Assume that a tunnel is dug through the Earth | vedclass

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

Question

(B) When a particle of mass $m$ is at a distance $r$ from the center of the Earth,the gravitational force acting on it is due only to the mass of the

Vedclass · Accepted Answer

$\left( \frac{4\pi}{3} mG\rho \right) r$

Vedclass · Answer

(B) When a particle of mass $m$ is at a distance $r$ from the center of the Earth,the gravitational force acting on it is due only to the mass of the Earth contained within a sphere of radius $r$. Let this mass be $M'$.The volume of this inner sphere is $V' = \frac{4}{3} \pi r^3$.Since the density $\rho$ is uniform,the mass $M'$ is given by $M' = \rho V' = \rho \left( \frac{4}{3} \pi r^3 \right)$.The gravitational force $F$ on the particle of mass $m$ at distance $r$ is given by Newton's law of gravitation:$F = \frac{G M' m}{r^2}$Substituting the value of $M'$:$F = \frac{G m}{r^2} \left( \rho \cdot \frac{4}{3} \pi r^3 \right)$Simplifying the expression:$F = \left( \frac{4}{3} \pi G \rho \right) m r$

Assume that a tunnel is dug through the Earth from the North Pole to the South Pole and that the Earth is a non-rotating,uniform sphere of density $\rho$. The gravitational force on a particle of mass $m$ dropped into the tunnel when it reaches a distance $r$ from the center of the Earth is:

Similar Questions

If a particle takes $t$ seconds less and acquires a velocity of $v \text{ m/s}$ more in falling through the same distance on two planets,where the accelerations due to gravity are $2g$ and $8g$ respectively,then:

The weights of an object are measured in a coal mine of depth $h_1$,then at sea level of height $h_2=0$,and lastly at the top of a mountain of height $h_3$ as $W_1, W_2$,and $W_3$ respectively. Which one of the following relations is correct? [$h_1 \ll R, h_3 \ll R, R=$ radius of the earth]

The speed with which the earth would have to rotate about its axis so that a person on the equator would weigh $\frac{3}{5}$ th as much as at present weight is ($g=$ gravitational acceleration,$R=$ equatorial radius of the earth).

What should be the angular speed with which the Earth must rotate on its axis so that a person on the equator would weigh $\frac{3}{5}$ th as much as their present weight?

The weight of a body at the centre of the earth is

Vedclass Test Series

Exam Paper Generator

Online Exam Module