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and Simple Harmonic ; without Slipping

8.01 W11D1 Next Reading Assignment: W11D2

Young and Freedman: 10.3-10.6

Simple : Approximation to Exact Period

Equation of motion: 2 2 d θ −lmg sinθ = ml 2 dt Approximation to exact period:

T  T (sinθ / θ )−3/8 = T + ΔT 0 0 0 0 Taylor Series approximation:

1 2 ΔT  T0 θ0 16 Mini-Experiment:

1. Pendulum: effect on period Table Problem: Torsional Oscillator

A disk with of I0 rotates in a horizontal plane. It is suspended by a thin, massless rod. If the disk is rotated away from its equilibrium by an θ, the rod exerts a restoring torque given by τ = - γθ . At t = 0 the disk is released from rest at an angular of θ0. Find the subsequent dependence of the θ(t).

6 Worked Example: Physical Pendulum use general case

A general physical pendulum consists of a body of m pivoted about a point S. The is a dcm from the pivot point. Find the period of the pendulum.

Concept Question: Physical Pendulum A physical pendulum consists of a uniform rod of length l and mass m pivoted at one end. A disk of mass m1 and radius a is fixed to the other end. Suppose the disk is now mounted to the rod by a frictionless bearing so that is perfectly free to . Does the period of the pendulum

1. increase? 2. stay the same? 3. decrease? Demo: Identical , Different Periods

Single pivot: body rotates about center of mass.

Double pivot: no about center of mass. Rolling

Reference frame fixed to ground Center of mass reference frame

Motion of point P on rim of rolling Relative of point P on rim:    v = v + V P cm,P cm Rolling Bicycle Wheel

Distance traveled in center of mass reference frame of point P on rim in time Δt:

Δs = RΔθ = Rω cm Δt Distance traveled in ground fixed reference frame of point P on rim in time Δt:

ΔXcm =Vcm Δt

Rolling Bicycle Wheel: Constraint relations Rolling without slipping:

Δs = ΔXcm

Rω cm = Vcm

Rolling and Skidding

Δs < ΔXcm

Rω cm < Vcm

Rolling and Slipping

Δs > ΔXcm

Rω cm > Vcm

Rolling Without Slipping: velocity of points on the rim in reference frame fixed to ground

The velocity of the point on the rim that is in contact with the ground is zero in the reference frame fixed to the ground. Concept Question: Rolling Without Slipping

When the wheel is rolling without slipping what is the relation between the final center-of-mass velocity and the final ?

1. .

2. .

3. .

4. .