Learning Object 1

1. Learning Object 1- A Simple Pendulum
1. A grandfather clock with a pendulum of 1.5 meters has a 0.75 kg mass attached to it. The maximum
displacement from equilibrium is 7.0 cm.
a) Find the angular frequency of the pendulum.
b) Calculate the total energy of the pendulum (think about the relation between kinetic energy and
potential energy).
c) Find the period of oscillation of the pendulum.
d) What would happen to the period if the length of the pendulum was halved?
2. Suppose the clock is running slowly and the time to complete one cycle is taking longer than it should.
Should you shorten or lengthen the pendulum? Explain your reasoning.
3. Suppose the same grandfather clock as above is on an imaginary planet where the period of the
pendulum is measured to be 3.70 seconds. What is the acceleration due to gravity on this planet?
The Grandfather Clock was considered the most precise timekeeper in the
world from when it was invented in 1656 to the 1930s. The pendulum
underneath the clock oscillates with a certain period. The error of the clock
increases with amplitude and thus, the angle is limited to 2˚ -4˚. This small
angle allows for the pendulum to demonstrate simple harmonic motion. By
moving the weight along the length of the pendulum the period can also be
adjusted.

2. Solutions
1. a) The angular frequency is given by the formula =
=
.
.
= 2.56 Hz
b) To calculate the total energy of the pendulum, we can choose at point that minimizes the
terms we need to find. At the equilibrium position, the weight has maximum kinetic energy and
zero potential energy.
E = PE + KE
E = KE = mv²
First we need to find the velocity at the bottom, which is the maximum velocity.
= A
= (0.07) (2.56)
= 0.179 m/s
KE = mv²
= (0.75) (0.179) ²
= 0.012 J
c) To find the period, we use the formula T=2
=2
.
.
= 2.46 seconds/cycle
d) The period decreases by a factor of √2.
The period is proportional to the square root of the length as seen in equation T = 2 . So
if the length is decreased by a factor of the period also decreases, but by a factor of√2.

3. 2. We need to shorten the pendulum because the period needs to be decreased and both
parameters are proportional. A shorter length pendulum swings faster and thus completes one
cycle in a shorter amount of time. This will result in the clock speeding up.
3. The acceleration due to gravity depends on the length of the pendulum and the period. Thus,
we use the following equation to solve for g.
T = 2
3.70 = 2
.
g =
.
. ²
g = 4.33 m/s²