Physics - Circular Motion Question with Solution | TestHub

A) Position: The vector is $r$ at angle $\theta$.

$\vec{r}=r(\cos \theta \hat{i}+\sin \theta \hat{j})=\mathbf{ut}(\cos \omega \mathbf{t}\hat{\mathbf{i}}+\sin \omega \mathbf{t}\hat{\mathbf{j}})$. (Correct)

B) Speed: Use $v=\sqrt{{\dot{r}}^2+(r\dot{\theta }{)}^2}$.

$\dot{r}=u$ and $r\dot{\theta }=(ut)\omega$.

$v=\sqrt{u^2+u^2{\omega }^2{t}^2}=\mathbf{u}\sqrt{1+{\omega }^2{\mathbf{t}}^2}$. (Correct)

C) Tangential Acceleration ($a_t$):

$a_t=\frac{d}{dt}(u\sqrt{1+{\omega }^2{t}^2})=u\cdot \frac{1}{2\sqrt{1+{\omega }^2{t}^2}}\cdot 2{\omega }^{2}t=\frac{{\omega }^2\mathbf{tu}}{\sqrt{1+{\omega }^2{\mathbf{t}}^2}}$. (Correct)

Correct Options: (A), (B), and (C)