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Q. No.The law of gravitation states that the gravitational force between two bodies of mass \(m_1\) \(m_2\) is given by:
\(F=\dfrac{Gm_1m_2}{r^2}\)
\(G\) (gravitational constant) \(=7\times 10^{-11}~\text{N-m}^2\text{kg}^{-2}\)
\(r\) (distance between the two bodies) in the case of the Earth and Moon \(=4\times 10^8~\text{m}\)
\(m_1~(\text{Earth})=6\times 10^{24}~\text{kg}\)
\(m_2~(\text{Moon})=7\times 10^{22}~\text{kg}\)
What is the gravitational force between the Earth and the Moon?
1. \(1.8375 \times 10^{19}~\text{N}\)
2. \(1.8375 \times 10^{20}~\text{N}\)
3. \(1.8375 \times 10^{25}~\text{N}\)
4. \(1.8375 \times 10^{26}~\text{N}\)
| (a) | The universal law of gravitation is an assumption or hypothesis. |
| (b) | The universal law of gravitation can be proved. |
| (c) | The universal law of gravitation can be verified. |
1. (b) and (c) only
2. (a) and (b) only
3. (a), (b) and (c)
4. (a) and (c) only
(\(M:\) mass of the sphere, \(a:\) radius of the sphere)
| 1. | \(\dfrac {GM^2}{ 2a^2}\) | 2. | \(\dfrac {GM^2}{4a^2}\) |
| 3. | \({\sqrt 3}\dfrac {GM^2}{2a^2}\) | 4. | \(\dfrac {\sqrt {3}}{4}\dfrac {GM^2}{a^2}\) |
As observed from the earth, the sun appears to move in an approximately circular orbit. For the motion of another planet like mercury as observed from the earth, this would:
| 1. | be similarly true. |
| 2. | not be true because the force between the earth and mercury is not inverse square law. |
| 3. | not be true because the major gravitational force on mercury is due to the sun. |
| 4. | not be true because mercury is influenced by forces other than gravitational forces. |
Three equal masses of \(m\) kg each are fixed at the vertices of an equilateral triangle \(ABC.\) What is the force acting on a mass \(2m\) placed at the centroid \(G\) of the triangle?
(Take \(AG=BG=CG=1\) m.)
1. \(Gm^2(\hat{i}+\hat{j})\)
2. \(Gm^2(\hat{i}-\hat{j})\)
3. zero
4. \(2Gm^2(\hat{i}+\hat{j})\)
| (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 a sufficiently long time, we will leave the solar system |
Choose the correct alternatives:
| 1. | (a), (c) | 2. | (a), (d) |
| 3. | (c), (d) | 4. | (a), (b) |
Particles of masses \(2M, m\) and \(M\) are respectively at points \(A, B\) and \(C\) with \(A B = \dfrac{1}{2} \left( B C \right).\) The mass \(m\) is much-much smaller than \(M\) and at time \(t = 0\), they are all at rest as given in the figure. At subsequent times before any collision takes place,
1. \(m\) will remain at rest.
2. \(m\) will move towards \(M.\)
3. \(m\) will move towards \(2M.\)
4. \(m\) will have oscillatory motion.
Choose the wrong option.
| 1. | Inertial mass is a measure of the difficulty of accelerating a body by an external force whereas gravitational mass is relevant in determining the gravitational force on it by an external mass. |
| 2. | That the gravitational mass and inertial mass are equal is an experimental result. |
| 3. | That the acceleration due to gravity on the earth is the same for all bodies is due to the equality of gravitational mass and inertial mass. |
| 4. | Gravitational mass of a particle-like proton can depend on the presence of neighbouring heavy objects but the inertial mass cannot. |
Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the centre of mass causing translation and net torque at the centre of mass causing rotation around an axis through the CM. For the earth-sun system (approximating the earth as a uniform density sphere):
| 1. | the torque is zero. |
| 2. | the torque causes the earth to spin. |
| 3. | the rigid body result is not applicable since the earth is not even approximately a rigid body. |
| 4. | the torque causes the earth to move around the sun. |
The ratio of the weights of a body on the Earth’s surface to that on the surface of a planet is \(9:4\). The mass of the planet is \(\left(\dfrac{1}{9}\right)^\text{th} \) that of the Earth. If \(R\) is the radius of the Earth, what is the radius of the planet? (Take the planets to have the same mass density)
| 1. | \(\dfrac R9\) | 2. | \(\dfrac R2\) |
| 3. | \(\dfrac R 3\) | 4. | \(\dfrac R 4\) |
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