Question

A Ferris wheel of radius R speeds up with angular acceleration $\mathrm{\alpha }$starting from rest. Find an expression for the (a) velocity and (b) centripetal acceleration of a rider after the Ferris wheel has rotated through angle$∆\mathrm{\theta }$.

Short answer

a. The velocity of a rider after the Ferris wheel has rotated through angle $∆\mathrm{\theta }$is$\mathrm{v}=\sqrt{2\mathrm{\alpha }∆\mathrm{\theta }}\mathrm{R}$.

b. The centripetal acceleration of a rider after the Ferris wheel has rotated through angle$∆\mathrm{\theta }$is$\mathrm{a}=2\mathrm{\alpha }∆\mathrm{\theta R}$.

Step-by-step solution

Given information

A Ferris wheel of radius $\mathrm{R}$speeds up with angular acceleration $\mathrm{\alpha }$ starting from rest. We have to find an expression for the (a) velocity and (b) centripetal acceleration of a rider after the Ferris wheel has rotated through angle $∆\mathrm{\theta }$.

Part a) Step 1: Calculation

The formula is:

${{\mathrm{\omega }}_{\mathrm{f}}}^2={{\mathrm{\omega }}_{\mathrm{i}}}^2+2\mathrm{\alpha }∆\mathrm{\theta }$

Where,

$$\begin{gathered} {\mathrm{\omega }}_{\mathrm{f}}\mathrm{is}\mathrm{final}\mathrm{angular}\mathrm{velocity} \\ {\mathrm{\omega }}_{\mathrm{i}}\mathrm{is}\mathrm{inital}\mathrm{angular}\mathrm{velocity} \\ \mathrm{\alpha }\mathrm{is}\mathrm{angular}\mathrm{acceleration} \\ \mathrm{\theta }\mathrm{is}\mathrm{angle} \end{gathered}$$

By putting values in formula:

$$\begin{gathered} {{\mathrm{\omega }}_{\mathrm{f}}}^2=0+2\mathrm{\alpha }∆\mathrm{\theta } \\ {\mathrm{\omega }}_{\mathrm{f}}=\sqrt{2\mathrm{\alpha }∆\mathrm{\theta }} \\ \mathrm{v}={\mathrm{\omega }}_{\mathrm{f}}\times \mathrm{R} \\ \mathrm{v}=\sqrt{2\mathrm{\alpha }∆\mathrm{\theta }}\mathrm{R} \end{gathered}$$

Part b) Step 1: Calculation

For centripetal acceleration:

${{\mathrm{\omega }}_{\mathrm{f}}}^2={{\mathrm{\omega }}_{\mathrm{i}}}^2+2\mathrm{\alpha }∆\mathrm{\theta }$

Where,

$$\begin{gathered} {\mathrm{\omega }}_{\mathrm{f}}\mathrm{is}\mathrm{final}\mathrm{angular}\mathrm{velocity} \\ {\mathrm{\omega }}_{\mathrm{i}}\mathrm{is}\mathrm{inital}\mathrm{angular}\mathrm{velocity} \\ \mathrm{\alpha }\mathrm{is}\mathrm{angular}\mathrm{acceleration} \\ \mathrm{\theta }\mathrm{is}\mathrm{angle} \end{gathered}$$

Putting values in formula:

$$\begin{gathered} {{\mathrm{\omega }}_{\mathrm{f}}}^2=0+2\mathrm{\alpha }∆\mathrm{\theta } \\ {\mathrm{\omega }}_{\mathrm{f}}=\sqrt{2\mathrm{\alpha }∆\mathrm{\theta }} \\ a={{\mathrm{\omega }}_{\mathrm{f}}}^2\times \mathrm{R} \\ a={\left(\sqrt{2\mathrm{\alpha }∆\mathrm{\theta }}\right)}^2\mathrm{R} \\ a=2\mathrm{\alpha }∆\mathrm{\theta R} \end{gathered}$$