Gravitational Potential of a Solid Sphere

Q. From a solid sphere of mass $M$ and radius $R$, a spherical portion of radius $\frac{R}{2}$ 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. $\frac{-GM}{2R}$
2. $\frac{-GM}{R}$
3. $\frac{-2GM}{3R}$
4. $\frac{-2GM}{R}$

Q. How to find gravitational potential at a centre of the earth, at the surface of the earth is zero

Q.

A ball is allowed to fall from a height of 10m. If there is 40% loss of energy due to impact then after one impact ball will go up by??

Q.

In a gravitational field, at a point where the gravitational potential is zero

1. The gravitational field is necessarily zero

2. The gravitational field is not necessarily zero

3. Nothing can be said definitely about the gravitational field

4. None of these

Q. A particle of mass $M$ is placed at the centre of a uniform spherical shell of equal mass and radius $a$. Find the gravitational potential at a point $\text{P}$ at a distance $a/2$ from the centre.

1. $-\frac{3GM}{a}$
2. $-\frac{5GM}{a}$
3. $-\frac{5GM}{2a}$
4. $-\frac{2GM}{3a}$

Q. A mass of $50\text{kg}$ is placed at the center of a uniform spherical shell of mass $100\text{kg}$ and radius $50\text{m}$. If the gravitational potential at a point, $25\text{m}$ from the center is $V\text{kg/m}$. The value of $V$ is:

Q. Inside a fixed sphere of radius $R$ and uniform desnity $\rho$, there is a sphericfal cavity of radius R/2 such that the surface of cavity passes through the center of the sphere as shown in the figure. A particle of mass $m_0$ is released from rest from centre B of cavity which strikes the center A of sphere with a velocity which is equal to $\frac{\sqrt{2\pi G\rho R^2}}{\sqrt{\beta }}$. Then the value of $\beta$ will be.

Q.

In a hollow spherical shell, potential (V) changes with respect to distance (r) from centre as

1. 
   
2. 
   
3. 
   
4. 
   

Q. A particle of mass $M$ is at a distance $a$ from surface of a thin spherical shell of equal mass and having radius $a$. Choose the correct statement .

1. Gravitational field and potential both are zero at centre of the shell
2. Gravitational field is zero not only inside the shell but also at a point outside the shell also .
3. Inside the shell, gravitational field alone is zero
4. Neither gravitational field nor gravitational potential is zero inside the shell (except at center)

Q. Two identical spherical masses of mass $M$ are kept at some distance. Potential energy of the system when a mass $M$ is taken from the surface of one of the sphere to the other

1. increases continuously.
2. decreases continuously.
3. first increases, then decreases.
4. first decreases, then increases.

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 long metallic rod along the E-W direction falls under gravity.potential difference between its two ends

1. Be zero
2. Be constant
3. Increase with time
4. Decrease with time

Q. P.E -GMm/R+h
but the formula is mgh!!
​

Q. $1000$ small water drops each of capacitance $C$ are combined together to form one large spherical drop. The capacitance of the bigger drop is

1. $C$
2. $10C$
3. $100C$
4. $1000C$

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. 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. Gravitational potential due to solid sphere of uniform density and radius $2R$ at a point $P$ is $v_0$. Now a spherical cavity of radius $R$ is made in the sphere as shown in figure. Gravitational potential at $P$ due to sphere with cavity is

1. $\frac{5}{48}{v}_0$
2. $\frac{v_0}{5}$
3. $\frac{v_0}{48}$
4. $\frac{43}{48}{v}_0$

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. A solid insulating sphere of radius R is charged uniformly. At what distance from its surface is the electrostatic potential half of the potential at the centre?

1. $R/2$
2. $R/3$
3. $2R$
4. $R$

Q.

How energy and pressure have the same dimension?

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 spherical shell of mass $M/2$ is enclosing a solid sphere of mass $3M$ and radius $2R$. Then ratio of potential because of the spherical shell and the solid sphere at their common centre is

1. $2:27$
2. $1:6$
3. $27:2$
4. $2:3$

Q. Let $V_G$ and $E_G$ denote gravitational potential and gravitational field respectively, then choose the wrong statement.

1. $V_G=0,E_G=0$
2. $V_G\ne 0,E_G=0$
3. $V_G=0,E_G\ne 0$
4. $V_G\ne 0,E_G\ne 0$

Q.

As you have learnt in the text, a geostationary satellite orbits the earth at a height of nearly 36, 000 km from the surface of the earth. What is the potential due to earth’s gravity at the site of this satellite? (Take the potential energy at infinity to be zero). Mass of the earth = 6.0 × 10²⁴ kg, radius = 6400 km.

Q. The potential at the centre of the base of a solid hemisphere of mass $m$ and radius $R$ is

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

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. At some point the gravitional potential and also the gravitational field due to earth is zero. The point is

1. On earth's surface
2. Below earth's surface
3. At a height $R_e$ from earths surface ($R_e=$radius of the earth)
4. At infinity

Q. If potential at the surface of earth is assigned zero value, then potential at centre of earth will be (Mass = M, Radius = R):

1. 0
2. $\frac{-GM}{2R}$
3. $\frac{-3GM}{2R}$
4. $\frac{3GM}{2R}$

Q. A certain particle of mass $m$ has momentum of magnitude $mc.$ What are $\left(a\right)\beta ,\left(b\right)\gamma ,$ and $\left(c\right)$ the ratio $K/E_0?$

Q. Inside a hollow isolated spherical shell

1. everywhere gravitational potential is zero.
2. everywhere gravitational field is zero.
3. everywhere gravitational potential is same.
4. everywhere gravitational field is same.