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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. - PowerPoint PPT Presentation
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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?