Mass \(M\) is divided into two parts \(xM\) and \((1-x)M.\) For a given separation, the value of \(x\) for which the gravitational attraction between the two pieces becomes maximum is:
1. | \(\frac{1}{2}\) | 2. | \(\frac{3}{5}\) |
3. | \(1\) | 4. | \(2\) |
1. | \(32\) N | 2. | \(56\) N |
3. | \(72\) N | 4. | zero |
Two particles of equal masses go around a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is:
1. \(v = \frac{1}{2 R} \sqrt{\frac{1}{Gm}}\)
2. \(v = \sqrt{\frac{Gm}{2 R}}\)
3. \(v = \frac{1}{2} \sqrt{\frac{G m}{R}}\)
4. \(v = \sqrt{\frac{4 Gm}{R}}\)
If the radius of a planet is \(R\) and its density is \(\rho\), the escape velocity from its surface will be:
1. \(v_e\propto \rho R\)
2. \(v_e\propto \sqrt{\rho} R\)
3. \(v_e\propto \frac{\sqrt{\rho}}{R}\)
4. \(v_e\propto \frac{1}{\sqrt{\rho} R}\)
A body weighs \(200\) N on the surface of the earth. How much will it weigh halfway down the centre of the earth?
1. | \(100\) N | 2. | \(150\) N |
3. | \(200\) N | 4. | \(250\) N |
The earth is assumed to be a sphere of radius \(R\). A platform is arranged at a height \(R\) from the surface of the earth. The escape velocity of a body from this platform is \(fv_e\), where \(v_e\) is its escape velocity from the surface of the earth. The value of \(f\) is:
1. \(\sqrt{2}\)
2. \(\frac{1}{\sqrt{2}}\)
3. \(\frac{1}{3}\)
4. \(\frac{1}{2}\)
Two satellites of Earth, \(S_1\), and \(S_2\), are moving in the same orbit. The mass of \(S_1\) is four times the mass of \(S_2\). Which one of the following statements is true?
1. | The time period of \(S_1\) is four times that of \(S_2\). |
2. | The potential energies of the earth and satellite in the two cases are equal. |
3. | \(S_1\) and \(S_2\) are moving at the same speed. |
4. | The kinetic energies of the two satellites are equal. |
1. | \(\frac{GMm}{12R} \) | 2. | \(\frac{GMm}{R} \) |
3. | \(\frac{GMm}{8 R} \) | 4. | \(\frac{GMm}{2R}\) |
The figure shows the elliptical orbit of a planet \(m\) about the sun \(\mathrm{S}.\) The shaded area \(\mathrm{SCD}\) is twice the shaded area \(\mathrm{SAB}.\) If \(t_1\) is the time for the planet to move from \(\mathrm{C}\) to \(\mathrm{D}\) and \(t_2\) is the time to move from \(\mathrm{A}\) to \(\mathrm{B},\) then:
1. | \(t_1>t_2\) | 2. | \(t_1=4t_2\) |
3. | \(t_1=2t_2\) | 4. | \(t_1=t_2\) |
Three equal masses \(m\) are placed at the three vertices of an equilateral triangle of side \(r\). Work required to double the separation between masses will be:
1. | \(Gm^2\over r\) | 2. | \(3Gm^2\over r\) |
3. | \({3 \over 2}{Gm^2\over r}\) | 4. | None of the above |