LAB 7
Conservation of Energy
OBJECTIVES
1. Observe and verify that mechanical energy is transformed between potential and
kinetic energies.
2. Graphically interpret conservation of energy.
3. Model the mass-spring system using Excel.
EQUIPMENT
Motion sensor, spring, masses, meter sticks, supports, steel balls, string.
THEORY
The energy E of a system is the sum of its kinetic energy K and its potential energy U: E
= K + U. If only conservative forces do work within an isolated system then the energy E
of the system cannot change. This principle of conservation of energy is written as
Etotal  constant  K1  U1  K2  U2
PROCEDURE
Mass-Spring System
Attach a hanging mass to a spring and stretch it from its equilibrium position. When the
mass is released, energy will be transferred between spring potential US, gravitational
potential Ug, and kinetic energy K. Using Capstone, measure the motion of the springmass system. Then we want to calculate and display the three types of energies and the
total energy and model this using Excel.
Part 1: Measure the Spring Constant k
Hang a spring and use 50 g increments to generate at least three values for
corresponding displacements y and spring force FS. Then plot the spring force FS vs. y
to determine the slope to find the spring constant k.
Part 2: Data Taking and Analyze the Data
a. Hang 150 g from your spring, pull the mass down about 10 cm and release it to start
the mass-spring system oscillating. Let it oscillate a few times so the hanging mass
will move up-and-down without much side-to-side motion
b. Setup the motion sensor to measure the time t, vertical position y, and velocity vy for
several oscillations of the system. Import these values (t, y, vy) into Excel.
c. Setup to calculate Ug (mgh) and US (½ kd2) as follows.
Gravitational Potential Energy
To get the gravitational potential energy (Ug = mgh), the height h will be a changing
quantity that was measured as the mass oscillates. Define the equilibrium location (≡
xeq) of the spring to be the zero reference gravitational potential energy: Ug(h = xeq) =
0. Another words, one wants h to be measured zero at that location but Capstone
only gives us the distance y.
 Derive a formula that gives h in terms of y and xeq and enter this formula into
Excel.
 Measure xeq from the motion sensor and enter this as a constant
Spring Potential Energy
To get the spring potential energy (US = ½ kd2), one will need to measure d – the
distance the string is stretched at any time. Set d to be zero when the spring is
unstretched (the bottom of the hanging mass), but at that point Capstone measures
xunstretched ≡ xuns. At some time t, data studio measures y, but we want d.
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 Derive a formula that gives d in terms of y and xuns and enter this formula into
Excel.
 Measure xuns from the motion sensor and enter this as a constant.
d. In Excel, calculate Ug (mgh), US (½ kd2), K (½ mv2) and total energy E.
e. Plot on one graph Ug , US , K and (Ug + US + K) vs. time
Pat 3: Data Analysis
a. Identify the spring potential US, the gravitational potential Ug, the kinetic energy K,
and the total energy E on the plot.
b. Focus on one complete cycle of the mass-spring system and identify the two turning
points (lower and higher) and the equilibrium point on this plot. On your plot, show
where
 Ug is a minimum and a maximum?
 US is a minimum and a maximum?
 K is a minimum and a maximum?
Explain your reasoning.
c. Setup a data table such that
US
Ug
K
Etotal = US+Ug+K
Eavg
% diff
Lower Turning Point
Higher Turning Point
Equilibrium Point
Arbitrary Point
d. Calculate the total energy for the following four points in terms of potential and
kinetic energies:
 At each of the two turning points.
 At the equilibrium point.
 At an arbitrary point in-between a turning point and the equilibrium.
e. Compare Etotal with Eavg using a percent difference. How do they compare?
f. As an alternate method of verifying conservation of energy, draw energy-bar
diagrams for each of the four points in part (3c). Is energy conserved?
g. If there is enough time, tweak the spring constant value k and/or the unstretched
displacement parameter xuns to “tune” the model better. Did tweaking parameters
improve the outcomes of the total energy?
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