1. almost follow Hooke’s law
Pendulums
almost follow Hooke’s law
§ 13.6

2. 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

3. Simple Pendulum L q m Massless, inextensible string/rod Point-mass bob3

4. 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 Q.
8. The gravitational field g.

5. Simple Pendulum Force SFT = –wT = –mg sinq L T = wR + mv2/L q
w = mg q
wT = mg sinq
wR = mg cosq
SFT = –wT = –mg sinq

6. Simple Pendulum Torque
SFT = –wT = –mg sinq
= LFT = –L mg sinq
Restoring torque 6

7. Small-Angle Approximation
For small q (in radians)
q  sin q  tan q

8. Simple Pendulum t = –L mg sinq t  –L mg q = –kq k = Lmg I = mL2 Lmg
w2 = k/I =
= g/L
w is independent of mass m
(w is not the angular speed of the pendulum) 8

9. Board Work About how long is the pendulum of a grandfather clock?
Find the length of a simple pendulum whose period is 2 s.
About how long is the pendulum of a grandfather clock?

10. Think Question
An extended object with its center of mass a distance L from the pivot, has a moment of inertia
greater than
the same as
less than
a point mass a distance L from the pivot.

11. Poll Question
If a pendulum is an extended object with its center of mass a distance L from the pivot, its period is
longer than
the same as
shorter than
The period of a simple pendulum of length L. 11

12. Physical Pendulum
Source: Young and Freedman, Figure

13. Physical Pendulum k mgd = w = Fnet = –mg sinq tnet = –mgd sinq
Approximately Hooke’s law
t  –mgdq
w = k I
mgd I =
I = Icm + md 2

14. Example: Suspended Rod
Mass M, center of mass at L/2 L 2
Physical pendulum
Simple pendulum L 2
I =    ML2 1 3
I =    ML2 1 4
harder to turn
easier to turn

15. Damped and Forced Oscillations
Introducing non-conservative forces
§ 13.7–13.8

16. Damping Force
Such as viscous drag v
Drag opposes motion: F = –bv

17. Poll Question How does damping affect the oscillation frequency?
Damping increases the frequency.
Damping does not affect the frequency.
Damping decreases the frequency.

18. Light Damping x(t) = Ae cos(w't + f) – w' = If w' > 0: Oscillates
–bt 2m
x(t) = Ae
cos(w't + f) k – b2
w' = m
4m2
If w' > 0:
Oscillates
Frequency slower than undamped case
Amplitude decreases over time

19. Critical Damping – w' = If w' = 0: x(t) = (C1 + C2t) e–atk m
4m2 b2 –
If w' = 0:
x(t) = (C1 + C2t) e–at
No oscillation
If displaced, returns directly to equilibrium

20. Overdamping – w' = If w' is imaginary: x(t) = C1 e–a t + C2 e–a tk m
4m2 b2 –
If w' is imaginary:
x(t) = C1 e–a t + C2 e–a t 1 2
No oscillation
If displaced, returns slowly to equilibrium

21. Energy in Damping Damping force –bv is not conservative
Total mechanical energy decreases over time
Power
= F·v
= –bv·v
= –bv2

22. Forced Oscillation
Periodic driving force
F(t) = Fmax cos(wdt)

23. Forced Oscillation If no damping
If wd = w', amplitude increases without bound

24. Resonance If lightly damped: greatest amplitude when wd = w'
Critical or over-damping (b ≥ 2 km):
no resonance
Source: Young and Freedman, Fig