1. Pendulum

2. Force to Torque A pendulum pivots at the top of the string.
The forces on a pendulum are due to gravity and tension.
Tension exerts no torque
Gravity exerts a torque q L FT
mg sin q mg

3. Small Angles The moment of inertia for a single mass is I = mr2.
The angular acceleration is due to the torque.
Compare angle and sine
Angle(rad) Sine
1 ( )
2 ( )
5 ( )
10 ( )
15 ( )
20 ( )
30 ( )
For small angles sinq = q.

4. Simple Pendulum
Angular acceleration and angle are related as a simple harmonic oscillator.
k = mg/L
The angular frequency and period are q L m

5. Tarzan
Tarzan is going to swing from one branch to another 8 m away at the same height using a vine which is 25 m long.
How long does the swing take?
Tarzan forms a pendulum and the period will be
Using 25 m and 9.8 m/s2
T = 10. s
The other branch is half a period, t = 5.0 s.
Note that the mass or distance to the branch didn’t affect the time.
L = 25 m
2A = 8 m

6. Vine Tension
What is the maximum tension of the vine in the previous problem?
The maximum occurs at the bottom with maximum centripetal acceleration.
Find the tension using circular motion. L
v2/r FT A mg

7. Physical Pendulum Real pendulums have mass over the whole length.
Use the actual moment of inertia

8. Damped Harmonic Motion
Real oscillators lose amplitude each cycle.
Friction force exists
This is called damping
Underdamped oscillator
Greater damping can eliminate all oscillations.
Overdamped oscillator
Critical damping (A)

9. Energy Damping Real pendulums have friction.
The damping is due to the work from friction.
Measure energy loss at maximum amplitude

10. Resonance Work can also be done to increase the energy.
If it’s synchronized to the natural frequency then the system is in resonance.
Pushing a swing at each period
A little force can get a large amplitude