Download Lab 4 Conservation of Mechanical Energy

Lab 4 Conservation of Mechanical Energy
Objective:
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To measure kinetic and potential energies of a bouncing ball and test the hypothesis that
the total mechanical energy is conserved for a system involving only conservative forces.
Equipment:
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Ball
Motion sensor
Stand
Science Workshop and Computer Interface
Physical principles:
Kinetic Energy
A body which has a mass m and moves with a speed v has energy by virtue of its motion. This
energy is called kinetic energy and is defined as
1
KE = m v 2 .
2
(1)
Gravitational Potential Energy
A body moving in a force field has energy by virtue of its position. This energy is called
potential energy. The potential energy of an object at a point B with respect to a point A is the
work which must be done to move the object from A to B. In the case of the falling ball the work
is done against gravity, so that the gravitational potential energy is given by
PE = m g y
(2)
where y is the height of the ball above some reference level.
Conservation of Total Mechanical Energy
When no non-conservative forces (e.g., frictional forces) are present, the total mechanical energy
is conserved, it is a constant.
E = KE + PE = constant .
(3)
At the top of its bounce the ball will be briefly at rest with zero kinetic energy and a maximum in
potential energy. When the ball reaches the table top, the kinetic energy is a maximum and the
potential energy is a minimum. As the ball moves up and down the sum of the kinetic and
potential energies remains constant. While the ball hits the table top it compresses and the
kinetic energy is reduced to zero while much of this energy is stored in the ball as elastic
potential energy. The ball then rebounds and commences another bounce cycle.
Prediction:
Draw graphs in your journal of position vs. time, velocity vs. time, and acceleration vs. time for
three bounce cycles. In your predictions answer the following questions. Define the origin to be
at the motion sensor and the positive direction to be downward.
1.
Where will the displacement (position) be equal to zero?
2.
Where will the displacement (position) have maximum magnitude?
3.
Where will the velocity be equal to zero?
4.
Where will the velocity have maximum magnitude?
5.
Where will the acceleration be equal to zero?
6.
Where will the acceleration have maximum magnitude?
Explain your reasoning for each of these answers.
Procedure:
Setting up Science Workshop
1.
Plug in the yellow phone plug into digital channel 1 and the other phone plug into digital
channel 2.
2.
Click on the phone plug icon and drag it onto digital channel 1. Double click on Motion
Sensor and set the trigger rate to 50, then click on OK.
3.
Click on Sampling Options and set a stop time of 5 s. Click on OK twice to close the
dialogue box.
4.
Click on the graph icon and drag it onto the motion sensor icon. Select Position, Velocity
and Acceleration graphs. Then click on Display. Click on the Statistics icon, E.
5.
Click on the Calculator icon and define the kinetic energy by entering in the equation box
.5* followed by the value of the mass of the ball, then enter *, click on INPUT and select
Digital 1 and velocity. Enter ^2. In the calculation name and short name boxes enter
KE. In the Units box enter J. Press the enter key to save your definition.
6.
Repeat this process to define the potential energy. In the calculation box enter - and the
value of the mass of the ball, then *9.8* . Then click on INPUT and select Digital 1 and
Position. In the calculation and short name boxes enter PE . In the Units box enter J.
Press the enter key to save your calculation.
7.
Repeat this process to define the mechanical energy. In the calculation box enter + , then
click on INPUT and select Calculations and KE. Click on + then INPUT and select
Calculations and PE. In the calculation and short name boxes enter E . In the Units box
enter J . Press the enter key to save your calculations.
8.
Add graphs for each of the kinetic energy, potential energy and total energy by clicking
on the new graphs icon, then Calculations and then kinetic energy, etc.
Collecting Data
Position the ball about 50 cm above the table top and about 20 cm below the motion sensor.
Drop the ball just after your partner clicks on the record button.
Analyzing Data:
1.
2.
3.
4.
In the position graph click and drag to elect a region of smooth position variation for one
bounce. Click on the E statistics icon for the position graph, select Curve Fit, then
Polynomial Fit . Record the value of the constant a3 and compute the acceleration of
gravity from g = 2 a3.
In the velocity graph click and drag to elect a region of smooth velocity variation for one
bounce. Click on the E statistics icon for the position graph, select Curve Fit, then Linear
Fit. Record the value of the constant a2 = the acceleration of gravity.
Repeat step two for the portion of motion while the ball is in contact with the table and
record the value as the upward acceleration of the ball due to the table’s force on the ball.
Select a smooth section of the data on the graph of total energy for one bounce. Click on
the E statistics icon for the mechanical energy graph, select Mean and Standard
Deviation. Click on the graph setup icon and title the graph, including your name. Print
the graph. The standard deviation is an indicator of the amount of spread in the total
energy from the mean value. Calculate and record the percent error from the equation
%Err '
5.
stdev(E)
×100%
mean(E)
Repeat step 4 for two other bounces.
Questions:
1.
What was the acceleration of gravity as inferred from the position versus time graph?
2.
What was the acceleration of gravity as inferred from the velocity versus time graph?
3.
How did the acceleration of the ball while in table contact compare with the acceleration
of gravity?
4.
How did the acceleration of the ball change in time while in table contact?
5.
How does the variation of the total energy compare with the variation of the kinetic
energy and potential energy?
6.
Was the mechanical energy conserved during one bounce cycle of the ball?
7.
Was the mechanical energy conserved during several bounce cycles of the ball? Account
for the mechanical energy during table contact.
mball =
acceleration of the ball from the position graph: g = 2 a3 =
acceleration of the ball from the velocity graph: g = a2 =
Ebounce 1 =
Ebounce 2 =
Ebounce 2 =
stdev of E1 =
stdev of E2 =
stdev of E3 =