Q. No.A planet of mass \(m\) is moving around a star of mass \(M\) and radius \(R\) in a circular orbit of radius \(r.\) The star abruptly shrinks to half its radius without any loss of mass. What change will be there in the orbit of the planet?
| 1. | the planet will escape from the star. |
| 2. | the radius of the orbit will increase. |
| 3. | the radius of the orbit will decrease. |
| 4. | the radius of the orbit will not change. |
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. | (b) and (a) only |
| 3. | (a), (b), (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. | \(\dfrac {\sqrt {3}}{4}\dfrac {GM^2}{a^2}\) |
A star of mass \(M\) and radius \(R\) is made up of gases. The average gravitational pressure compressing the star due to the gravitational pull of the gases making up the star depends on \(R\) as:
| 1. | \(\dfrac{1}{R^{4}}\) | 2. | \(\dfrac{1}{R^{}}\) |
| 3. | \(\dfrac{1}{R^{2}}\) | 4. | \(\dfrac{1}{R^{6}}\) |
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. |
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