Question

A 55.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 540 kg · m2 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? counterclockwise or clockwise
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

The turntable rotates in the counterclockwise direction when the woman walks clockwise due to conservation of angular momentum. Angular speed is determined by dividing the woman's angular momentum by the moment of inertia of the turntable. Work done is calculated as the kinetic energy imparted to the system.

Step-by-step explanation:

Answer to Student's Physics Question

When the 55.0-kg woman starts walking clockwise on the rim of the stationary horizontal turntable, by the principle of conservation of angular momentum, the turntable will rotate in the opposite direction, which is counterclockwise.

The woman and the turntable system constitute an isolated system with no external torques, so the total angular momenta before and after she walks must be equal. Because the woman is walking clockwise, to conserve angular momentum, the turntable must spin counterclockwise.

To determine the angular speed (ω) of the turntable, we can set the angular momentum of the woman (Lwoman) equal to the angular momentum of the turntable (Lturntable). The angular velocity of the woman walking can be calculated using the formula ω = v / r, where v is the linear speed and r is the radius of the turntable.

To calculate the angular momentum of the woman, we use Lwoman = m × v × r, where m is the mass of the woman. Substituting her values, we get Lwoman = 55.0 kg × 1.50 m/s × 2.00 m. Considering the conservation of angular momentum, Lturntable = -Lwoman, where the negative sign signifies the opposite direction of rotation (counterclockwise for the turntable).

The moment of inertia of the turntable (Iturntable) is given as 540 kg · m². The angular speed of the turntable is therefore ωturntable = Lturntable / Iturntable. Plugging in the known values, the calculation will yield the angular speed.

For part (b), the work done by the woman can be calculated as the kinetic energy imparted to the system since the system was initially at rest. The kinetic energy (KE) of a rotating object is given by KE = (1/2) × I × ω2. To find the work done by the woman, we add the kinetic energy of the woman and the turntabl