Gravitation

Four spheres each of mass m form a square of side d (as shown in figure). A fifth sphere of mass M is situated at the centre of square. The total gravitational potential energy of the system is

The centre of mass of an extended body on the surface of the earth and its centre of gravity

(a) Are always at the same point for any size of the body.

(b) Are always at the same point only for spherical bodies.

(c) Can never be at the same point.

(d) Is close to each other for objects, say of sizes less than 100 m. (e) both can change if the object is taken deep inside the earth.

Which of the following are true?

(a) A polar satellite goes around the earth’s pole in northsouth direction.

(b) A geostationary satellite goes around the earth in east-west direction.

(c) A geostationary satellite goes around the earth in west-east direction.

(d) A polar satellite goes around the earth in east-west direction

Supposing Newton’s law of gravitation for gravitation forces F₁ and F₂ between two masses m₁ and m₂ at positions r₁ and r₂ read F₁=-F₂=- $\frac{{\mathit{r}}_{12}}{{\mathit{r}}_{12}^3}\mathit{G}{\mathit{M}}_{\mathit{o}}^2\left(\frac{{\mathit{m}}_1{\mathit{m}}_2}{{\mathit{M}}_{\mathit{o}}^2}\right)$ ⁿ where M_o is a constant of dimension of mass, r₁₂ = r₁ – r₂ and n is a number. In such a case,

(a) The acceleration due to gravity on earth will be different for different objects.

(b) None of the three laws of Kepler will be valid.

(c) Only the third law will become invalid.

(d) For n negative, an object lighter than water will sink in water.

There have been suggestions that the value of the gravitational constant G becomes smaller when considered over very large time period (in billions of years) in the future. If that happens, for our earth,

(a) Nothing will change.

(b) We will become hotter after billions of years.

(c) We will be going around but not strictly in closed orbits.

(d) After sufficiently long time we will leave the solar system.

If the sun and the planets carried huge amounts of opposite charges,

(a) All three of Kepler’s laws would still be valid.

(b) Only the third law will be valid.

(c) The second law will not change.

(d) The first law will still be valid.

If the mass of sun were ten times smaller and gravitational constant G were ten times larger in magnitudes-

(a) Walking on ground would became more difficult.

(b) The acceleration due to gravity on earth will not change.

(c) Raindrops will fall much faster.

(d) Airplanes will have to travel much faster.

If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law-

(a) Planets will not have elliptic orbits.

(b) Circular orbits of planets is not possible.

(c) Projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.

(d) There will be no gravitational force inside a spherical shell of uniform density.

Which of the following options are correct?

(a) Acceleration due to gravity decreases with increasing altitude.

(b) Acceleration due to gravity increases with increasing depth (assume the earth to be a sphere of uniform density).

(c) Acceleration due to gravity increases with increasing latitude.

(d) Acceleration due to gravity is independent of the mass of the earth.

Particles of masses 2M, m and M are respectively at points A, B and C with AB = ½ (BC). m is much-much smaller than M and at time t = 0, they are all at rest (Fig. 8.1). At subsequent times before any collision takes place:

(a) m will remain at rest.

(b) m will move towards M.

(c) m will move towards 2M.

(d) m will have oscillatory motion.