A laboratory turntable rotates without friction about a vertical axis

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

Problem

A laboratory turntable rotates without friction about a vertical axis. Between

t = 0\,\text{s}

and

t = 4\,\text{s}

its angular position is given by

\theta(t) = 0.50\,t^2

, where

\theta

is in radians and

t

is in seconds. Two identical metal beads are glued to the surface: bead P at radius

r_P = 0.15\,\text{m}

and bead Q at radius

r_Q = 0.30\,\text{m}

. (a) Derive an expression for the angular velocity

\omega(t)

of the turntable as a function of time. (b) Derive expressions for the instantaneous linear speeds

v_P(t)

and

v_Q(t)

of beads P and Q, respectively. (c) At

t = 4.0\,\text{s}

, calculate the numerical values of

v_P

v_Q

, and the tangential accelerations

a_{t,P}

and

a_{t,Q}

. (d) Determine the total linear displacement of each bead between

t = 0\,\text{s}

and

t = 4\,\text{s}

. Show all necessary steps. (e) Explain why the ratio

v_Q/v_P

remains constant for all times and discuss how this result follows from the physical setup.