- The Physical Pendulum - other oscillators

Serway 15.4, 15.5

Practice: chapter 15, problems 39, 40, 41, 59, 63, 67, 72

Physics 1D03 6 Simple pendulum: a particle on a massless string. θ

“Physical” pendulum: any rigid body, pivoted at P, and free to swing back and forth.

P To find the period: θ 1) Consider the torque due to gravity CM 2) Write τ(θ) = Iα = I (d 2θ/dt 2)

3) SHM if τ is proportional to −θ mg

Physics 1D03 7 Example: a metre stick, pivoted at one end. What is its period of oscillation?

1 2 “Uniform thin rod, pivot at end”: I = /3 ML

Calculate torque about the end:

L τ = Iα θ L 1 − Mg sin θ ⋅ = ML 2 ⋅α 2 3 d 2θ 3g and so α = = − sin θ Mg dt 2 2L

Note, this does not describe SHM!

Physics 1D03 8 But for small oscillations, sin θ ≅ θ L θ d 2θ 3g so ≅ − θ = −ω 2θ dt 2 2L Mg 3g The angular frequency is ω = 2L

2π 2L and the period is T = = 2π ω 3g

2 This is like a simple pendulum of length /3L.

Physics 1D03 9 Any swinging object can be analysed in a similar way; we just need to know its moment of inertia, I, about the pivot point. d 2θ I = −Mgd sin θ ≈ −(Mgd )θ pivot dt 2 d Mgd ω = in general CM I θ Mg For a simple pendulum, I = Md 2 g and ω = d

θ (t) = θ o cos( ωt + φ)

phase amplitude constant (2 π / period)

Physics 1D03 10 Quiz: A geologist is camped on top of a large deposit of nickel ore, in a location where the gravitational field is 0.01% stronger than normal. The period of his pendulum will be

A) 0.01% longer B) 0.005% longer C) 0.01% shorter D) 0.005% shorter E) 0.02% shorter

Physics 1D03 - Lecture 34 11 Question: What happens to the period of P the metre stick when the pivot is moved closer to the centre?

A) The period gets longer. mg B) The period gets shorter. C) It’s hard to tell.

Physics 1D03 12 4

T (sec) 3

2

1

0 0 0.1 0.2 0.3 0.4 0.5

Physics 1D03 14 Example: The same general approach allows us to analyse other systems:

Suppose a 2-kg pendulum bob is suspended from a 1-meter light rod, with a spring (k=10N/m) attached at h=60cm. What is its oscillation period?

h L

k M

Physics 1D03 19 Example: Torsion pendulum

Show that the angular frequency of torsional (twisting) oscillations of a disk welded to a rod is given by

κ ω = I where κ is the “torsion constant” (torque per radian of twist) of the rod, and I is the moment of inertia of the disk.

Physics 1D03 20 Example:

A particle of mass m is attached to the middle of a light wire of length L stretched between two walls,

under tension F T. What is the period of transverse oscillations?

Physics 1D03 21 Physics 1D03 22 Exercise: The wrecking ball on a 9.0-metre cable is released at 20 o and hits the wall at 10 o on the other side of the vertical. How much time elapses?

Physics 1D03 26 Summary

The pendulum motion of any rigid body is approximately SHM, if the amplitude is small, with angular frequency

Mgd ω = I

Physics 1D03 31