Gravitation practice · Mixed difficulty

A boy can jump to a height $h$ on ground level. What should be the radius of a sphere of density $d$ such that on jumping on it, he escapes out of the gravitational field of the sphere

For elliptical orbits, in the equation $T^2=\left(\frac{4 \pi^2}{{GM}_s}\right) R^3, R$ refers to

The effect of rotation of the earth on the value of acceleration due to gravity is

A planet is revolving round the sun. Its distance from the sun at Apogee is $r_A$ and that at Perigee is $r_{{r}}$. The mass of planet and sun is $m$ and $M$ respectively, $v_A$ and $V_P$ is the velocity of planet at Apogee and Perigee respectivley and ${T}$ is the time period of revolution of planet round the sun.
(a) $T^2=\frac{\pi^2}{2 G m}\left(r_A+r_p\right)^2$
(b) $T^2=\frac{\pi^2}{2 G M}\left(r_A+r_p\right)^3$
(c) $v_A r_A=v_p r_p$
(d) $v_A < v_p, r_A > r_p$

When a satellite moves around the earth in a certain orbit, the quantity which remains constant is:

An object of mass $X$ is split into two portions $y$ and $(X-Y)$, which are then placed at some distance. Then what will be the ratio of $Y / X$ maximizes the gravitational force between the two portions?

A planet has twice the density of earth but the acceleration due to gravity on its surface is exactly the same as on the surface of the earth. Its radius in terms of radius of earth $R$ will be

The angular momentum of the earth revolving round the sun is proportional to $r^n$, where $r$ is the distance between the centres of earth and the sun. The value of $n$ is

Assertion: If rotation of earth about its own axis is suddenly stops then acceleration due to gravity will increase at all places on the earth.
Reason: At height $\mathrm{h}$ from the surface of earth, acceleration due to gravity is $\mathrm{g}_{\mathrm{h}}=\mathrm{g}\left(1-\frac{2 \mathrm{h}}{\mathrm{R}_{\mathrm{e}}}\right)$.

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): Angular momentum conservation can be used to explain Kepler's second law of planetary motion.

Reason (R): Areal velocity of a planet revolving around the sun is equal to its angular momentum.

In the light of the above statements, choose the correct answer from the options given below:

A body moves in a circular orbit of radius $r$ under the action of a central force. Potential due to the central force is given by $V(r)$ – $kr$ ($k$ is a positive constant). Period of revolution of the body is proportional to:

The escape velocity of a sphere of mass $m$ from earth having mass $M$ and radius $R$ is given by

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

Escape velocity when a body of mass $\mathrm{m}$ is thrown vertically from the surface of the earth is $\mathrm{v}$, what will be the escape velocity of another body of mass $4 \mathrm{~m}$ is thrown vertically

Assertion: A satellite can not be set on a stable orbit in a plane not passing through the earth's centre.
Reason: For the orbital motion of satellite, the centripetal force is provided by gravitational pull of earth on satellite.

Statement I: The weight of a body at the centre of earth is zero

Statement II: The mass of a body decreases with increase in depth below the surface of earth.

Which is constant for a satellite in orbit

If the total energy of a satellite is negative. It
means

The moon does not fall to Earth because:

Assertion: For a satellite revolving very near to earth&#39;s surface the time period of revolution is given by $1 \mathrm{~h} \, 24 \mathrm{~min}$.
Reason: The period of revolution of a satellite depends only upon its height above the earth&#39;s surface.

In a certain region of space, the gravitational field is given by $-k/r$, where $r$ is the distance and $k$ is a constant. If the gravitational potential at $r = {r_0}$ be ${V_0}$, then what is the expression for the gravitational potential V?

Two bodies of masses $m$ and $M$ are placed a distance $d$ apart. The gravitational potential at the position where the gravitational field due to them is zero is $V$, then

The period of revolution of planet A around the sun is 8 times that $B$. The distance of a from the sun is how many times greater than that of $B$ from the sun?

Which of the following is always positive?

Two satellites ${S_1}$ and ${S_2}$ revolve around a planet in coplanar circular orbits in the same sense. Their periods of revolution are 4 hr and 6 hr respectively. The radius of the orbit of ${S_1}$ is 104 km. The angular speed of ${S_2}$ as observed by an astronaut in ${S_1}$ (in radius)

The mass and diameter of a planet have twice the value of the corresponding parameters of Earth. Acceleration due to gravity on the surface of the planet is:

Mark the correct statements.

I. Gravitational potential at the centre of curvature of a thin hemispherical shell of radius $R$ and mass $M$ is equal to $G M / R$

II. Gravitation field strength at a point lying on the axis at a distance of ' $a$ ' on a thin, uniform circular ring of radius $\mathrm{R}$ and mass $\mathrm{M}$ is equal $\frac{G M R}{\left(a^2+R^2\right)^{3 / 2}}$

III. Newton's law of gravitation for gravitational force between two bodies is applicable only when bodies have sphrical symmetric distribution of mass

Average density of the earth

The escape velocity of a body from the earth's surface is $v_e$. If the same body is projected from the surface of the earth with the velocity $2 v_e$, then the velocity of the body when it reaches to infinity, i.e., beyond the earth's gravitational field is

The value of g on the earth’s surface is $980\mathrm{\;cm}{\mathrm{s}^{ - 2}}$. Its value at a height of 64 km from the earth’s surface is