AP Physics
Lab: Changes in Potential Energy
Name ____________________________
Date ________________ Per. ______
Suppose a spring is hung vertically with a mass attached to its lower end. The force
exerted on the spring by the mass will cause the spring to stretch and the mass will come
to rest at its equilibrium position. If the mass is lifted a few centimeters above its
equilibrium position and dropped, it will oscillate with simple harmonic motion. At the top
and bottom of its motion the mass will be momentarily at rest and thus have zero kinetic
energy. Choosing the lowest point in the oscillation as the reference level for gravitational
potential energy, the total mechanical energy at this point will all be in the form of elastic
potential energy. Similarly, at the top of the oscillation, the total mechanical energy will
again be all potential with most of it in the form of gravitational potential energy. The
purpose of this experiment will be to compare the CHANGE in gravitational potential energy
of the system, ∆Ug, with the CHANGE in elastic potential energy, ∆Ue, as the mass
oscillates between these two extreme positions.
I. Determining ∆Ue
The EXTENSION of a spring is its length over and above its natural or relaxed length.
For example, suppose a spring having a relaxed length L is hung vertically and a mass M
attached to its lower end. The mass exerts a downward force of magnitude Mg on the
spring, causing its length to become L + X. The resulting change in length X is the
EXTENSION of the spring.
A. Suspend a spring vertically and hang various masses from its lower end. Use masses
ranging from a minimum of 0.2 kg up to and including 1.5 kg. In TABLE I (p. 4) record
the masses used and the resulting position of the lower end of the spring ("Meter Stick
Reading"). Complete the table by calculating the force exerted on the spring by each
mass (use g = 9.8 m/s2) and the resulting extension.
B. Graph the FORCE, F(X), exerted on the spring versus its EXTENSION, X.
C. Calculate the best fit line for the data. Write the equation of the best fit line below and
plot the equation on your FORCE-EXTENSION graph. Include the units of each
constant in the equation for F(X).
F(X) = _________________________
D. Let k represent the constant of proportionality between F and X (the slope in the
equation above). k is called the spring constant of the spring. It is a measure of how
difficult it is to stretch (or compress) the spring. The larger k, the more difficult it is to
stretch (or compress) the spring. What is the value of k for the spring used in this
experiment?
k = _____________ __________
(units)
E. Using the equationrof F(X) and the definition of potential energy,
r
∆U = (F external )⋅ ds ,
2
∫
derive the expression for the elastic potential energy of the spring, Ue(X), as a function
of its extension. Let X = 0 be the reference level for Ue [i.e., let Ue(0) = 0]. In your
derivation, show each step, including the evaluation of the dot product. Include the units
of each constant in your final equation for Ue(X).
Ue(X) = _________________________
F. Using your equation of Ue(X) derived in part E, calculate the elastic potential energy
stored in the spring when the extension of the spring is 0.2 meter.
Ue(0.2 m) = _______________
G. Given a Force-Distance graph for any type of system (not necessarily a spring-mass
system), how would you use the graph to calculate the WORK done on the system as
the system is moved between one position and another?
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H. Using the method stated in your answer to part G, read the appropriate values from
your graph and calculate the WORK done on the spring as the spring is stretched from
0 to 0.2 meter.
W (0 → 0. 2 )m = _______________
If you have done parts C through H correctly, the answers to F and H should be the
same within a few per cent. We would expect them to be since the work done on the
spring by the mass (i.e., the area under the graph from X = 0 to 0.2 m) is equal to the
resulting increase in Ue. As a further confirmation of this relationship, complete Parts I and
J on the next page.
3
I. Using your equation for Ue(X), calculate the change in the elastic potential energy of the
spring as the extension increases from 0.1 to 0.3 meter. That is, calculate
∆Ue (0.1 m → 0 .3 m) = Ue (0. 3 m) − Ue (0.1 m) .
∆Ue(0.1 m → 0.3 m) = _______________
J. Shade in the area under your graph between 0.1 and 0.3 meter. The shaded area is
the ∆Ue of the spring as it is stretched from 0.1 to 0.3 meter. Read the appropriate
values from your graph necessary to calculate the area of this shaded portion and
thereby determine ∆Ue(0.1 m → 0.3 m).
∆Ue(0.1 m → 0.3 m) = _______________
% deviation between your answers in Parts I and J = ______%
II. Determination of ∆Ug
In the drawing at the right, note that the coordinate system is chosen so that X = 0 is the
position of the lower end of the unstretched spring. When a mass is attached the spring
stretches until it comes to rest at the equilibrium position, Xeq. At this point the extension of
the spring is Xeq. If the mass is lifted from Xeq to a position where the
extension of the spring is Xi and released, it falls, coming to rest when
the end of the spring is at Xf. As it falls from Xi to Xf the DECREASE
in gravitational potential energy is given by:
∆Ug = − mg∆h = −mg( Xf − Xi ).
Notice that as Ug decreases, Ue increases. At the highest point in the
oscillation the extension of the spring is Xi and at the lowest point it is
Xf. Therefore, as the mass falls, the ∆Ug can be calculated from the
above expression and ∆Ue can be found using either of the methods
from Part I or Part J.
4
Using the masses and the initial extensions listed in TABLE II, drop the mass and
determine the final extension of the spring Xf at the bottom of the oscillation. Remember, Xi
and Xf are measured from X = 0 NOT from Xeq. Using the average value of Xf obtained
from several trials, calculate the DECREASE in gravitational potential energy, ∆Ug. Show a
sample calculation of ∆Ug in the space provided below TABLE II.
III.
Comparison of ∆Ue and ∆Ug
Transfer your values of Xi, (Xf)ave, and ∆Ug from TABLE II into TABLE III . Calculate
∆Ue as the mass falls between Xi and (Xf)ave as you did in either Part I-I or I-J. Record
your calculated values in TABLE III . Show a sample calculation of ∆Ue in the space
provided below TABLE III .
A. Compare your values for ∆Ue and ∆Ug in Table III. Do they differ? By what %?
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B. Was total mechanical energy conserved in this experiment? Explain.
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DATA TABLES
Caution! - Since each of the springs available have different spring constants, the
SAME spring must be used throughout this experiment. Be sure you have a way of
identifying your spring.
TABLE I -
Mass
(kg)
0
Meter Stick
Reading (cm)
Force
(N)
Extension
(m)
No Data
No Data
5
TABLE II -
Trial #
Mass (kg)
Xi (m)
1
1.0
0.05
2
1.0
0.05
3
1.0
0.05
1
1.0
0.10
2
1.0
0.10
3
1.0
0.10
Xf(m)
(Xf)ave (m)
∆Ug (J)
Sample calculation of ∆Ug:
∆Ug = _______________
TABLE III -
Mass (kg)
Xi (m)
1.0
0.05
1.0
0.10
Sample calculation of ∆Ue:
∆Ue = _______________
(Xf)ave (m)
∆Ue (J)
∆Ug (J)