Conservation of Energy

Energy Conservation
Introduction and Theory
The energy that an object has due to its motion is called kinetic energy and is defined as
K = Z In v2 (1)
where m is the object’s mass and visits speed. An object will also have gravitational potential energy
depending upon its location relative to a reference point. Near the surface of the Earth this can be
written as
U = m g y (2)
where g is the acceleration due to gravity and y is the vertical position of the object relative to an
arbitrary reference point.
Another form of potential energy is elastic energy contained in a deformed elastic object such as a
stretched (or compressed) spring. Mathematically this kind of energy can be expressed as
Usp = % kxz (3)
where k is the spring constant and X is the distance the spring is stretched.
The total mechanical energy of an object is defined as the sum of kinetic energy and any potential
energy associated with that object:
E = K + U + USp (4)
When there is no energy input or energy lost from a system its total mechanical energy will remain
constant. In everyday life this law of conservation of energy is most easily observed by transformation of
potential energy €“ either gravitational or that of a loaded spring €“ into kinetic energy of a moving body.
Likewise, a moving object can transform its kinetic energy into potential energy by climbing a slope or
stretching/compressing a spring.
Objective
To investigate whether or not energy is conserved in three simple mechanical systems:
swinging pendulum
oscillating mass on a spring
basketball shot.
Equipment
Computer with Data Studio and LoggerPro software, Science Workshop interface with Motion Detector
and Rotary Motion Sensor, pendulum assembly, spring, hooked mass, pre-recorded video clip of
basketball shot.
Procedure
Part 1. Swinging pendulum.
The pendulum is attached to the rotary motion sensor and set up to measure the angular position of the
swinging mass as a function of time. Using trigonometry you will convert the angular position of the
pendulum into X-y coordinates in the Cartesian system with the origin at the lowest point of the swing.
This data will allow you to calculate the x and y components of the linear velocity of the pendulum bob
and its total speed along the path of swing needed to determine the kinetic energy. The potential
energy of the moving mass will be estimated with respect to the rest position of the pendulum (the
origin of the system of coordinates).
Measure the length L of the pendulum = the distance between the pivot and the center of the bob.
the pre-set experiment file as directed on the whiteboard. Run 1-point calibration (Adjust Offset
Only) to set the zero for angle measurements when the pendulum is at rest at the lowest point of
swing. To accomplish this the Set-up widow and click the calibration tag. In the new calibration
window mark the 1 point calibration option, then enter 0 (zero) in the angle field and press Read
from Sensor, and then OK. Close the Set-up window. Start the data recording (which will automatically
stop after 7 s), displace the pendulum to about 40 ° (as read in the digital display) and release it. If your
recorded graph does not look like a nice sine wave centered on the time axis you need to repeat the
whole procedure including the calibration part. Any time you wish to record a new run you have to
repeat the calibration!
On the recorded graph of angle vs. time highlight and copy two periods of the pendulum data. Paste it
into Graphical Analysis. In the pull down File menu in GA in the Settings for Untitled File select degrees
option and choose 3 points for derivatives calculation. Now create the following calculated columns:
Pendulum position in horizontal direction, X = L*sin (angle), where L is the length of the
pendulum,
Pendulum position in vertical direction, y = L*(1 €“ cos (angle)), as measured from the lowest
point of swing,
Horizontal component of the linear velocity of the pendulum, vX = derivative X (find this
function in the calculus packet listed under Functions option for new calculated column),
Vertical component of the linear velocity of pendulum, vy= derivative y,
Kinetic energy per unit mass, K/m =% * (vxz + vyz),
Potential energy per unit mass, U/m = 9.8 * y,
In the same graph window show plots of K/m, U/m and E/m vs. time. This can be done by choosing the
More option after clicking on the name of the vertical axis of the graph, and then check marking the
appropriate boxes corresponding to selected variables. Apply a linear fit to the total mechanical energy
plot (include uncertainties) and conclude whether the mechanical energy was conserved over the two
swings.
Insert a new graph of K/m vs. y and use this plot to estimate the gravitational acceleration. The
explanation of your reasoning for using this method to estimate 9 should be included in the lab report.
Resize both graph windows and the data table to fit everything on one page and print it out for your lab
report.
Part 2. Oscillating spring-mass system.
Check the following link to understand the simplified analysis applied to our experimental system.
201/phy 201 7/lab211 7x.pdf
A motion detector situated directly under, and carefully aligned with, the mass-spring system measures
the distance h from the sensor to the closest object (load on the spring) as a function of time. When the
suspended mass-spring system is at rest this distance equals heq. According to the provided link, all we
need to do in order to test the conservation of energy in the case of an oscillating spring is to calculate
and evaluate the following expression: %*mv2 + %*ky2 , where k is the spring constant (given on the
whiteboard) and y = h €“ heq signifies the position of the mass with respect to the system equilibrium.
Use the same experiment file as in Part 1. the Setup window and change the sample rate to 100
Hz. With the mass suspended from the spring and hanging still take one set of data. The recording stops
automatically after 7 s. Apply statistics to your graph of position vs. time to establish the equilibrium
distance heq. Next, lift the mass until the spring is not stretched and release the system by quickly
moving your hand out of the way. Make sure there is no sideways motion of the mass-spring system.
Start a new run.
Copy the recorded data from the graph and paste it into Graphical Analysis. Add new columns to
calculate v (as before use 3 points for derivative calculation), the kinetic energy of the oscillating mass,
the component associated with the spring elastic potential energy and finally the total mechanical
energy. Referring again to Eq.6 in the cited link, the Conservation of Energy principle is satisfied if the
sum of the first two terms stays constant. Present all three energies vs. time plots in the same GA
window. Apply a linear fit to the total energy data, include uncertainties for the fit parameters. Why
does your total energy data appear to oscillate slightly? Resize the data table and the graph window
so they nicely fit on one page and print it out for your records.
Part 3. Basketball shot.
In this part of the lab you will verify the conservation of energy law for a basketball undergoing a
projectile motion. You will analyze a pre-recorded video clip using the video analysis feature of the
LoggerPro software.
Logger Pro. Now on the main menu bar click INSERTéMOVIE and the clip Basketball Shot
from your PHY 122 lab folder.
Before you begin you will need to the video analysis tool bar and set the size scale of the video. To
do this click the icon with red dots and a right arrowhead in the lower right corner of the video frame
->i . . E]
that looks like this and then clIck on the set scale Icon (With Image of yellow ruler). Place
the mouse pointer at one end of the meter stick seen in the video frame. Left click and then drag the
mouse across the stick. When you release the mouse button the Scale window will asking you to
input the size of the object you just specified. Enter 1 meter. Now click the Set Origin button 3 and
set the origin of a system of coordinates for your measurements at any point on the lower left part of
the video frame. Since the video is slightly crooked you may want to rotate your system of coordinates
(grab it by the yellow dot and move it around) and align the x axis with any horizontal object seen in the
picture.
You are now ready to collect data. On the right hand side of the video frame click the icon with a single
I.
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red dot on it . This allows you to mark positions on the video. LoggerPro will record the X and y
coordinates of the points you select, as well as the velocities in the x and y directions. The video may be
a little fuzzy, but mark the location of the center of the basketball as best as you can. If you want to
erase a data point you can click on the arrow icon on the right side of the video frame , and once
the mouse pointer is an arrow (instead of the crosshairs), you can left click on the point you want to
remove and then hit the delete key. You should mark about 15 data points starting from the moment
the ball was released by the player.
Once your data is collected and knowing that the mass of the ball is 0.62 kg, insert calculated columns
for the kinetic energy of the ball, its potential energy, and total mechanical energy. Select 3 point
derivatives calculation as before. Basically you should follow the procedure described in Part 1, except
that this time you are to calculate the actual energies and not the energies per unit mass.
Just like in Part 1, you will want to do a linear fit to your total energy plot and comment on the results.
Insert a new graph of K vs. y and use this plot to estimate the gravitational acceleration. The explanation
of your reasoning for using this method to estimate 9 should be included in the lab report.
Resize both the graph window and the data table to fit everything on one page and print it out.
All printouts will need to be submitted with your Lab report.
Your Data Analysis should include answers to the following questions:

Is the mechanical energy in all investigated cases constant? Explain how you arrived at your answer. If it
is not constant, formulate a reasonable hypothesis to explain why.
What is the minimum value of the pendulum’s gravitational potential energy? Where is the arbitrary
reference level that was used to measure the y coordinate located? Discuss how the values of PE/m,
KE/m, and ME/m would change if this reference level was moved to where the rotary motion is.
Where in its motion does the basketball have its maximum kinetic energy? Minimum kinetic energy?
Maximum potential energy? Minimum potential energy? Clearly label these points on your graph.
Would your answers change if the values of y were measured from the ceiling.