Question

A 65.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 480 kg middot m^2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. (a) In what direction does the turntable rotate? clockwise counterclockwise With what angular speed does the turntable rotate? rad/s (b) How much work does the woman do to set herself and the turntable into motion? J

Answer 1

  Set the initial angular momentum of the system equal to the final angular momentum of the system. Call the angular momentum of the turntable L₁ and that of the woman - L₂, then:

(L₁ + L₂)(i) = (L₁ + L₂)(f)

L is equal to the product of moment of inertia (I) and angular velocity (ω), and since initially the system is at rest, we write:

0 = I₁ω₁ + I₂ω₂
ω₁ = - I₂ω₂ / I₁

You are given the woman’s speed as 1.5m/s( it should be -1.5m/s as she is moving clockwise), but angular speed is:

ω = v / r

So replacing ω₂ with v / r and I₂ with mr², we have:

ω₁ = -mr²v/r / I₁
= -mrv / I₁
= -(65kg)(2.0m)(-1.5m/s) / 480kg∙m²
= 0.406rad/s (that is 0.406rad/s counter-clockwise)

The work she does to get the turntable to this speed is found from the work-energy theorem, where the net work is equal to the change in kinetic energy:

ΣW = ΔKE
= KE(f) - KE(i)
= [0.5I₁ω₁² + 0.5mv₂²] - 0
= [0.5(480kg∙m²)(0.406rad/s)² + 0.5(65kg)(-1.5m/s)²]
= 112.6J