Question

You need to design an industrial turntable that has diameter d and has a kinetic energy K when turning at angular speed ω.

A) What must be the moment of inertia of the turntable about the rotation axis? Find an expression in terms of quantities given in the problem.
B) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass? Find an expression in terms of quantities given in the problem.

Answer 1

The moment of inertia of the industrial turntable should be I = 2K/ω^2. If it's a uniform solid disk, its mass must be m = 8K/(ω^2 d^2) to achieve the specified kinetic energy at the given angular speed.

Step-by-step explanation:

To find the moment of inertia I of the industrial turntable with kinetic energy K when turning at an angular speed ω, we can use the relationship between kinetic energy and moment of inertia in rotational motion, i.e., K = 1/2 I ω^2. Solving for I gives I = 2K/ω^2.

For part B, assuming the turntable is designed as a uniform solid disk, the moment of inertia is given by I = 1/2 m d^2/4, where m is the mass of the disk. Setting this equal to the expression found in part A, we can solve for the mass: m = 8K/(ω^2 d^2).