Poll Question

The period of a spring simple harmonic oscillator depends on:(Add together the numbers for all correct choices and text in the sum.)

1. The spring constant k.2. The mass m.4. The maximum amplitude A.8. The gravitational field g.

Pendulums

almost follow Hooke’s law

§ 14.4–14.6

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 sin

Restoring torque

L

m

Small-Angle Approximation

For small (in radians) sin tan

Simple Pendulum

= –L mg sin–L mg = – = LmgI = mL2

L

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

is independent of mass m

( is not the angular speed of the pendulum)

Board WorkFind 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 inertiaA. greater thanB. the same as C. less thana 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 isA. longer thanB. the same as C. shorter thanThe 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

Physical Pendulum

• What is the period when d >> R?