Question

The earth’s radius is 6400 km. It makes one rotation about its own axis in 24 hours. The centripetal acceleration of appoint on its equator is nearly –
\[\begin{align}
  & \text{A) 340cm}{{\text{s}}^{-2}} \\
 & \text{B) 34cm}{{\text{s}}^{-2}} \\
 & \text{C) 3}\text{.4cm}{{\text{s}}^{-2}} \\
 & \text{D) 0}\text{.34cm}{{\text{s}}^{-2}} \\
\end{align}\]

Answer 1

Hint: We need to understand the relation between the radius of the planet earth, its speed of rotation and the force acting which causes the rotation of the planet on its axis. This can be used to find the centripetal acceleration of the earth’s rotation.

Complete Step-by-Step Solution:
We know that the rotation of the earth about its axis is due to the centripetal force on the planet. The radius of the planet determines the centripetal force. We know that the rotation of any object with a constant time period will have a constant angular velocity. The centripetal acceleration is necessary to maintain the rotational motion.

Now, we know that the time period of rotation of the earth once around its axis is 24 hours. We need to convert to the standard units before finding the acceleration.
i.e.,
\[\begin{align}
  & 24hrs=24\times 60\times 60s \\
 & \therefore 24hours=86400s \\
\end{align}\]

Now, we can find the velocity of the earth’s rotation using the relation between the time period and the circumference of the earth as –
\[\begin{align}
  & \text{Velocity, v}=\dfrac{2\pi R}{T} \\
 & \Rightarrow v=\dfrac{2\pi \times 640000000}{86400} \\
 & \therefore v=46542.11cm{{s}^{-1}} \\
\end{align}\]

Now, we can find the centripetal acceleration using the relation connecting the velocity and the radius of the rotating mass. The ratio of the square of the velocity and the radius of the rotating particle gives the centripetal acceleration of the particle along its radius. i.e.,
\[\begin{align}
  & \text{Angular acceleration,}{{a}_{c}}=\dfrac{{{v}^{2}}}{R} \\
 & \Rightarrow {{a}_{c}}=\dfrac{{{(46542.11cm{{s}^{-1}})}^{2}}}{6400\times {{10}^{5}}cm} \\
 & \therefore {{a}_{c}}=3.4cm{{s}^{-2}} \\
\end{align}\]

We can see that the centripetal acceleration of the earth in terms of the centi-meter per seconds squared is 3.4

Hence,the correct answer is given by option C.

Note:
The angular acceleration of the earth’s rotation is similar to the circular motion of any other body. The circular motion when uniform will not have an angular acceleration, but the centripetal or the acceleration along the radius is necessary to maintain the motion.