vertically up from the motor. A horizontal rod is mounted on the vertical rod. A long string is tied to the end of this horizontal rod. A rubber stopper is tied to the end of the string. Essentially, this is a swing ride at an amusement park. As the motor picks up speed, the angle of rotation will increase.

 

Above, you see a ring stand with a horizontal piece of paper. This was used to measure, the vertical distance from the ground to the rubber stopper. The piece of paper was raised until the stopper just grazed it as it passed by.

 

 

Required Measurements:

• H = 2 m +/- .001
• L = 1.664 m +/- .001
• R = .87 m +/- .001
• theta: varies depending on angular speed
• h: varies depending on angular speed

To find theta, we used:

• theta = arccos((H-h)/L)

In order to determine a relationship between theta and omega, we collected values of h at a variety of values of w. So, for each run we calculated theta  and omega. These values will then be used in two expressions found  below. In which, we compared a theoretical value of omega to the experimented value of omega. Here, the theoretical value of omega depends upon theta. Now, by graphing these values we will be able to determine a relationship between omega and theta based on the slope of the graph.

Collecting Data:
 In all, we had six runs. We collected theta, omega, h, and period (T) for each run.

• h (m): measured height when stopper hit paper
• period (s): Time for 10 rotations/ 10
• theta (degrees): arccos((H-h)/L)
• omega (rad/sec): (2*pi)/T

Run 1:

• 37.67 sec for 10 rotations, T = 3.767 sec
• w = 1.67 rad/sec
• h= .473 +/- .005 m
• theta = 23.4

Run 2:

• 32.75 sec for 10 rotations, T = 3.275 sec
• w = 1.92 rad/sec
• h= .624 +/- .005 m
• theta = 34.2

Run 3:

• 28.04 sec for 10 rotations, T = 2.804 sec
• w = 2.24 rad/sec
• h= .850 +/- .005 m
• theta = 46.3

Run 4:

• 23.01 sec for 10 rotations, T = 2.301 sec
• w = 2.73 rad/sec
• h= 1.185 +/- .005 m
• theta = 60.7

Run 5:

• 19.30 sec for 10 rotations, T = 1.930 sec
• w = 3.26 rad/sec
• h= 1.408 +/- .005 m
• theta = 69.2

Run 6:

• 15.30 sec for 10 rotations, T = 1.530 sec
• w = 4.11 rad/sec
• h= 1.610 +/- .01 m
• theta = 76.4

To recap:

Analyzing Data:

Next, we setup a FBD for the rubber stopper.

We solved for forces in the y and x-direction.
R=.87 m +/- .001
Theta: Calculated
w:Calculated

• Fx: Tsin(theta)=m*R*w^2
• Fy: Tcos(theta)=mg

Then we solved the Fy equation for w and the Fx equation for T. We combined them by plugging in T and simplified the equation.

• w = sqrt((g*tan(theta))/(R+ Lsin(theta)))

Now, we can setup a relationship between w and Theta by graphing w vs. w.

• Theoretical Values = y-axis: w (theta) = sqrt((g*tan(theta))/(R+ Lsin(theta)))
• Experimented Values = x-axis: w =  (2*pi)/T

By graphing this setup, we are implying that w and theta are proportional to one another. This means that the slope of the graph should equal 1 or something very close to it.
These are the points for our graph:

 

We plotted the points and took a proportional fit of the graph to determine the slope.

• Slope: .9935 +/- .006345

 As you can see the slope of our graph is .9935. This shows us that we are about 1.3% off from theoretical values because they are not exactly proportional. This is due to the amount of uncertainty from measuring the height of the stopper as well as measuring the periods. Overall, we determined that as angular speed increases, the angle of rotation will also increase.