Question

A solid sphere of uniform density and radius $R$ applies a gravitational force of attraction equal to $F_{1}$ on a particle placed at $P$ , distance $2 \, R$ from the centre O of the sphere. A spherical cavity of the radius $R/2$ is now made in the sphere as shown in the figure. The sphere with the cavity now applies a gravitational force $F_{2}$ on the same particle placed at $P$ . The ratio $F_{2}/F_{1}$ will be

• A. $\frac{1}{2}$
• B. $\frac{7}{9}$
• C. $3$
• D. $7$

Answer 1

The Correct Option is B

Gravitational force due to solid sphere, $\left(\textit{F}\right)_{1}=\frac{\textit{GMm}}{\left(2 \textit{R}\right)^{2}}$ where M and m are mass of the solid sphere and particle respectively and R is the radius of the sphere. The gravitational force on the particle due to sphere with cavity = force due to a solid sphere - force due to sphere creating a cavity, assumed to be present above at that position. i.e., $\quad F_{2}=\frac{G M m}{4 R^{2}}-\frac{G(M / 8) m}{(3 R / 2)^{2}}=\frac{7}{36} \frac{G M m}{R^{2}}$ So $\frac{F_{2}}{F_{1}}=\frac{\frac{7 G M m}{36 R^{2}}}{\left(\frac{G M m}{4 R^{2}}\right)}=\frac{7}{9}$

Answer 2

The method for computing the gravitational force between two bodies includes the gravitational constant, indicated by the symbol G.

The proportionality constant in Newton's universal law of gravitation connects the gravitational force between two bodies to the product of their masses and the inverse of the square of their separation.

The English physicist Henry Cavendish conducted the first experimental measurement of the gravitational constant in 1798.

His gravitational constant measuring experiment is well-known as the Cavendish experiment.

Sir Issac Newton proposed that all particles or objects in the universe attract each other in the same manner as the earth attracts the apple. The force of attraction between any two bodies of this universe is called Gravitation or Gravitational Force.

“According to the Universal Law of Gravitation, every object in the universe attracts every other object with a force that is directly proportional to the product of the masses of the objects and inversely proportional to the square of the distance between them.”

Mathematically, F ∝  m₁m₂/r²

Where

F is the gravitational force

m1 and m2 are the masses of the two objects

r is the distance between the bodies

On removing the proportionality sign, we get

F = G x m₁m₂/r²

Where G is a constant of proportionality, known as a Gravitational constant.