A body moves on a circle with angular velocity that depends…

A body moves on a circle with angular velocity that depends on the angle as $\omega(\theta) = \omega_0 - k\theta$, where $\omega_0$ is the initial angular velocity and $k = 2\ \mathrm{s^{-1}}$; $\theta = 0$ at $t = 0$. Find (a) $\theta(t)$; (b) $\omega(t)$; (c) the time after which the angular velocity drops to $\omega_0/2$.

Explanation of Method

Angular velocity that decreases with the angle.

Step 1: (a) Angle as a function of time

Write $\omega = d\theta/dt$, separate the variables and integrate from 0: \[\int_0^\theta \frac{d\theta}{\omega_0 - k\theta} = \int_0^t dt \;\Longrightarrow\; -\frac1k \ln\frac{\omega_0 - k\theta}{\omega_0} = t.\]

\[ \omega_0 - k\theta = \omega_0 e^{-kt}. \]

\[\theta(t) = \frac{\omega_0}{k}\left(1 - e^{-kt}\right)\]

Step 2: (b) Angular velocity as a function of time

The left side of the last equation is exactly $\omega$: $\omega(t) = \omega_0\,e^{-kt}$.

$\text{fraction}$ $t\ (\mathrm{s})$ $0$ $0.35$ $1$ $2$ $0.5$ $1$ $\omega/\omega_0 = e^{-kt}$ $\theta \big/ \tfrac{\omega_0}{k} = 1 - e^{-kt}$ $\omega = \tfrac{\omega_0}{2} \text{ at } t = \tfrac{\ln 2}{k} \approx 0.35\ \mathrm{s}$

angular velocity (fraction of ω₀)
angle (fraction of ω₀/k)

The angular velocity decays exponentially: it halves every 0.35 s and never quite reaches zero. The angle approaches the limit $\omega_0/k$ but never passes it.

Step 3: (c) Half the initial angular velocity

\[ \omega_0 e^{-kt} = \frac{\omega_0}{2} \;\Longrightarrow\; t = \frac{\ln 2}{k} = \frac{0.693}{2}. \]

\[t \approx 0.35\ \mathrm{s}\]

Check

$ \tfrac{d\theta}{dt} = \tfrac{\omega_0}{k} \cdot k e^{-kt} = \omega_0 e^{-kt} = \omega $, and $ \omega_0 - k\theta = \omega_0 - \omega_0(1 - e^{-kt}) = \omega_0 e^{-kt}\ \checkmark $

Result

\[\boxed{\,\theta(t) = \frac{\omega_0}{k}\left(1 - e^{-kt}\right)\,}\]

\[\boxed{\,\omega(t) = \omega_0\,e^{-kt}\,}\]

\[\boxed{\,t \approx 0.35\ \mathrm{s}\,}\]

Remark

Notes. The body never quite stops: the angle approaches $\omega_0/k$ without reaching it. The constant $k$ has units $\mathrm{s^{-1}}$ (the statement writes rad/s, which is the same thing since the radian is dimensionless). The half-time does not depend on $\omega_0$.