= Torque-driven circular swept area
A path along a <circle> can have variable <areal velocity>. As an illustrative controlled trajectory, keep a <point mass> on a <circle> of radius $R$ with <angle>
$$
\theta(t)=\omega_0t+\frac12\alpha t^2.
$$
Its <acceleration in polar coordinates> and the required net <force> are
$$
\mathbf a=-R(\omega_0+\alpha t)^2\mathbf e_r+R\alpha\mathbf e_\theta,
\qquad \mathbf F_{\mathrm{net}}=m\mathbf a,
$$
where $\mathbf e_r$ and $\mathbf e_\theta$ are the radial and transverse <unit vectors>. A circular guide can supply the radial <force>, while a tangential drive supplies $mR\alpha$. Hence
$$
\tau_n=mR^2\alpha,\qquad
\dot A=\frac{R^2}{2}(\omega_0+\alpha t),\qquad
\ddot A=\frac{R^2\alpha}{2}.
$$
For $R=m=\omega_0=\alpha=1$ in illustrative units, the three equal <time> intervals $[0,0.5]$, $[0.5,1]$ and $[1,1.5]$ sweep <areas> $0.3125$, $0.4375$ and $0.5625$. Their difference comes from the transverse <force>'s positive <torque>, even though the radius and path shape stay fixed. The <area> under each <areal velocity> graph segment equals the corresponding swept <area>.
\Image[/torque-driven-circular-area-rate.png]
{title=A transverse <force> gives increasing <areal velocity> and unequal swept <areas> in equal <times>}
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