Transcript Pendulums - Richard Barrans’s web site

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.
2.
4.
8.
The spring constant k.
The mass m.
The maximum amplitude A.
The gravitational field g.
Pendulums
almost follow Hooke’s law
§ 14.4–14.6
Angular Oscillators
• Angular Hooke’s law:
t = –kq
• Angular Newton’s second law:
t = Ia
• So
–kq = Ia
• General Solution:
q = Q cos(wt + f)
• where w2 = k/I; Q and f are constants
Simple Pendulum
L
q
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.
2.
4.
8.
The length L.
The mass m.
The maximum amplitude Q.
The gravitational field g.
Simple Pendulum Force
L
T = wR + mv2/L
q
m
SFT = –wT = –mg sinq
w = mg q
wR = mg cosq
wT = mg sinq
Simple Pendulum Torque
SFT = –wT = –mg sinq
t = LFT = –L mg sinq
Restoring torque
L
q
m
Small-Angle Approximation
For small q (in radians)
q  sin q  tan q
Simple Pendulum
L
q
t = –L mg sinq
t  –L mg q = –kq
k = Lmg
I = mL2
Lmg
2 = k/I =
w
= g/L
m
2
mL
w is independent of mass m
(w 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 sinq
tnet = –mgd sinq
Approximately Hooke’s law
t  –mgdq
k
w= I =
I = Icm + md 2
mgd
I
Example: Suspended Rod
Mass M, center of mass at L/2
Physical pendulum
L
2
L
Simple pendulum
L
2
1
I = 3 ML2
1
I = 4 ML2
harder to turn
easier to turn
Physical Pendulum
• What is the period when d >> R?