A satellite of mass $m$ revolves around the earth of radius $R$ at a height $x$ from its surface. If $g$ is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is
(1) $g x$
(2) $\frac { g R } { R - x }$
(3) $\frac { g ^ { 2 } } { R + x }$
(4) $\left( \frac { g R ^ { 2 } } { R + x } \right) ^ { 1 / 2 }$

The time period of an earth satellite in circular orbit is independent of
(1) the mass of the satellite
(2) radius of its orbit
(3) both the mass and radius of the orbit
(4) neither the mass of the satellite nor the radius of its orbit.

Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
(1) $R ^ { \left( \frac { n + 1 } { 2 } \right) }$
(2) $R ^ { \left( \frac { n - 1 } { 2 } \right) }$
(3) $R ^ { n }$
(4) $\mathrm { R } ^ { \left( \frac { \mathrm { n } - 2 } { 2 } \right) }$

A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is $12 \mathrm { rad } \mathrm { s } ^ { - 1 }$, the magnitude of its angular momentum about a point on the ground right under the center of the circle is:
(1) $14.4 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
(2) $11.52 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
(3) $20.16 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
(4) $8.64 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet? [Given: Mass of planet $= 8 \times 10 ^ { 22 } \mathrm {~kg}$, Radius of planet $= 2 \times 10 ^ { 6 } \mathrm {~m}$, Gravitational constant $\mathrm { G } = 6.67 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } / \mathrm { kg } ^ { 2 }$ ]
(1) 17
(2) 9
(3) 13
(4) 11

A body A of mass $m$ is moving in a circular orbit of radius $R$ about a planet. Another body B of mass $\frac { m } { 2 }$ collides with A with a velocity which is half $\left( \frac { \vec { v } } { 2 } \right)$ the instantaneous velocity $\vec { v }$ of A. The collision is completely inelastic. Then, the combined body:
(1) continues to move in a circular orbit
(2) Escapes from the Planet's Gravitational field
(3) Falls vertically downwards towards the planet
(4) starts moving in an elliptical orbit around the planet

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius $\mathrm { R } _ { \mathrm { e } }$. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become $\sqrt { \frac { 3 } { 2 } }$ times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is $R$. Value of $R$ is:
(1) $4 \mathrm { R } _ { \mathrm { e } }$
(2) $2.5 \mathrm { R } _ { \mathrm { e } }$
(3) $3 R _ { e }$
(4) $2 \mathrm { R } _ { \mathrm { e } }$

The time period of a satellite in a circular orbit of the radius $R$ is $T$. The period of another satellite in a circular orbit of the radius $9R$ is:
(1) $9T$
(2) $27T$
(3) $12T$
(4) $3T$

A vehicle of mass 200 kg is moving along a levelled curved road of radius 70 m with angular velocity of $0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centripetal force acting on the vehicle is:
(1) 560 N
(2) 2800 N
(3) 2240 N
(4) 14 N

A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution?
(1) 25
(2) 50
(3) 100
(4) 20

Q3. A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius 9 m and completes 120 resolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in $\mathrm { m } / \mathrm { s } ^ { 2 }$ ):
(1) $57600 \pi ^ { 2 } \mathrm {~ms} ^ { - 2 }$
(2) Zero
(3) $4 \pi ^ { 2 } \mathrm {~ms} ^ { - 2 }$
(4) $16 \pi ^ { 2 } \mathrm {~ms} ^ { - 2 }$

Q6. A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is : ( Given $=$ Radius of geo-stationary orbit for earth is $4.2 \times 10 ^ { 4 } \mathrm {~km}$ )
(1) $1.4 \times 10 ^ { 4 } \mathrm {~km}$
(2) $1.05 \times 10 ^ { 4 } \mathrm {~km}$
(3) $8.4 \times 10 ^ { 4 } \mathrm {~km}$
(4) $1.68 \times 10 ^ { 5 } \mathrm {~km}$