Pendulums

almost follow Hooke’s law

§ 13.6

Angular Oscillators

• Angular Hooke’s law:

= –• Angular Newton’s second law:

= I• So

– = I• General Solution:

= cos(t + )• where 2 = /I; and are constants

Simple Pendulum

L

m

• Massless, inextensible string/rod• Point-mass bob

Poll Question

The period of a simple pendulum depends on:(Add together the numbers for all correct choices and enter the sum.)

1. The length L.

2. The mass m.

4. The maximum amplitude .

8. The gravitational field g.

Simple Pendulum Force

FT = –wT = –mg sin

L

m

T = wR + mv2/L

w = mg

wT = mg sin

wR = mg cos

Simple Pendulum Torque

FT = –wT = –mg sin = LFT = –L mg sinRestoring torque

L

m

Small-Angle Approximation

For small (in radians)

sin tan

Simple Pendulum

= –L mg sin–L mg = – = Lmg

I = mL2

L

m 2 = /I = = g/L Lmg mL2

is independent of mass m

( is not the angular speed of the pendulum)

Board Work

Find the length of a simple pendulum whose period is 2 s.

About how long is the pendulum of a grandfather clock?

Think Question

An extended object with its center of mass a distance L from the pivot, has a moment of inertia

A. greater than

B. the same as

C. less than

a point mass a distance L from the pivot.

Poll Question

If a pendulum is an extended object with its center of mass a distance L from the pivot, its period is

A. longer than

B. the same as

C. shorter than

The period of a simple pendulum of length L.

Physical Pendulum

Source: Young and Freedman, Figure 13.23.

Physical Pendulum

Fnet = –mg sin

net = –mgd sin

Approximately Hooke’s law

–mgd

= I

mgdI

=

I = Icm + md 2

Example: Suspended Rod

Mass M, center of mass at L/2

I =    ML213 I =    ML21

4

LL2

Physical pendulum Simple pendulum

L2

harder to turn easier to turn

Damped and Forced Oscillations

Introducing non-conservative forces

§ 13.7–13.8

Damping Force

Such as viscous drag

v

Drag opposes motion: F = –bv

Poll Question

How does damping affect the oscillation frequency?

A. Damping increases the frequency.

B. Damping does not affect the frequency.

C. Damping decreases the frequency.

Light Damping

x(t) = Ae cos('t + )

–bt2m

If ' > 0:

• Oscillates

• Frequency slower than undamped case

• Amplitude decreases over time

' = km 4m2

b2–

Critical Damping

If ' = 0:

x(t) = (C1 + C2t) e–at

• No oscillation

• If displaced, returns directly to equilibrium

' = km 4m2

b2–

Overdamping

• No oscillation

• If displaced, returns slowly to equilibrium

' = km 4m2

b2–

If ' is imaginary:

x(t) = C1 e–a t + C2 e–a t1 2

Energy in Damping

• Damping force –bv is not conservative

• Total mechanical energy decreases over time

• Power = –bv2= F·v = –bv·v

Forced Oscillation

Periodic driving force

F(t) = Fmax cos(dt)

Forced Oscillation

If no damping

If d = ', amplitude increases without bound

Resonance

If lightly damped:

greatest amplitude when d = '

Source: Young and Freedman, Fig. 13.28

Critical or over-damping (b ≥ 2 km):

no resonance