Question

A square weightless frame ABCD with the side a is rotating around the side AB with the constant angular velocity ω1. A uniform disk of radius R and mass m is rotating around the frame’s diagonal AC with the constant angular velocity ω2. The center of the disk coincides with the center of the frame. Assuming ω1 = ω2 = ω find

a. The kinetic energy of the system.
b. The magnitude of the torque that needs to be applied to the axis AB to keep it (the axis AB) still.

Answer 1

The kinetic energy of the system is one quarter the product of the mass, the square of disk radius, and the square of angular velocity (¼ mR²ω²); and the magnitude of torque required to keep the axis still is theoretically zero if no external forces are considered.

Step-by-step explanation:

The student's question involves calculating the kinetic energy of a composite system and the torque required to keep an axis still. The system consists of a square, weightless frame with a uniform disk rotating around the frame's diagonal. Both the frame and disk have an angular velocity ω.

• a. Kinetic Energy of the System: The total kinetic energy (KE) is the sum of the kinetic energy of the frame's rotation and the disk's rotation. As the frame is weightless, it contributes no kinetic energy. The kinetic energy of the disk is given by KE = ½ I ω², where I is the moment of inertia of the disk, and ω is the angular velocity. The moment of inertia for a uniform disk rotating about an axis through its diameter is I = ½ mR², hence KE = ½ (½ mR²) ω² = ¼ mR²ω².

• b. Torque Required to Keep Axis AB Still: Torque (τ) can be calculated using the equation τ = I α, where α is the angular acceleration. To maintain a constant angular velocity (ω), the angular acceleration must be zero, thus the torque required is also zero. However, this assumes no external forces or friction. If there are external influences, the torque needed would counter those exactly to maintain a constant angular velocity.