CPS Question

The period of a simple harmonic oscillator depends on:(Add together the numbers for all correct choices and enter 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

§ 13.4–13.6

Angular Oscillations

• Angular Hooke’s law:

= –• Angular Newton’s second law:

= I• So

– = I• General Solution:

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

Group Work

• MasteringPhysics problem 13.34

Simple Pendulum

L

m

CPS 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

Fnet = –wT = –mg sin

L

m

T = wR

w = mg

wT = mg sin

wR = mg cos

Small-Angle Approximation

For small (in radians)

sin tan

Simple Pendulum

Fnet = –wT = –mg sin= –L mg sin–L mg = – = Lmg

I = mL2

L

m

2 = /I = = g/L Lmg mL2

is independent of mass m

Group Work

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

About how long is the pendulum of a grandfather clock?

Group Work

• Do the lab.

CPS 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

=

New Powers

• Now you can do MasteringPhysics problem 13.52

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

Example Problem

A pendulum is constructed of a circular disk pivoting at a point midway between its center of mass and its edge.

What must its radius be for it to have a period of 1 s?

Physical Pendulum

• What is the period when d >> R?