Question

If the angular velocity of the earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximate.

• A. $60$ minutes
• B. does not change
• C. $84$ minutes
• D. $1200$ minutes

Answer 1

The correct option is C

$84$ minutes

Step 1: Given data

Acceleration due to gravity, $g=10m{s}^{-2}$

The radius of the earth, $R=6400\times {10}^{3}m$

$\mathrm{\pi }=3.14$

Step 2: Formulas used

For a body to achieve weightlessness, the force due to gravity must equal the centripetal force generated by earth's rotation. That is,

$F_g=F_c$ where $F_g$ is the gravitational force and $F_c$ is the centripetal force.

We know that $F_g=mg$ where $m$ is the mass of the body and $g$ is the acceleration due to gravity.

We know that $F_c=m{\omega }^{2}r$ where $\omega$ is the angular velocity and $r$ is the distance of object from the axis of rotation.

Time period of rotation is given as,
$T=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$

Step 3: Calculate angular velocity.

As established,
$\begin{array}{ccc}& F_g=F_c& \\ \Rightarrow & mg=m{\omega }^{2}r& \left[\because F_g=mg,F_c=m{\omega }^{2}r\right]\\ \Rightarrow & \omega =\sqrt{\frac{g}{r}}& \\ \Rightarrow & \omega =\sqrt{\frac{10}{6400\times {10}^3}}& \left[\because r=R=6400\times {10}^{3}m,g=10m{s}^{-2}\right]\\ \therefore & \omega =\frac{1}{800}{s}^{-1}& \end{array}$

Step 4: Calculate the duration of day

The duration of the day is the time period of the rotation of the earth.

Substituting the expression for angular velocity in the time period equation,

Thus,
$\begin{array}{rcl}T& =& \frac{2\mathrm{\pi }}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$800$}\right.}}\\ & =& 2\mathrm{\pi }\times 800\\ \therefore T& \approx & 5027\mathrm{s}\end{array}$

Converting to minutes,
$5027s\approx 84\text{minutes}$

Therefore, if the angular velocity of the earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately $84\text{minutes}$.

Hence, option C is correct.