Lesson - Energy Conservation in Simple Harmonic Motion
(Pendulum)

The applet simulates the motion of a simple pendulum and
energy conservation in this system.

**Prerequisites**

The applet simulates the motion of a simple pendulum in its simple-harmonic-motion (SHM) approximation. Students should be familiar with the general properties of SHM and the sense in which SHM is a good approximation to the motion of a simple pendulum at small amplitude. In particular, they should be familiar with the SHM-approximation to the displacement and velocity of a simple pendulum as a function of time. Students should be familiar with the concepts of kinetic and potential energy and energy conservation.

**Learning Outcomes**

Students will learn how to use energy conservation to obtain information about the potential energy of a simple pendulum. They will be able to describe the time dependence of the potential energy when the pendulum is oscillating and to explain how the potential energy depends on the displacement of the pendulum from its equilibrium position. They will learn that the energy of the simple pendulum is proportional to the square of the amplitude of oscillation of the pendulum.

**Instructions**

Students should know how the applet functions, as described in Help and ShowMe.

The applet should be open. The step-by-step instructions in the following text are to be done in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited.

Contents

Kinetic Energy as a Function of Time

Potential Energy as a Function of Time

Potential Energy as a Function of Position

Energy as a Function of Amplitude

Kinetic Energy as a Function of Time

Figure 1 below illustrates the *x*-axis that will be
used to describe the motion of the simple pendulum.

**Figure 1**

We will use the bob's *x*-coordinate as the pendulum's
position. The origin of the *x*-axis, *x* = 0, is
at the bob's equilibrium position at the bottom of the swing.

For small amplitudes, the time dependence of the bob's
*x*-coordinate is to a good approximation SHM. In the
simulation and in this lesson, the time dependence of the
bob's *x*-coordinate is taken to be SHM exactly.

The kinetic energy *KE* of a simple pendulum is a
function of the time *t*. Let's find out how *KE*
depends on *t*.

**Exercise 1**. RESET the applet. Set the amplitude to
*A* = 0.50 m and the length of the pendulum to *L* =
3.00 m. Keep the mass of the pendulum bob and the magnitude of
the acceleration due to gravity at their default values of
*m* = 0.50 kg and *g* = 9.8 m/s^{2},
respectively.

PLAY the motion from the pendulum's default position, which has
the pendulum bob at its right-most position, at *x* =
*A*, and observe the motion. Sketch graphs of both the
bob's position (*x*) and velocity (*v _{x}*)
vs. time (

When done, display the applet's *x* vs. *t* and
*v _{x}* vs.

**Exercise 2**. Continuing from Exercise 1, REWIND the applet
and PLAY the motion again. Observe the size of the kinetic
energy column during the motion, and sketch the kinetic energy
*KE* of the oscillating pendulum vs. *t*. You may
find it helpful to Step through the motion. On the graph, mark
the points where the bob passes its far right, far left, and
equilibrium positions. Make sure your graph is consistent with
the *v _{x}* vs.

**Exercise 3**. Write down general expressions in symbols
for the bob's position *x*(*t*) and velocity
*v _{x}*(

Using the expression for *v _{x}*(

**Answer**. For the kinetic energy, you should obtain

*KE*(*t*) =
(*m*/2)*A*^{2}ω^{2} *sin*^{2}
ω*t*(1)

**Exercise 4**. REWIND the applet, and STEP through the
motion until the bob is as close to its equilibrium point as
possible (the energy column would be entirely red at the
equilibrium point). Record the value of the kinetic energy at
this point from the Data box. Calculate this value using
Equ.(1) above and compare your result to the value of
*KE* in the Data box.

Calculate the kinetic energy at another instant during the pendulum's motion. Also, click Rewind, change some of system parameters, and Step the system to a time of your choice. Calculate the kinetic energy at these times from Equ.(1). Compare your values with those displayed in the Data box.

Potential Energy as a Function of Time

The potential energy of the simple pendulum is strictly speaking the gravitational potential energy of the pendulum-earth system. We will refer to it as the "potential energy of the pendulum" for short. Similarly, we will refer to the energy of the pendulum-earth system as the "energy of the pendulum".

The mechanical energy *E*, *energy* for short, of
the simple pendulum stays constant during the pendulum's
motion (if the motion is undamped, as it is in the applet).
One says the energy is *conserved*. The energy is the
sum of kinetic and potential energy,

*E* = *KE* + *PE*. (2)

PLAY the motion and observe how there is an ongoing conversion of potential into kinetic energy or vice versa during the motion such that the sum of the two energies remains constant.

In this section, you will use Equ.(2) together with Equ.(1) to obtain information about the time dependence of the potential energy.

**Exercise 1**. RESET the applet. Set the amplitude to
*A* = 0.50 m and the length of the pendulum to *L* =
3.00 m. Keep the mass of the pendulum bob and the magnitude of
the acceleration due to gravity at their default values of
*m* = 0.50 kg and *g* = 9.8 m/s^{2},
respectively. These are the same settings as at the beginning of
the preceding section.

PLAY the motion, and sketch a graph of the potential energy as function of time. Mark the points where the bob passes its far right, far left, and equilibrium positions. When done, compare your sketch to the graph drawn by the applet.

**Exercise 2**. Continuing from Exercise 1, determine the
value of the potential energy at the bob's far right and far
left positions and the value of the mechanical energy as
follows.

The potential energy of any system is defined up to an
additive constant whose value one can choose freely to suit
one's convenience. In the applet, as is commonly done for the
present system, this constant is chosen so that the potential
energy of the pendulum is 0 when the pendulum is at its
equilibrium position. Therefore, at this point, the
mechanical energy *E* is equal to the kinetic energy
*KE*.

When the bob is either at its far right or far left
positions, the bob is momentarily at rest and therefore its
kinetic energy equal to zero. Thus, Equ.(2) implies that
*E* = *PE* at this point.

Therefore, since *E* is constant during the motion, the
potential energy at the far right or far left is equal to the
kinetic energy when the bob is passing its equilibrium
position, which is the maximum kinetic energy:

*PE _{right,left}* =

Use Equ.(3) combined with Expression (1) for the kinetic
energy, to derive a general expression for the potential
energy at the far right or far left of the motion and for the
energy *E*. Hint: what is the maximum value of
*sin*^{2} ω*t*?

**Answer**. Since the maximum value of the sine function
and that of its square are equal to 1, Equ.(1) implies

*PE _{right,left}* =

Use Equ.(4) to calculate the values of
*PE _{right,left}* and

**Exercise 3**. In the Lesson "Simple Harmonic Motion
(Pendulum)" accompanying the Pendulum applet it is shown that
the angular frequency ω is related to
the magnitude due to gravity *g* and the length *L* of
the pendulum by

ω^{2} =
*g* / *L* .(5)

Substitute this expression for ω^{2} into Equ.(4) to obtain an
expression for the energy in terms of *m*, *g*,
*L*, and *A*.

The result is

*PE _{right,left}* =

Substitute the present values into Equ.(6) and check if the
resulting value of *PE _{right,left}* agrees with
that obtained in Exercise 2.

**Exercise 4**. Substitute Expression (1) for the kinetic
energy into Equ.(2) to obtain an expression for the potential
energy at time *t*. Into the resulting equation for
*PE*(*t*), substitute Expression (4) for *E*
and simplify. You should obtain the following equation for
*PE*(*t*):

*PE*(*t*) =
(*m*/2)*A*^{2}ω^{2} [1 - *sin*^{2}
ω*t*] .(7)

Using Equ.(5), this can be rewritten as

*PE*(*t*) =
(1/2)(*mg*/*L*)*A*^{2} [1 -
*sin*^{2} ω*t*]
.(8)

Simplify Equ.(8) by using the standard trigonometric identity

*sin*^{2} θ + *cos*^{2} θ
= 1 .(9)

Equ.(8) will simplify to

*PE*(*t*) =
(1/2)(*mg*/*L*)*A*^{2}
*cos*^{2} ω*t*
.(10)

Check that the graph you sketched in Exercise 1 is consistent with this equation. Explain in words why you think the two are consistent. Discuss whether Equ.(10) has the right behavior at the far right, far left, and equilibrium positions of the motion.

PLAY and PAUSE the motion at some instant, and use Equ.(10)
to calculate the value of *PE* at this instant. Compare
your result to that shown in the Data box.

Potential Energy as Function of Position

In Exercise 3 of the section "Kinetic Energy as a Function of
Time", you were asked to write down a general expression for
the position of the pendulum bob as a function of time,
assuming the motion starts at *t* = 0 with the bob in
its far right position.

The equation is

*x*(*t*) = *A* *cos*
ω*t* .(11)

Substitute this expression for *x*(*t*) into
Equ.(10) to obtain an expression for the potential energy as
function of position, instead of time. The result will be

*PE*(*x*) = (1/2)(*mg*/*L*)
*x*^{2} .(12)

**Exercise 1**. Sketch *PE* vs. *x* between
*x* = -*A* and *x* = *A*. What is the
name of this kind of a curve?

**Exercise 2**. Discuss if Equ.(12) is consistent with
Equ.(6).

**Exercise 3**. RESET the applet. Set the amplitude to
*A* = 0.50 m and the length of the pendulum to *L* =
3.00 m. Keep the mass of the pendulum bob and the magnitude of
the acceleration due to gravity at their default values of
*m* = 0.50 kg and *g* = 9.8 m/s^{2},
respectively. These settings were used previously.

Drag the pendulum bob all the way to the left, by clicking on
the bob and dragging. The bob should be at *x* = -0.50
m. Calculate the corresponding value of *PE* from
Equ.(12), and compare your result to the value in the Data
box.

Drag the weight to the right to *x* = -0.40 m, and again
calculate *PE*. Observe the blue potential energy column
and the rate at which it changes. Continue doing this for
*x* = -0.30 m, *x* = -0.20 m, *x* = -0.10 m,
*x* = 0, and then for positive *x* in steps of 0.10
m. Make a table of the values of *PE* vs. *x*.
Discuss where the rate at which *PE* changes is
greatest: near the far right of the oscillation, or near the
far left or near the equilibrium point.

For each 0.10-m interval, calculate the average slope
Δ*PE* / Δ*x* of the potential energy curve over
the interval from *x* to *x* + 0.10 m, and compare
this average slope to the *x*-component of the net force
acting on the bob at the midpoint of this interval, i.e., at
*x* + 0.05 m. To calculate the net force, use the
expression for the acceleration *a _{x}* derived
in an Appendix to the Lesson accompanying the applet "Simple
Harmonic Motion (Pendulum)",

*a _{x}* = - (

and multiply by the mass *m*.

**Answer**. You should find that

*F _{x,mid}* = - Δ

for each of the 0.10-m intervals. Note the minus sign in this equation. When the slope of the potential energy curve is positive, the net force is negative. Does that make sense?

You could make the *x*-interval smaller and smaller and
would find Equ.(14) to be true always. In the limit of
vanishing interval size, the average slope over an interval
would become the slope at a point. This way one can prove
that

Thex-component of the net force acting on the pendulum bob at a given displacementxfrom equilibrium is equal to the negative slope at that point of the potential energy taken as a function ofx.

The mathematical symbol for slope (also called
*derivative*), as applied to the present case, is
*dPE* / *dx*. With this notation,

*F _{x}* = -

Energy as a Function of Amplitude

Equ.(6) above gives the energy of the simple pendulum in
terms of the amplitude *A*, the mass *m*, the
magnitude *g* of the acceleration due to gravity, and
the length *L* of the pendulum. Note that the energy
depends on the **square** of the amplitude, not the first
power of the amplitude.

**Exercise 1**. RESET the applet, and set the amplitude to
*A* = 0.17 m, the length of the pendulum to *L* =
3.00 m, and the mass of the pendulum bob to *m* = 1.00
kg. Keep the magnitude of the acceleration due to gravity at
its default value of *g* = 9.8 m/s^{2}. Display
the Data box.

Record the value of the energy *E*. Vary the amplitude
with the slider from 0.17 m to 0.51 m, i.e., by a factor of
3, and record the corresponding values of *E* for
several values of the amplitude along the way. Plot a graph
of *E* vs. *A* and demonstrate that it exhibits the
quadratic dependence on *A* from Equ.(6).

__Comment__. The pendulum's potential energy is really
gravitational potential energy. The gravitational potential
energy of the bob is equal to *mgh* where *h* is
the bob's elevation above the bob's bottom position. One can
show that this expression is consistent with Expression (12)
for the potential energy by showing that, for small
amplitudes,

*mgh* = (1/2)(*mg*/*L*)
*x*^{2} .(16)