The Simple Pendulum

• The simple pendulum is an example of simple harmonic

motion

• Consists of small object suspended from the end of a cord.

Assumptions:

– Cord doesn’t stretch

– Mass of cord is negligible

What causes it to swing back and forth?

The oscillation

period of a spring

depends on how

springy it is and

what mass is on it.

eff

springk

mT π2=

Analogy with springs

z

The Simple Pendulum

• The simple pendulum is an example of simple harmonic motion

Gravity causes torque, restoring force for oscillations:

F = -mg sin θ

If θ is small (small amplitude oscillations):

L

x

L

z≈=θsin

xL

mgFpendulum −=

Pendulum = Simple Harmonic Motion

xL

mgFpendulum −=

Restoring force is proportional to

negative of displacement (Fspring= -kx)

Effective “spring constant” is keff = mg/L

eff

springk

mT π2=

g

LTpendulum π2=

bob

Pendulum = Simple Harmonic Motion

xL

mgFpendulum −=

Restoring force is proportional to

negative of displacement (Fspring= -kx)

Effective “spring constant” is keff = mg/L

eff

springk

mT π2=

g

LTpendulum π2=

For a pendulum clock, the timing mechanism is designed by adjusting L

bob

Period of simple pendulum is independent of mass or amplitude;

instead depends on the length of cord

Let’s Discuss The Homework

A man enters a tall tower, needing to know its height.

He notes that a long pendulum extends from the

ceiling almost to the floor and that its period is 26.0 s.

(a) How tall is the tower?

(b) If this pendulum is taken to the Moon, where the

free-fall acceleration is 1.67 m/s2, what is the period

there? (Do similar on next slide.)

g

LT π2=

Pendulum: Calculation

A particular grandfather clock has a period of 2

seconds on Earth. If this clock were placed on

Mars, what would its period of oscillation be?

(Acceleration due to gravity on Mars is 3.7 m/s2.)

g

LT π2=

In case you are curious, the

acceleration due to gravity depends

on the mass and radius of the object

(Section 7.5).

Pendulum: Conceptual

If a pendulum clock keeps perfect time at the base of a mountain, will it also keep

perfect time when it is moved to the top of the mountain? If not, will the pendulum

period be slightly larger or smaller?

A) The same B) Larger C) Smaller

g

LT π2=

T will be bigger so it will take longer to complete an oscillation.

effk

mT π2=

spring pendulumx

L

mgF −=

Q150

Damped Oscillations

Why does a child stop swinging

if not continuously pushed?

Damped Oscillations

Why does a child stop swinging

if not continuously pushed?

When work is done by a dissipative force (friction or air

resistance), not all of the mechanical energy is conserved.

This means not all of her potential energy

at the top of each swing is converted into

kinetic energy so her next swing is not

as high.

The period of oscillations stays the same.

The amplitude decreases with time.

A.

B.

C.

Q144

A.

B.

C.

Q145

g

LTpendulum π2=

A. t = T/4

B. t = T/2

C. t = 3T/4

D. t = T

At which of the following times does the object

have the most negative velocity vx?

Q146

This is an x-t graph

for an object in

simple harmonic

motion.

A. t = T/8

B. t = T/4

C. t = 3T/8

D. t = T/2

E. more than one of the above

At which of the following times is the

kinetic energy of the object the greatest?

Q147

This is an x-t graph

for an object in

simple harmonic

motion.

A. t = T/8

B. t = T/4

C. t = 3T/8

D. t = T/2

E. more than one of the above

At which of the following times is the potential

energy of the spring the greatest?

Q148

This is an x-t graph

for an object in

simple harmonic

motion.

Practice

A 163 g block connected to a light spring

with a force constant of k = 7 N/m is free

to oscillate on a horizontal, frictionless

surface. The block is displaced 3 cm from

equilibrium and released from rest.

What is its period and angular frequency?

eff

springk

mT π2=

m

kƒ2 =π=ω

tAx ωcos=

tAv ωω sin−=

tAa ωω cos2−=