Assume that the Earth is a solid sphere of un | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) When a particle is at a distance $x$ from the center of the Earth, the gravitational force acting on it is directed towards the center and is give

Vedclass · Accepted Answer

$1$ hour $24$ minutes

Vedclass · Answer

(B) When a particle is at a distance $x$ from the center of the Earth, the gravitational force acting on it is directed towards the center and is given by $F = -\frac{GMm}{R^3}x$.Since $F = ma$, the acceleration $a$ is given by $a = -\frac{GM}{R^3}x$.Using the relation $g = \frac{GM}{R^2}$, we can write the acceleration as $a = -\frac{g}{R}x$.Comparing this with the standard equation for simple harmonic motion, $a = -\omega^2 x$, we get $\omega^2 = \frac{g}{R}$, or $\omega = \sqrt{\frac{g}{R}}$.The time period $T$ is given by $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{R}{g}}$.Substituting the given values: $R = 6400 \, km = 6400 	imes 10^3 \, m$ and $g = 10 \, m/s^2$.$T = 2 	imes 3.14 	imes \sqrt{\frac{6400 	imes 10^3}{10}} = 2 	imes 3.14 	imes \sqrt{640000} = 2 	imes 3.14 	imes 800 \, s$.$T = 5024 \, s$.Converting to minutes: $T = \frac{5024}{60} \approx 83.73 \, minutes$, which is approximately $1$ hour and $24$ minutes.

Similar Questions

At what depth below the surface of the earth will the acceleration due to gravity be half of its value at $1600 \ km$ above the surface of the earth?

The time period of a simple pendulum is $T_1$ when on the Earth's surface and $T_2$ when taken to a height $h = 2R$ above the Earth's surface, where $R$ is the radius of the Earth. The ratio $T_1 : T_2$ is:

$A$ body weighs $500 \, N$ on the surface of the earth. How much would it weigh halfway below the surface of the earth (in $, N$)?

The mass of the moon is $\frac{1}{81}$ of the earth,but the gravitational pull (acceleration due to gravity) is $\frac{1}{6}$ of the earth. This is due to the fact that:

If $R_{E}$ is the radius of the Earth,then the ratio between the acceleration due to gravity at a depth $r$ below and a height $r$ above the Earth's surface is: (Given: $r < R_{E}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module