A small motor drives a thin rod to rotate about its center

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

Problem

A small motor drives a thin rod to rotate about its center. The angular velocity

\omega

of the rod as a function of time

t

is described by the following piecewise-linear graph: • At

t = 0\,\text{s}

\omega = +4\,\text{rad/s}

. •

\omega

increases uniformly to

+10\,\text{rad/s}

t = 3\,\text{s}

. •

\omega

then decreases uniformly, reaching

-2\,\text{rad/s}

t = 7\,\text{s}

. • From

t = 7\,\text{s}

t = 9\,\text{s}

\omega

remains constant at

-2\,\text{rad/s}

. Take the positive direction to be counter-clockwise. (a) Determine the rod’s angular displacement

\Delta\theta

during the interval

0\,\text{s} \le t \le 9\,\text{s}

. Show your work explicitly. (b) Calculate the average angular velocity

\omega_{\text{avg}}

for the same time interval. (c) At what instant within the 0–9 s interval does the rod reach its greatest angular position (maximum

\theta

)? Justify your answer using information from the graph.