Gravitational Potential of a Solid Sphere

Q. Consider a thin uniform annular disc of mass $m=4\text{kg}$ having inner and outer radius 3 m and 4 m respectively. A point mass $m=2\text{kg}$ is taken from point P to infinity so that at infinity it has kinetic energy of 20 J. If work done by external agent in this process is $W=\{20+\frac{16G}{7}(N\sqrt{2}-5\left)\right\}J$ (G is universal gravitational constant in SI unit). Find the value of N

Q. If the potential at the surface of a planet of mass M and radius R is assumed to be zero. The potential at the center of planet is .

1. $\frac{GM}{2R}$
2. $\frac{-GM}{2R}$
3. zero
4. not defined.

Q. A diametrical tunnel is dug across earth. A ball is dropped into tunnel from one side. The velocity of the ball when it reaches other en​​​d is.​

1. $\sqrt{0.5gR}$
2. $\sqrt{gR}$
3. $\sqrt{2gR}$
4. zero

Q. The potential at any internal point of earth at a distance r from the centre is $\frac{-GM}{2R^3}(3R^2-r^2)$.

1. False
2. True

Q. If the potential at the surface of a planet of mass M and radius R is assumed to be zero. Then the potential at infinity is.

1. $\frac{GM}{R}$
2. $\frac{\_GM}{R}$
3. not defined.
4. zero

Q. Figure shows the variation of kinetic energy, potential energy and total energy as a function of the radius of a body in circular motion. Then

1. A, B, C represent total energy, kinetic energy and potential energy respectively.
2. A, B, C represent kinetic energy, potential energy and total energy respectively.
3. A, B, C represent kinetic energy, total energy and potential energy respectively.
4. A, B, C represent potential energy, kinetic energy and total energy respectively.

Q. There is a spherical cavity inside a uniform solid sphere as shown in the figure. Then, for the magnitude of gravitational potential difference V, which of the following is true?

1. $V_{AB}=V_{CD}$
2. $V_{AB}>V_{CD}$
3. $V_{AB}<V_{CD}$
4. $V_{AC}=0$

Q. A sphere of uniform density has a spherical cavity inside it, then

1. field inside cavity will be constant.
2. region of cavity may be equipotential.
3. region of cavity may not be equipotential.
4. field inside cavity may be zero.

Q. From a solid sphere of mass $M$ and radius $R$, a spherical portion of radius $\left(\frac{R}{2}\right)$ is removed as shown in the figure. Taking gravitational potential $V=0$ at $r=\infty$, the potential at the centre of the cavity thus formed is (G = gravitational constant)

1. $-\frac{-2GM}{R}$
2. $-\frac{GM}{R}$
3. $-\frac{GM}{2R}$
4. $-\frac{-2GM}{3R}$

Q. Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the mid point of the line joining the centres of the spheres ? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable ?