Question

What happens to the centripetal acceleration of a revolving body if you double the orbital speed v and half angular velocity ω

• A. It remains unchanged
• B. Is is halved
• C. It is doubled
• D. It is quadrupled

Answer 1

The correct option is A

It remains unchanged

Step 1: Definition

1. Centripetal acceleration is the attribute of an object's velocity in a circular route.
2. Centripetal acceleration is an acceleration vector pointing towards the center of the circle.

Mathematically, centripetal acceleration is represented as

${\mathrm{a}}_{\mathrm{c}}=\frac{{\mathrm{v}}^2}{\mathrm{r}}$

Here,

${\mathrm{a}}_{\mathrm{c}}$ is the centripetal acceleration.

${\mathrm{v}}^2$ is the velocity.

r is the radius.

Also,

${\text{a}}_{\mathrm{c}}={\mathrm{\omega }}^2\mathrm{r}$

Step 2: The centripetal acceleration of a revolving body if you double the orbital speed v and half angular velocity $\mathrm{\omega }$

Since we know,

$$\begin{gathered} \mathrm{v}=\mathrm{\omega r} \\ {\mathrm{a}}_{\mathrm{c}}=\mathrm{\omega }\times \mathrm{v} \\ {\mathrm{a}}_{\mathrm{c}}\propto \mathrm{\omega v} \end{gathered}$$

The change in centripetal acceleration is represented as,

$$\begin{gathered} {\mathrm{a}}_{\mathrm{c}}\text{'}=\frac{\mathrm{\omega }}{2}\times 2\mathrm{v} \\ \frac{{\mathrm{a}}_{\mathrm{c}}}{{\mathrm{a}}_{\mathrm{c}}\text{'}}=\frac{2\mathrm{v}\times {\displaystyle \frac{\mathrm{\omega }}{2}}}{\mathrm{v\omega }} \\ {\mathrm{a}}_{\mathrm{c}}={\mathrm{a}}_{\mathrm{c}}\text{'} \end{gathered}$$

Therefore, the centripetal acceleration of a revolving body if you double the orbital speed v and half angular velocity $\mathrm{\omega }$ it remains unchanged.