## Question:

pin C through the spiral groove described by the equation r = a$\theta$ . If the angular velocity is constant at $\theta$ , determine the radial and transverse components of velocity and acceleration of the pin.

## Step-by-step

Time Derivatives: Since $\dot {\theta}$ is constant, then $\ddot{\theta}$ =0

r = a$\theta$  $\dot {r}$ = a$\dot{\theta}$    $\ddot {r}$ = a$\ddot{\theta}$=0

${v}_{r}$ = $\dot {r}$ = a$\dot{\theta}$

${v}_{\theta}$  =  r$\dot{\theta}$ =a$\theta\dot{\theta}$

${a}_{r}$ = $\ddot {r}$ – r${\dot{\theta}}^{2}$ = 0 – a$\theta{\dot{\theta}}^{2}$ = -a$\theta{\dot{\theta}}^{2}$

${a}_{r}$ = r$\ddot{\theta}$ + 2$\dot{r}\dot{\theta}$ = 0 + 2(a$\dot{\theta}$)($\dot{\theta}$) = 2a$\dot{\theta}$