A bead can slide without friction along a radial slot cut into a flat

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

0.25\,\text{kg}

bead can slide without friction along a radial slot cut into a flat, rotating disk. The bead begins at a distance

r_0 = 0.30\,\text{m}

from the disk’s center at time

t = 0

. A small electric motor slowly pulls the bead inward so that its radial position as a function of time is given by

r(t) = r_0 - k t, \quad \text{for } 0 \le t \le t_\text{max}

, where

k = 0.040\,\text{m}/\text{s}

and

t_\text{max}

is the instant the bead reaches the center. Assume the disk itself has negligible mass. Address the following questions with complete reasoning. (1) Part 1: (a) Derive an expression for the bead’s moment of inertia about the disk’s rotation axis as a function of time

t

while the bead moves inward. (2) Part 2: (b) Calculate the bead’s moment of inertia at

t = 5.0\,\text{s}

. (3) Part 3: (c) The system starts with an angular speed

\omega_0

t = 0

. Assuming no external torques act on the system, obtain an expression for the angular speed

\omega(t)

as the bead moves inward. (4) Part 4: (d) Describe qualitatively how the angular speed changes over the time interval

0 \le t \le t_\text{max}

, relating your explanation to the time dependence of the moment of inertia you found in part (a).

Related theory

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