1. Aim: How do we explain the pendulum?

2. Simple Pendulum
The simple pendulum exhibits simple harmonic motion within certain limits.

3. Differential Equation for Pendulum
Ft = mat -mgsinθ=md2s/dt2 d2θ/dt2 =(-g/l)sinθ And for small angles sinθ=θ, So d2θ/dt2 = (-g/l)θ

4. Solving the differential equation for a pendulum
We let ω2 =g/l so ϴ=ϴmaxcos(ωt+Ф) where ϴmax is the maximum angular position

5. The period of a pendulumum
The period and frequency of a simple pendulum oscillating at small angles depend only on the length of the string and the free-fall acceleration. T=2π√(l/g)

6. Thought Question 1
A simple pendulum is suspended from the ceiling of a stationary elevator, and the period is determined. Describe the changes, if any, in the period when the elevator (a) accelerates upward, (b) accelerates downward, and (c) moves with constant velocity. a) The period decreases T=2π√[L/(g+a)] b) The period increases T=2π√[L/(g-a)] c) The period would stay the same

7. Thought Question 2
A grandfather clock depends on the period of a pendulum to keep correct time. Suppose a grandfather clock is calibrated correctly and then a mischievous child slides the bob of a pendulum downward on the oscillating rod. Does the grandfather clock run (a) slow, (b) fast, or (c) correctly? The length of the pendulum increases So the period increases

8. Thought Question 3
You set up two oscillating system: a simple pendulum and a block hanging from a vertical spring. You carefully adjust the length of the pendulum so that both oscillators have the same period. You now take the two oscillators to the Moon. Will they still have the same period as each other? What happens if you observe the two oscillators in an orbiting space shuttle? On the moon, the oscillators will have different periods because the pendulum’s period will change but the vertical spring’s period will not. The two oscillators will only have the same period on the orbiting space shuttle if the centripetal acceleration of the shuttle is equal to 10m/s2

9. Problem 1-A Measure of Height
A man enters a tall tower, needing to know its height. He notes that a long pendulum extends from the ceiling almost to the floor and that its period is 12.0s. How tall is the tower? T=2π√(L/g) 12=2π√(L/10) L=36 m

10. Vertical Spring
Does gravity change the period of a vertical spring?

11. Answering the Question
At rest, Fg = Fs so mg=k∆x or ∆x=mg/k When we displace the spring from its rest position, ma=-k(x+∆x) + mg ma=-k(x +mg/k)+mg so ma=-kx and we see that gravity disappears

12. Alternative Approach
d2x/dt2 +k/m x = g x’=x-mg/k d2x’/dt2 + (k/m )x’ = 0 All that gravity changes is the equilibrium point of the spring