Lab 15(14th): 2D Collision

Title: Collisions in Two Dimensions

Purpose: Determine whether or not momentum and total energy are conserved in a collision between two objects.

Theory:
During collision of an object, assuming no external forces, overall direction and magnitude of momentum is expected to be conserved. Kinetic energy is not expected to be conserved as the collision is non-elastic. These hypothesis can be tested by taking velocity measurements of the two objects prior to collision and after collision. Total momentum and kinetic energy can then be calculated according to the equations p = mv and Ek = 1/2 mv^2 and initial and final values can be compared.

Apparatus:

• Smooth glass panel
• Glass marble (2)
• Steel ball bearing
• Slow-motion video recording device
• Camera positioning rig
• Meter stick

Procedure:

Level the glass panel by adjusting the legs beneath the panel so that a ball remains stationary when placed. Position the video recording device in the rig set up above the glass panel so that most if not all of the glass panel is in frame. Position a meter stick on the glass panel within the video frame. Begin recording at at least 120 frames per second and collide the steel ball bearing with a stationary marble. Perform the experiment again by colliding two of the same type of marble. Perform video analysis using LoggerPro, skipping 2-4 frames for every data point to compensate for the increased FPS. Define an axis as parallel as possible to the initial motion of the moving ball prior to collision.

Data and Graphs:

Photo of experimental setup and the method used to produce data of position and velocity

Graphs of x and y velocities for a steel ball colliding with a marble

Graphs of x and y velocities for two marbles colliding

Table of x and y velocities, calculated change in momentum, and change in kinetic energy

Mmarble	5,2	g
Msteel	28	g
Δp = m(vf-vi)
Velocity (m/s)
Experiment 1	Steel		Marble
	X1	Y1	X2	Y2
Initial	0,3190	0	0	0
Final	0,2817	-0,0281	0,08207	0,1310
Δp (kgm/s)	-0,00101		0,000804
ΔKE (J)	-0,101		6,21E-05
Experiment 2	Marble 1		Marble 2
	X1	Y1	X2	Y2
Initial	0,5181	0	0	0
Final	0,1499	-0,1233	0,2804	0,09052
Δp (kgm/s)	-0,00168		0,00153
ΔKE (J)	-0,268		0,000226

Data Analysis:

Using position graphs produced by video analysis, x and y velocities for the two balls in each experiment were found by using LoggerPro's linear fit option for graphical analysis. The slope of each graph was taken to be its velocity in that direction of motion. The magnitude of each ball's initial and final velocity was then found and used to calculate change in momentum and change in kinetic energy. 

Conclusion:

Momentum is not entirely conserved during the collision as the stationary object experienced a smaller change in momentum than did the initially moving object. In the first experiment, the second ball experienced a change in momentum of +0,00804kgm/s while the first ball experienced a change in momentum of -0,00101kgm/s, an approximately 21% difference. The second experiment produced more similar changes in momentum at -0,00168kgm/s for the first ball, giving a difference of approximately 9%.  Kinetic energy was not conserved as expected with the first ball losing 0,101J of kinetic energy and the second ball gaining only 0,0000621J of kinetic energy. 

Lab 14(13th): Impulse-Momentum

Title: Impulse-Momentum activity

Purpose: Verify that the impulse-momentum relationship holds true in a variety of situations.

Theory:
Impulse and momentum are directly related to each other where Ft=mv for an object such that some value of impulse will produce the same numerical change in momentum on an object. This relationship can be derived using kinematics equations where a given force will accelerate an object of mass m at a rate of F/m, the total change in velocity over a time interval t will be equal to (F*t)/m, and then the change in momentum is m((F*t)/m) which equals F*t, or impulse. By independent measurements of force, time, mass, and velocity of an object, impulse and momentum may be calculated and compared to verify their relationship in elastic and inelastic collisions.

Apparatus:

• Spring loaded cart
• Cart with force sensor attachment
• Cart track
• C-clamp
• Clamp
• Metal rod 
• Motion detector

Procedure:

Level the cart track horizontally with one end near the edge of a table. Measure the mass of the cart with the force sensor attached. Using the C-clamp, metal rod, and clamp, secure the spring loaded cart at one end of the cart track so that the force sensor on the other cart will collide with the extended spring. Position the motion detector behind the force sensor cart. Lightly push the cart and collect data for position and force vs time. With the same setup, add more mass to the cart and collect the appropriate data. Then, replace the spring loaded cart with a clay mass and attach a sharpened screw to the force sensor. Push the cart so that it collides and sticks into the clay while collecting the appropriate data.

Data and Graphs:

Photo of the experimental setup

Mass of cart, initial and final velocities used to calculate change in momentum

Mcart	0,648	kg
Mcart2	1,148	kg
Velocities (m/s)
Experiment 1
Vi	Vf
0,421	-0,296
Experiment 2
Vi	Vf
0,404	-0,262
Experiment 3
Vi	Vf
0,619	0

Graph of Force vs time and Velocity vs time for first collision, integral of force taken to find impulse

Graph of Force vs time and Velocity vs time for faster elastic collision

Note: graphs not aligned by Time axis

The clay target used to simulate an inelastic collision

Graph of Force vs time and Velocity vs time for inelastic collision with clay

Comparison of calculated momentum and force integrals from graphs

Δp=m Δv %error = (true value - experimental value)/true value *100 Calculated momentum (kgm/s) Force integral (Ns) % Error Experiment 1 -0,465 -0,4866 -4,52 Experiment 2 -0,765 -0,8252 -7,35 Experiment 3 -0,401 -0,4070 -1,45

Data Analysis:

The initial and final velocities of the carts for each experiment were found by locating the points at which a force was exerted and the points at which the force stopped being exerted. From these values, the change in momentum was calculated according to the equation Δp=m Δ. Percentage error values were calculated using the force integral given by LoggerPro as the "true" value.

Conclusion:

For the first and third experiments, the percentage error between the force integral and the calculated momentum were both under 5% while the second experiment had a larger error at -7.35%. Overall, the calculated moments were all slightly lower values compared to impulse. Because the duration of each collision was very short at approximately 0.2 seconds for elastic collisions and 0.1 seconds for the inelastic collision, friction does not have a significant distance to act over and is thus negligible. The lower values of change in momentum may then be due to inaccuracies in finding the initial and final velocities. The force sensor was capable of producing a larger number of samples per second than the motion detector. This means that the force graphs will produce more accurate results as the entire collision will be more accurately described. In terms of finding the velocity of the object, this means that a corresponding point on the force graph may not have an exact time equivalent on the velocity graph, leading to uncertainties in calculated momentum. Assuming uncertainties in velocity of +-0.01m/s and an uncertainty in mass of 1g, the propagated uncertainties are shown below.

(dp/p) = (dm/m) * ((dv1/v1) + (dv2/v2))
dp = p ((dm/m) * ((dv1/v1) + (dv2/v2)))
Uncertainty in Momentum
Experiment 1
0,00719
Experiment 2
0,00893
Experiment 3
0,0100

The only value of impulse to fall within the uncertainty range of momentum is that of experiment 3, suggesting that is it probable that at least one of our idealized assumptions is false, that our method of data collection is inadequate, or that our experimental setup is flawed. 

Lab 13 (12th): Magnetic Potential Energy

Title: Magnetic Potential Energy Lab

Purpose:
Determine a relationship for the repulsive force between two identical magnets while also verifying that conservation of energy applies to magnetic fields.

Theory:
Forces due to gravitational and electric fields follow an inverse square law. Since magnetic forces also behave similarly in that they exert a force in a field, it is believed that a magnetic force should also behave according to some inverse power of distance in the form F=C(r^n) where C is an experimental constant for some particular situation, r is the distance between the magnets, and n is an experimental constant. By graphing measurements of force on one magnet and the resulting distance between the magnets, a model can be produced to predict the magnetic force at various distances r. As our experiment involves repulsive magnetic forces, we may test conservation of energy by graphing the motion of an object as it moves towards the magnet and finding the velocities at which it approaches and then retreats after being repelled.

Apparatus:

• Strong magnet (2)
• Air track and cart
• Motion detector
• Digital scale
• Angle measurement device

Procedure:

Attach one magnet to the cart and one magnet to one end of the air track, facing towards the cart. Tilt the air track and measure its degree of inclination. With the air track on, release the cart and allow it to reach an equilibrium position. Measure the distance between the magnet on the cart and the magnet attached to the track. Repeat these steps for a total of 6 different angle measurements. Level the track and orient the two magnets towards each other at some relatively short distance. Measure the distance between the motion sensor and the area of the cart from which position is measured and measure the distance between the two magnets. Subtract the distance between the magnets from the distance between the cart and the motion detector to obtain a value used to convert position as measured by the motion sensor to distance between the magnets. With the air track on, slide the cart towards the fixed magnet and produce a graph of velocity vs time.

Data and Graphs:

Photo of experimental setup

Sample calculation of gravitational force equated to magnetic force

Fgrav = Fmagnet = Mcart*G*sin(θ) = 0.354*9.81*sin(0.035) = 0.118N

Sample calculation of magnetic potential energy Umagnet

Umagnet = integral F(r)dr = integral (2.729*10^-5)/(r^-2.371) = (-2.29*10^-5)/(1.371*(r^-1.371))

Sample calculation of distance between magnets r, used to create PE graph

r = measured position (from detector) - constant value = 20cm - 10.6cm = 9.4cm

Table of collected data and calculated values

Mass	345	g
θ (°)	r (cm)	θ (rad)	Fgrav (N)
2,0	2,86	0,035	0,118
3,8	2,18	0,066	0,224
5,9	1,86	0,103	0,348
6,9	1,73	0,120	0,407
8,8	1,54	0,154	0,518
11,3	1,40	0,197	0,663

Graph of Force vs radius

Graph of kinetic, potential, and total energy (above) and of velocity (below)

Data Analysis:

The graph of force vs radius produced the experimental constants for our equation for magnetic force. Using LoggerPro's power fit ability, an equation was produced for the curve. The constant value C is equal to 2.729*10^-5 and the power n is equal to -2.371. Using the new equation of F=2.729*10^-5r^(-2.371), the total magnetic potential energy was calculated using the integral of force with respect to distance. From the velocity graph, a graph of kinetic energy vs time was produced. Using our equation for magnetic potential energy, a graph of potential energy vs time was produced. Total energy was then found by adding kinetic and potential energy. 

Conclusion:

The model produced for magnetic force was F=2.729*10^-5r^(-2.371) where r is the distance between two repelling magnets. This matches our prediction of an inverse power relationship between magnetic force and distance. Using this equation, the total potential energy shown in the first graph was calculated at  4.17*10^-3J. Based on the graphs produced of KE, PE, and total energy, it was concluded that energy is conserved in magnetic potential energy since no sudden decrease in total energy was observed. The gradual decrease in total energy is due to slight friction and air resistance as the cart moved along the track. Although calculations were not made to determine the exact total energy before and after the magnetic repulsion, it is assumed that these frictional forces are responsible for the slight decrease in total energy during the magnetic repulsion. 

Lab 11: Work-KE

Title: Work-Kinetic Energy Theorem Activity

Purpose:
The purpose of this lab is to determine the relationship between work and kinetic energy based on measurements of force and distance traveled.

Theory:
Work is represented by the equation W=Fx where a force F acts parallel to some displacement x. Work is also a measure of the change in the internal energy of an object where work done on an object increases its internal energy and work done by and object decreases its internal energy. If the gravitational potential energy of an object is held constant by maintaining the same height relative to its initial position, the work done on an object can be assumed to approximately equal the change in kinetic energy if losses in energy due to non-conservative forces are minimal. In a similar way, work is also equal to change in spring potential energy where if gravitational potential energy and kinetic energy are held constant, the work done on a spring is equal to the change in spring potential energy.

Apparatus:

• Low friction cart and track
• Low friction pulley
• String
• Force sensor
• Motion detector
• Digital scale
• 100g mass (1)
• Small spring

Procedure:

Attach the force sensor to the cart and point the motion detector towards the cart's direction of motion. Zero the force sensor. Simulate a constant force by hanging a 100g mass from a string attached to the force sensor and produce graphs of force and velocity over time. Calculate values for kinetic energy according to the equation KE=1/2mv^2 and layer a graph of kinetic energy over a graph of force. Taking the integral of the force graph should produce the calculated value of kinetic energy at the given time interval. Conduct this setup and graphical analysis again for a system where instead of the 100g mass, a spring is attached to the force sensor and anchored at the other end and the cart is slowly pushed at a constant velocity. Again, the integral of the graph of force vs time should equal the predicted spring potential energy. The third setup will be identical to the second, however instead of slowly moving the cart, the cart is to be pulled to some distance, stretching the spring, and is then to be released. Repeat the same process of graphical analysis.

Data and Graphs:

Mass of the cart: 0.667kg

Photo of first experimental setup

Photo of data collected from first experiment

Graphical analysis using integrals of force as a function of time

Experimental setup for 2nd and 3rd experiments

Graphical analysis of data from 2nd experiment

Data Analysis:

Graphical analysis was carried out as described in procedures. Integrals of force vs time were taken and compared to calculated values for kinetic energy.

Conclusion:

In the first experiment, calculated kinetic energy and integral of force were taken at two different times. For the first selected time, the integral of force was 0.141J while the calculated kinetic energy was 0.130J. At the second selected time, the integral of force was 0.219J while the calculated kinetic energy was 0.196. This difference is most easily explained by the presence of friction forces which were assumed negligible for our experiment. This is because while the total force acting on the object remains mostly constant, some energy is lost as heat due to friction and thus lowers the actual velocity and kinetic energy of the object. For this experiment, a value of kinetic energy produced by the integral of force is considered the "true" value of kinetic energy if our system satisfied our assumptions. The value of kinetic energy calculated from position vs time is then the actual value which accounts for friction and any other counteracting forces. Similar results were obtained for both other experiments. Based on the overall results, work appears to be directly related to change in kinetic energy under conditions where all other internal energies of the object are held constant. 

Lab 10: Work and Power

Title: Work and Power

Purpose:
The goal of this lab is to physically demonstrate then calculate work and power first by lifting an object directly upwards then by walking and running up stairs.

Theory:
Work is a function of force exerted in the direction of motion over some distance. Power is a function of work per unit time. By lifting a known mass a known distance while also timing the process, work and power may be calculated.

Apparatus:

• Meter stick
• Stopwatch
• 9kg weight
• Rope
• Long wooden plank
• Pulley

Procedure:

Have a group of 3 students. Find a flight of stairs and an easily accessible moderately high balcony area. Set up a pulley system on the balcony to lift the 9kg weight. Measure the height from the ground to where the 9kg weight is to be lifted. Take turns lifting and timing each other for a total of 1 trial per person. Proceed to the stairs and find the total height by measuring the height of a single step and finding the number of steps. Have each group member walk up the stairs while timed. Then have each group member run up the stairs while timed. Estimate the mass of each group member.

Data and Graphs:

Photo of pulley setup

Stairs on which walk/run were timed

Collected data and calculations

h (cm)	17,3	M1 (kg)	68	g (m/s^2)	9,81
steps	26	M2 (kg)	108
h total	449,8	M3 (kg)	59
	Time (s)			Work (J)			Power (W)
Trial	Lifting 9kg	Walking	Running	Lifting 9kg	Walking	Running	Lifting 9kg	Walking	Running
1	8,08	14,16	4,60	397,1	3001	3001	49,15	211,9	652,3
2	7,62	13,27	4,99	397,1	4766	4766	52,12	359,1	955,0
3	16,26	14,75	4,60	397,1	2603	2603	24,42	176,5	566,0

Data analysis:

Work was calculated according to the equation W=mgh where m is mass of the object, g is the gravitational constant, and h is the change in height from the original height. Power was calculated by dividing work by time taken to complete the task.

Conclusion:

1. In calculating total work done, kinetic energy was assumed to be negligible. Kinetic energy is equal to 1/2mv^2 and is thus most dependent on the speed at which the object moves. Lifting and walking experiments were conducted at low velocities while the running experiment is likely to have greater error due to kinetic energy. For lifting the 9kg mass, 397.1J of work as potential energy was done on the masses while only 1.57J of work was done for trial 2, the fastest trial. For trial 1 of running up the stairs, 3001J of work was done as potential energy while 98.9J of work was done as kinetic energy. The percentage errors for each of these two trials are .365% and 3.30% respectively. Based on these calculations we can expect our error to be less than 5% for all trials.
2. Based on personal data in trial 3 for work per flight of stairs, a climb rate of approximately 11 steps per second is needed to match the power output of a 1100W microwave.
3. The total amount of steps required to cook a potato for 6 minutes in the same microwave would be 3955 steps.
4. The power output needed for 12.5MJ of energy in 10 minutes is 20.8kW.
   The number of people needed to supply this amount of power at 100W per person is 209 people.
   To generate 12.5MJ of energy with a single person supplying 100W, a total of 1.45 days would be needed.

Lab 9: Centripetal force with a motor

Title: Centripetal force with a motor

Purpose:
This lab aims to produce a relationship between theta and omega in a rotating system where the rotating object is affected only by gravity and centripetal force.

Theory:
As an object rotates in a circle, it will be affected by a centripetal force directed towards the center of its path. If this object is attached to a freely moving string, then the centripetal force can be related to the angle theta of the string from vertical as gravity, a component of the tension in the string, is constant and can thus be used with trigonometric identities to create a model describing angular velocity as it relates to theta. By finding the height to which the string rises above its base length while stationary, it is possible to find the angle at which the attached object rotates, calculate a value for centripetal force, and then calculate angular velocity of the object.

Apparatus:

• Powered rotating stand
• Adjustable power supply
• 2 meter stick (1)
• Stopwatch (1)
• String
• Small object (1)
• Stand with clamp and paper

Procedure:

Measure the overall height H of the rotating stand. Measure out a length l of string that nearly matches the height of the stand but does not touch the ground when tied to the top of the stand. Measure the length R of the horizontal rod on top of the powered stand from its axis of rotation to where the string is attached. Set the power supply to some output and measure the time taken for 10 rotations. Move the stand with paper clamped to it near the path of the rotating object, slowly moving the paper up until the object collides. Measure the height from the ground to the spot of collision on the paper.

Data and Graphs:

Table of collected data and calculations

H	1,79	m	rev	10
R	0,749	m	g	9,81
l	1,585	m
t (s)	ω (rad/s)	y (m)	ly (m)	θ (deg)	ω (calc)
32,92	1,908623	0,385	1,405	27,57133	1,858734
28,21	2,22729	0,623	1,167	42,58472	2,224776
22,93	2,740159	0,935	0,855	57,35501	2,710945
19,46	3,228769	1,16	0,63	66,57949	3,205976
16,41	3,828876	1,334	0,456	73,27986	3,795436
13,8	4,553033	1,422	0,368	76,57475	4,235706

Photo of experimental setup

Photo of derivation of formula relating angular velocity to angle theta

Data Analysis:

Calculations used to reach a value of omega from theta are shown in photos above.

Conclusion:

Our model produced values of omega that were near true values of omega, with only 2.06% error on average. These results suggest that this model is very accurate given the error prone experimental setup, particularly as the rotating stand was found to be unstable, tending to bend, tilt, and otherwise deviate from ideal conditions. The method for obtaining the height of the object as it rotated could also have been improved by using a more precise method of incrementing the height of the paper. 

Lab 8: Centripetal Acceleration vs Angular Frequency

Title: Centripetal Acceleration vs Angular Frequency

Purpose:
This lab aims to determine a relationship between the centripetal force on an object and the speed at which it rotates.

Theory:
As an object moves in a circle, it will experience centripetal force accelerating it towards the center of its path. This force changes proportionally with the angular velocity of the object as well as its mass and path radius in the form F=mrw^2. In a system rotating parallel to the ground, the only forces acting on the rotating object will be gravity and centripetal force. Assuming friction is negligible, the measured centripetal force can be expected to behave according to our model. This means that when changing only one variable of mass, radius, or angular velocity, a graph of force vs one changing variable should produce a linear graph with slope equal to the product of the constants.

Apparatus:

• Rotating table (1)
• Wireless force sensor (1)
• Measuring tape (1)
• 100g mass (3)
• String
• Stopwatch (1)
• Adjustable power supply

Procedure:
Prepare the experimental setup as shown in the photo below. Set the power supply to one value and measure a length of string. Collect data for force and angular velocity at constant radius while varying mass at 100g intervals up to 300g. Collect data again for 3 trials while varying radius, keeping mass and power setting the same. Collect another 3 trials while varying the power supply, keeping all else constant.

Data and Graphs:

Photo of experimental setup

Photo of data collection procedure for Force

Photo of data collection procedure for angular velocity

Data table of measured and calculated values

Graphs of Force v Mass*w / Radius*w / w

Analysis:
Angular velocity was calculated based on time taken for 10 revolutions. The slope of each graph represents the constant values in each relationship. Plots of force were always taken against w^2 due to highly inconsistent angular speeds at each trial.

Conclusion:
The Force vs Mass*w^2 graph produced a slope of 0.454, near the actual radius of 0.47m. The plot of Force vs w^2 graph produced a value for m*r of 0.095kgm compared to actual m*r value of 0.078kgm. The Force vs Radius*w^2 graph produced a slope of 0.263kg  with an actual mass of 0.2 kg. Of these graphs, only Force vs Mass*w^2 produced a theoretical constant that was deemed acceptable with a 3.4% error compared to predicted m*r with an error of 21.8% and predicted m with 31.6% error. The huge error in Force vs radius*w^2 may be attributed to the second data point which does not appear to follow the proportional relationship with force, having a value of 1.651N, very near the first data point of 1.659N despite a significantly changed radius. As force measurements were taken as an average over a period of time, the uneven design of the rotating tabletop likely did not contribute much to our error in measurement. It is also unlikely that the force sensor was improperly calibrated enough to produce this value. This suggests that our model may not be entirely valid for our experimental system and that an idealized assumption made during experimentation was false.