A disk-shaped turntable of negligible mass has two identical small blocks
PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion
Problem
A disk-shaped turntable of negligible mass has two identical small blocks, each of mass m, rigidly attached to light rods so that initially each block moves in a circle of radius r about the central vertical axis. A motor mounted beneath the turntable delivers a constant counterclockwise torque of magnitude τ0 to the system. After spinning up from rest, a mechanism simultaneously draws both blocks straight inward until each now moves in a circle of radius 2r. Assume the blocks move radially inward very quickly compared to the rotation period, so any change in angular speed during the inward motion can be neglected for the purpose of computing the new rotational inertia.
(a) Derive an expression (in terms of m and r) for the rotational inertia of the system about the central axis before the blocks are drawn inward.
(b) Using your result from part (a), obtain an algebraic expression for the magnitude of the angular acceleration α1 of the system while the blocks are at radius r.
(c) Repeat parts (a) and (b) for the configuration after the blocks have been moved to radius 2r, obtaining expressions for the new rotational inertia and the new angular acceleration α2.
(d) Determine the ratio α1α2 and interpret the physical meaning of this ratio in the context of the motor delivering the same constant torque throughout the process.
(e) Immediately after the blocks reach radius 2r, the motor continues to apply the torque τ0. Compare the instantaneous power delivered by the motor at that moment to the power being delivered just before the blocks moved. Justify your answer qualitatively without introducing new symbols.