Question

For the weight of a body of mass $5\ \text{kg}$ to be zero on equator of the earth, angular velocity of the earth must be $\left[\text{Radius of earth} = 6400\ \text{km},\ g = 10\ \text{m/s}^2\right]$

• A. $\dfrac{1}{80}\ \text{rad/s}$
• B. $\dfrac{1}{400}\ \text{rad/s}$
• C. $\dfrac{1}{800}\ \text{rad/s}$
• D. $\dfrac{1}{1600}\ \text{rad/s}$

Hint:

Zero apparent weight occurs when centrifugal force exactly balances gravitational force.

Answer 1

The Correct Option is C

Step 1: Condition for zero apparent weight at equator.
For apparent weight to be zero, centrifugal force must balance gravitational force: \[ mg = m\omega^2 R \] Step 2: Cancel mass and substitute values.
\[ g = \omega^2 R \] Step 3: Solve for angular velocity $\omega$.
\[ \omega = \sqrt{\dfrac{g}{R}} \] Step 4: Substitute numerical values.
\[ R = 6400\ \text{km} = 6.4 \times 10^6\ \text{m} \] \[ \omega = \sqrt{\dfrac{10}{6.4 \times 10^6}} \] Step 5: Simplify.
\[ \omega = \dfrac{1}{800}\ \text{rad/s} \]