B M Sharma Solutions for Chapter: Gravitation, Exercise 2: CONCEPT APPLICATION EXERCISE

As seen in the figure shown below, two spheres of mass $m$ and a third sphere of mass $M$ form an equilateral triangle and the fourth sphere of mass $m_4$ is at the centre of the triangle. The net gravitational force on that central sphere from the three other spheres is zero.
(a) What is $M$ in terms of $m$?
(b) If we double the value of $m_4,$ then what is the magnitude of the net gravitational force on the central sphere?

Three identical particles, each of mass $m$, are located in space at the vertices of an equilateral triangle of side length $a$. They are revolving in a circular orbital under mutual gravitational attraction. Find the speed of each particle.

Two planets $X$and $Y$ travel counterclockwise in circular orbits about a star as shown in figure. The radii of their orbits are in the ratio $3: 1$. At one moment, they are aligned as shown in Fig. (a), making a straight line with the star. During the next five years, the angular displacement of planet $X$ is $90.0°$as shown in Fig. (b). What is the angular displacement of planet $Y$ at this moment?

A thin rod with mass $M$ is bent in a scmicirclc of radius $R$ (figure).

(a) What is its gravitational force (both magnitude and direction on a particle with mass $m$ at $P$, the center of curvature?
(b) What would be the force on the particle if the rod were a complete circle?

A binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (figure). Assume the orbital speed of each star is $\left|\overrightarrow{v}\right|=200 \mathrm{km}/\mathrm{s}$ and the orbital period of each is $20$ days. Find the mass $M$ of each star.

Three identical points of mass $m$ each are kept at the vertices $A, B$, and $C$ of a cube having side length ' $a$ ' (see figure). Another identical mass is placed at the center point $D$ of the cube.

(a) Where will you place a fifth identical mass so that the net gravitational force acting on mass at $D$ becomes zero?
(b) Calculate the net gravitational force acting on the mass at $D$.

In the figure shown below, a spherical hollow inside a lead sphere of radius $R=2.00 \mathrm{cm}\text{,}$ the surface of the hollow passes through the centre of the sphere and touches the right side of the sphere. The mass of the sphere before hollowing was $M=2.95 \mathrm{kg}$. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass $m=0.950 \mathrm{kg}$ that lies at a distance $d=9.00 \mathrm{cm}$ from the centre of the lead sphere, on the straight line connecting the centres of the spheres and of the hollow?

Two small balls of masses $m$ and $2m$ are placed at a separation $d$. A third ball of mass $m$ is placed at an unknown location on the line joining the first two balls such that the net gravitational force experienced by the first ball is $\frac{6\mathrm{G}{m}^2}{d^2}$. What is the location of the third ball?