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# The solution to Rotational Motion

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A 62.99 kg woman stands at the rim of a horizontal turntable having a moment of inertia of 495 kg·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 (clockwise or counterclockwise) and with what angular speed (in rad/s) does the turntable rotate?

(b) How much work (in joules) does the woman do to set herself and the turntable in motion?

https://brainmass.com/physics/angular-momentum/solution-rotational-motion-253062

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a) The woman and turntable form an isolated system, as there is no external torque being applied on it. Hence, by the law of conservation of angular momentum, the net angular momentum of the system must remain constant. As the woman and the turntable are initially at rest, the initial net angular momentum of the system is zero. As the woman sets into motion with a speed of 1.5 m/s (relative to the ground) clockwise, the turntable starts rotating in the opposite direction (i.e., counterclockwise) with an angular speed such that the sum of the angular momenta of the woman and the turntable remains zero.

The angular momentum of the woman is LW = mvr, where m is the mass of the woman, v is her speed, and r is the radius of the circular path the woman is moving in.

L_W = 62.99 x 1.5 x 2 = 189 kg m^2/s

The angular momentum of the turntable is L_T = Iω = 495ω.

By the law of conservation of angular momentum, L_W + L_T = 189 + 495ω = 0.

Thus ω = -189/495 = -0.382 rad/sec.

The turntable rotates counterclockwise with an angular speed of 0.382 rad/sec.

b) The kinetic energy of the woman is ½ mv2 = ½ x 62.99 x 1.5^2 = 70.87 J.

The kinetic energy of the turntable is ½ Iω^2 = ½ x 495 x 0.382^2 = 36.12 J.

The total kinetic energy is KE = 70.87 + 36.12 = 107 J.

The woman has to do 107 J of work to set herself and the turntable in motion.

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