A body rotating with constant angular velocity ω_0 = 4 rad/s…

A body rotating with constant angular velocity $\omega_0 = 4\ \mathrm{rad/s}$ begins to decelerate at $\alpha = 0.5\ \mathrm{rad/s^2}$. Find its angular velocity (a) after $t = 1\ \mathrm{s}$; (b) after turning through $\theta = \pi/3\ \mathrm{rad}$; (c) after $N = 2$ revolutions; and (d) the time after which it stops.

Explanation of Method

Angular velocity after a time, an angle and two revolutions.

Step 1: Accelerating or decelerating?

Part (d) asks when the body stops, so the angular acceleration must slow it down: $\alpha = -0.5\ \mathrm{rad/s^2}$. The formulas are

\[ \omega = \omega_0 - |\alpha|\,t, \]

\[ \omega^2 = \omega_0^2 - 2|\alpha|\,\theta. \]

Step 2: (a) After 1 s

\[ \omega = 4 - 0.5 \cdot 1. \]

\[\omega = 3.5\ \mathrm{rad/s}\]

Step 3: (b) After $\pi/3$ rad

\[ \omega = \sqrt{16 - 2 \cdot 0.5 \cdot \tfrac{\pi}{3}} = \sqrt{16 - 1.047}. \]

\[\omega \approx 3.87\ \mathrm{rad/s}\]

Step 4: (c) After 2 revolutions

Two revolutions are $4\pi \approx 12.57\ \mathrm{rad}$. The body can turn through $\tfrac{\omega_0^2}{2|\alpha|} = 16\ \mathrm{rad}$ (about 2.5 revolutions) before stopping, so it does complete 2 revolutions:

\[ \omega = \sqrt{16 - 2 \cdot 0.5 \cdot 4\pi} = \sqrt{16 - 12.57}. \]

\[\omega \approx 1.85\ \mathrm{rad/s}\]

Step 5: (d) Time to stop

\[ 0 = 4 - 0.5\,t. \]

\[t = 8\ \mathrm{s}\]

$\omega\ (\mathrm{rad/s})$ $t\ (\mathrm{s})$ $0$ $1$ $4.29$ $1.85$ $3.5$ $4$ $\text{(a) } 3.5\ \mathrm{rad/s}$ $\text{(c) after 2 rev: } 1.85\ \mathrm{rad/s}$ $\text{area} = 4\pi\ \text{(2 rev)}$ $8\ \text{(d) stops}$

Angular velocity falls from 4 rad/s at 0.5 rad/s². The shaded area is the angle turned: 2 revolutions ($4\pi$ rad) are completed at about 4.3 s. Part (b), $\pi/3$ rad, is reached very early, after only about 0.27 s.

Result

\[\boxed{\,\omega = 3.5\ \mathrm{rad/s}\,}\]

\[\boxed{\,\omega \approx 3.87\ \mathrm{rad/s}\,}\]

\[\boxed{\,\omega \approx 1.85\ \mathrm{rad/s}\,}\]

\[\boxed{\,t = 8\ \mathrm{s}\,}\]

Remark

About the original solution. It used $+0.5\ \mathrm{rad/s^2}$ (speeding up) in parts (a)–(c) but $-0.5$ in part (d), which is inconsistent: a body that speeds up never stops. With the speeding-up reading, the answers to (a)–(c) would be 4.5, 4.13 and 5.34 rad/s, and part (d) would have no answer. The decelerating reading above is the only one that fits all four parts.