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?