Question:
If the earth suddenly shrinks to \(\dfrac{1}{64}\) of its original volume, while keeping the same mass, then the duration of the day will be:
[Assume earth is a perfect sphere]
If the earth suddenly shrinks to \(\dfrac{1}{64}\) of its original volume, while keeping the same mass, then the duration of the day will be:
[Assume earth is a perfect sphere]
[Assume earth is a perfect sphere]
Show Hint
If no external torque acts on a rotating body, angular momentum remains conserved:
\[
I\omega=\text{constant}
\]
A decrease in radius reduces the moment of inertia and increases the angular speed.
Updated On: Jun 26, 2026
- \(24\) hours
- \(1.5\) hours
- \(16\) hours
- \(48\) hours
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Find the new radius of the Earth.
The volume of a sphere is \[ V=\frac{4}{3}\pi R^3 \] Given that the new volume is \[ V'=\frac{V}{64} \] Therefore, \[ \frac{4}{3}\pi R'^3=\frac{1}{64}\left(\frac{4}{3}\pi R^3\right) \] \[ R'^3=\frac{R^3}{64} \] Taking cube root on both sides, \[ R'=\frac{R}{4} \] Hence, the radius becomes one-fourth of its original value.
Step 2: Use conservation of angular momentum.
Since no external torque acts on the Earth, \[ I\omega=\text{constant} \] For a uniform solid sphere, \[ I=\frac{2}{5}MR^2 \] Since the mass remains unchanged, \[ I\propto R^2 \] Thus, \[ \frac{I'}{I}=\left(\frac{R'}{R}\right)^2 =\left(\frac{1}{4}\right)^2 =\frac{1}{16} \] Therefore, \[ I'=\frac{I}{16} \] Using conservation of angular momentum, \[ I\omega=I'\omega' \] \[ I\omega=\frac{I}{16}\omega' \] \[ \omega'=16\omega \] Thus, the angular speed becomes \(16\) times the original angular speed.
Step 3: Relate angular speed to time period.
The time period of rotation is \[ T=\frac{2\pi}{\omega} \] Hence, \[ T'=\frac{2\pi}{\omega'} =\frac{2\pi}{16\omega} =\frac{T}{16} \] The present duration of a day is \[ T=24 \text{ hours} \] Therefore, \[ T'=\frac{24}{16} \] \[ T'=1.5 \text{ hours} \]
Step 4: Final conclusion.
Therefore, the duration of the day will be \[ \boxed{1.5 \text{ hours}} \]
The volume of a sphere is \[ V=\frac{4}{3}\pi R^3 \] Given that the new volume is \[ V'=\frac{V}{64} \] Therefore, \[ \frac{4}{3}\pi R'^3=\frac{1}{64}\left(\frac{4}{3}\pi R^3\right) \] \[ R'^3=\frac{R^3}{64} \] Taking cube root on both sides, \[ R'=\frac{R}{4} \] Hence, the radius becomes one-fourth of its original value.
Step 2: Use conservation of angular momentum.
Since no external torque acts on the Earth, \[ I\omega=\text{constant} \] For a uniform solid sphere, \[ I=\frac{2}{5}MR^2 \] Since the mass remains unchanged, \[ I\propto R^2 \] Thus, \[ \frac{I'}{I}=\left(\frac{R'}{R}\right)^2 =\left(\frac{1}{4}\right)^2 =\frac{1}{16} \] Therefore, \[ I'=\frac{I}{16} \] Using conservation of angular momentum, \[ I\omega=I'\omega' \] \[ I\omega=\frac{I}{16}\omega' \] \[ \omega'=16\omega \] Thus, the angular speed becomes \(16\) times the original angular speed.
Step 3: Relate angular speed to time period.
The time period of rotation is \[ T=\frac{2\pi}{\omega} \] Hence, \[ T'=\frac{2\pi}{\omega'} =\frac{2\pi}{16\omega} =\frac{T}{16} \] The present duration of a day is \[ T=24 \text{ hours} \] Therefore, \[ T'=\frac{24}{16} \] \[ T'=1.5 \text{ hours} \]
Step 4: Final conclusion.
Therefore, the duration of the day will be \[ \boxed{1.5 \text{ hours}} \]
Was this answer helpful?
0
0
Top AP EAPCET Physics Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The radii of the two columns in a U - tube are r1 and r2. When a liquid of density p (angle of contact is $ 0{}^\circ $ ) is filled in it, the level difference of the liquid in the two arms is h. The surface tension of the liquid is: (g = acceleration due to gravity)
- A rigid metallic sphere is spinning around its own axis in the absence of external torque. If the temperature is raised, its volume increases by $9 \%$. The change in its angular speed is
View More Questions
Top AP EAPCET Earth Satellites Questions
- Satellites which give high value data for remote sensing, meteorology, and environmental studies are
- The acceleration due to gravity at a height $h$ above the Earth's surface is the same as that at a depth $d$ below the surface. If both $h$ and $d$ are much smaller than the radius of Earth $R$, then the relation between $h$ and $d$ is:
- A 1200 kg artificial satellite is in an orbit of radius $2R_{E}$ about the earth. The energy required to transfer it to an orbit of radius $3R_{E}$ is ($g=10\text{ms}^{-2}$, $R_{E}=6400 \text{ km}$)
- A rocket is fired vertically from the surface of the earth with quarter the escape speed. If R is radius of the earth, the maximum altitude reached by the rocket is:
- If a body is thrown vertically upwards from a height of 0.5 R (R is the radius of the earth) with a velocity equal to the escape velocity of a body from the surface of the earth, then the velocity of the body when it escapes from the gravitational influence of the earth is: (g is the acceleration due to gravity on the surface of the earth)
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions