Q. No.A body of mass \(m\) is taken from the surface of earth to a height equal to twice the radius of earth \(\left(R_e\right).\) The increase in potential energy will be:
(\(g\) is acceleration due to gravity at the surface of earth)
1. \(\dfrac{1}{2} m g R_e\)
2. \(\dfrac{3}{4} m g R_e\)
3. \(\dfrac{1}{4} m g R_e\)
4. \(\dfrac{2}{3} m g R_e\)
(\(g\) is acceleration due to gravity at the surface of earth)
1. \(\dfrac{1}{2} m g R_e\)
2. \(\dfrac{3}{4} m g R_e\)
3. \(\dfrac{1}{4} m g R_e\)
4. \(\dfrac{2}{3} m g R_e\)
Subtopic: Gravitational Potential Energy |
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Given below are two statements:
In the light of the above statements, choose the correct answer from the options given below:
| Assertion (A): | The kinetic energy needed to project a body of mass \(\text m\) from earth surface to infinity is \(\dfrac{mgR}{2},\) where \(R\) is the radius of earth. |
| Reason (R): | The maximum potential energy of a body is zero when it is projected to infinity from earth surface. |
| 1. | Both (A) and (R) are True and (R) is the correct explanation of (A). |
| 2. | Both (A) and (R) are True but (R) is not the correct explanation of (A). |
| 3. | (A) is True but (R) is False. |
| 4. | (A) is False but (R) is True. |
Subtopic: Gravitational Potential Energy |
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Three identical spheres of mass \(m, \) are placed at the vertices of an equilateral triangle of length \(a\). When released, they interact only through gravitational force and collide after a time \(T=4 \) seconds. If the sides of the triangle are increased to length \(2a\) and also the masses of the spheres are made \(2m,\) then they will collide after: (in seconds)
1. \(8\)
2. \(19\)
3. \(35\)
4. \(27\)
1. \(8\)
2. \(19\)
3. \(35\)
4. \(27\)
Subtopic: Gravitational Potential Energy |
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Match the following physical quantities related to the Earth–Sun system:
(Given: Mass of sun = \(M_s,\) Mass of earth =\(M_e,\) Radius of earth = \(R,\) Distance between sun and earth =\(a\))
Codes:
(Given: Mass of sun = \(M_s,\) Mass of earth =\(M_e,\) Radius of earth = \(R,\) Distance between sun and earth =\(a\))
| \(\mathrm{(A)}\) | The kinetic energy of the earth | \(\mathrm{(I)}\) | \(-\dfrac{G M_s M_e}{a}\) |
| \(\mathrm{(B)}\) | The potential energy of the earth and the sun | \(\mathrm{(II)}\) | \(\dfrac{G M_s M_e}{2 a}\) |
| \(\mathrm{(C)}\) | The total energy of the earth and the sun | \(\mathrm{(III)}\) | \(\dfrac{G M_e}{R}\) |
| \(\mathrm{(D)}\) | Escape energy from the surface of the earth per unit mass | \(\mathrm{(IV)}\) | \(-\dfrac{G M_s M_e}{2 a}\) |
Codes:
| 1. | \(\mathrm{A\text-II,B\text-I,C\text- IV,D\text- III}\) |
| 2. | \(\mathrm{A\text-I,B\text-II,C\text- III,D\text- IV}\) |
| 3. | \(\mathrm{A\text-III,B\text-IV,C\text- I,D\text- II}\) |
| 4. | \(\mathrm{A\text-IV,B\text-III,C\text- II,D\text- I}\) |
Subtopic: Gravitational Potential Energy |
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A particle is released from a height equal to radius of earth, \(R.\) Its velocity when it strikes the ground is:
1. \(\sqrt{gR}\)
2. \(\sqrt{{{gR}\over{2}}}\)
3. \(\sqrt{2gR}\)
4. \(\sqrt{4gR}\)
1. \(\sqrt{gR}\)
2. \(\sqrt{{{gR}\over{2}}}\)
3. \(\sqrt{2gR}\)
4. \(\sqrt{4gR}\)
Subtopic: Gravitational Potential Energy |
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A particle is released at a height equal to the radius of the earth above the surface of the earth. Its velocity when it hits the surface of the earth is equal to:
(Where, \(M_e\)=mass of earth, \(R_e\)= radius of the earth.)
1. \(\sqrt{\left[{\frac{2{GM}_{e}}{{R}_{e}}}\right]}\)
2. \(\sqrt{\left[{\frac{{GM}_{e}}{2{R}_{e}}}\right]}\)
3. \(\sqrt{\left[{\frac{{GM}_{e}}{{R}_{e}}}\right]}\)
4. \(\sqrt{\left[{\frac{2{GM}_{e}}{3{R}_{e}}}\right]}\)
(Where, \(M_e\)=mass of earth, \(R_e\)= radius of the earth.)
1. \(\sqrt{\left[{\frac{2{GM}_{e}}{{R}_{e}}}\right]}\)
2. \(\sqrt{\left[{\frac{{GM}_{e}}{2{R}_{e}}}\right]}\)
3. \(\sqrt{\left[{\frac{{GM}_{e}}{{R}_{e}}}\right]}\)
4. \(\sqrt{\left[{\frac{2{GM}_{e}}{3{R}_{e}}}\right]}\)
Subtopic: Gravitational Potential Energy |
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An object of mass \(m\) is placed at a height \(R_{e}\) from the surface of the earth. What is the increase in potential energy of the object if the height of the object is increased to \(2R_{e}\) from the surface? (\(R_{e}:\) Radius of the earth)
1. \({\dfrac{1}{3}{mgR}_{e}}\)
2. \({\dfrac{1}{6}{mgR}_{e}}\)
3. \({\dfrac{1}{2}{mgR}_{e}}\)
4. \({\dfrac{1}{4}{mgR}_{e}}\)
1. \({\dfrac{1}{3}{mgR}_{e}}\)
2. \({\dfrac{1}{6}{mgR}_{e}}\)
3. \({\dfrac{1}{2}{mgR}_{e}}\)
4. \({\dfrac{1}{4}{mgR}_{e}}\)
Subtopic: Gravitational Potential Energy |
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Four spheres, each of mass \(m,\) are placed at the corners of a square of side \(d.\) A fifth sphere of mass \(M\) is located at the centre of the square. What is the total gravitational potential energy of the system?
| 1. | \( -\dfrac{{Gm}}{{d}}\left [(4+\sqrt{2}) {m}+4 \sqrt{2} {M} \right ] \) |
| 2. | \( -\dfrac{{Gm}}{{d}}\left [(4+\sqrt{2}) {M}+4 \sqrt{2} {m} \right ] \) |
| 3. | \( -\dfrac{{Gm}}{{d}}\left[3 {m}^2+4 \sqrt{2}{M}\right] \) |
| 4. | \(-\dfrac{{Gm}}{{d}}\left[6{m}^2+4 \sqrt{2}{M}\right]\) |
Subtopic: Gravitational Potential Energy |
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An object of mass \(1\) kg is taken to a height from the surface of the Earth which is equal to three times the Earth's radius. The gain in potential energy of the object will be:
(Given: \(g=10\) ms–2 and radius of Earth \(=6400\) km)
1. \(48\) MJ
2. \(24\) MJ
3. \(36\) MJ
4. \(12\) MJ
(Given: \(g=10\) ms–2 and radius of Earth \(=6400\) km)
1. \(48\) MJ
2. \(24\) MJ
3. \(36\) MJ
4. \(12\) MJ
Subtopic: Gravitational Potential Energy |
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A body is projected vertically upwards from the surface of the earth with a velocity equal to one-third of escape velocity. The maximum height attained by the body will be:
(take the radius of the Earth \(R=6400~\text{km}\) and \(g=10~\text{m/s}^2\) )
1. \(800~\text{km}\)
2. \(1600~\text{km}\)
3. \(2133~\text{km}\)
4. \(4800~\text{km}\)
(take the radius of the Earth \(R=6400~\text{km}\) and \(g=10~\text{m/s}^2\) )
1. \(800~\text{km}\)
2. \(1600~\text{km}\)
3. \(2133~\text{km}\)
4. \(4800~\text{km}\)
Subtopic: Gravitational Potential Energy |
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