Torque and Simple Harmonic Motion; Rolling without Slipping 8.01 W11D1 Next Reading Assignment: W11D2 Young and Freedman: 10.3-10.6 Simple Pendulum: 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: amplitude effect on period Table Problem: Torsional Oscillator A disk with moment of inertia I0 rotates in a horizontal plane. It is suspended by a thin, massless rod. If the disk is rotated away from its equilibrium position by an angle θ, the rod exerts a restoring torque given by τ = - γθ . At t = 0 the disk is released from rest at an angular displacement of θ0. Find the subsequent time dependence of the angular displacement θ(t). 6 Worked Example: Physical Pendulum use general case A general physical pendulum consists of a body of mass m pivoted about a point S. The center of mass is a distance 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 spin. Does the period of the pendulum 1. increase? 2. stay the same? 3. decrease? Demo: Identical Pendulums, Different Periods Single pivot: body rotates about center of mass. Double pivot: no rotation about center of mass. Rolling Bicycle Wheel Reference frame fixed to ground Center of mass reference frame Motion of point P on rim of rolling bicycle wheel Relative velocity 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 angular velocity? 1. 2. 3. 4. .