Question

What will be the angular velocity of earth for which a particle at equator just flies off?
A. 17 times of present value
B. 18 times of present value
C. 19 times of present value
D. 20 times of present value

Answer 1

Hint: Acceleration due to gravity ‘g’ depends on angular speed of rotation of earth. If a particle is supposed to ‘fly off’ it means that there will be no net force acting on the particle. The magnitude of gravitational force would be equal to the magnitude of the centrifugal force. Calculate the magnitudes of the gravitational force and of the centrifugal force that act on the particle on the Equator.

Complete answer:
Let us assume that earth is rotating with angular velocity $\omega$
The radius of earth is R and a particle of mass m is placed on its equator. Then, the force acting on the particle due to gravity will be ${{F}_{g}}=mg$ towards the earth.
When a body rotates, every particle in the body makes circular motions about the axis of rotation. In this question, the earth is under rotation with a constant angular velocity $\omega$, then the mass moves in a circular path with an angular velocity $\omega$. At equator the radius of circular path is equivalent to radius of earth R.
The centrifugal force acting on the particle is given by ${{F}_{cf}}=m{{\omega }^{2}}R$
Since the magnitude of gravitational force would be equal to the magnitude of the centrifugal force. Therefore,
${{F}_{cf}}={{F}_{g}}$
$\Rightarrow m{{\omega }^{2}}R=mg$
As g = 10m/s and R = 6400 km = 6400000 m
$\Rightarrow {{\omega }^{2}}=\dfrac{g}{R}=\dfrac{10}{6400000}\Rightarrow \omega =\sqrt{\dfrac{10}{6400000}}$
On solving this, we get
$\omega =1.25\times {{10}^{-3}}rad/s$
At present, time period of one revolution is 24 hr, which implies that, its angular velocity
${{\omega }_{0}}=\dfrac{2\pi }{T}=\dfrac{2\pi }{24\times 3600}=7.27\times {{10}^{-5}}rad/s$
Therefore, $\dfrac{\omega }{{{\omega }_{0}}}=\dfrac{1.25\times {{10}^{-3}}}{7.27\times {{10}^{-5}}}=17$
The Earth rotates roughly 17 times slower than the minimum speed to fly off the Earth.

Thus option A is correct.

Note:
Weight and mass have different meanings. Weight is the resultant force on the particle due to gravity while its mass is the amount of stuff it has.
Acceleration due to gravity increases as we move from the pole of earth to its equator. The acceleration due to gravity also varies with height of the particle.