A sphere of mass M and radius R 2 has a concentric cavity of radius R 1 as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as 0 ≤ r ≤ ∞
The radius of the earth is R. For a satellite to appear stationary, it must be placed in orbit around the earth at a height of about (given R = 6380 km) nR above the surface of earth.Then the value of n is
The value of g at a particular point is 10 ms -2 . Suppose the earth shrinks uniformly to half of its present size without losing any mass. The value of g at the same point (assuming that the distance of the point from the centre of the earth does not change) will now be
A tunnel is dug along a diameter of the earth. If M e and R. are the mass and radius, respectively, of the earth, then the force on a particle of mass m placed in the tunnel at a distance r from the centre is
A body weighs 64 N on the surface of the earth. What is the gravitational force (in N) on it due to the earth at a height equal to one-third of the radius of the earth ?
A planet revolves about the sun in elliptical orbit. The areal velocity d A d t of the planet is 4.0 × 10 16 m 2 / s . The least distance between planet and the sun is 2 × 10 12 m . Then the maximum speed of the planet in km/s is
Gravitational acceleration on the surface of the planet is 6 11 g , where g is the acceleration due to gravity on the surface of earth. The average mass density of the planet is 2/3 time that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms -1 , then find the escape speed on the surface of the planet (in kms -1 ).
Suppose earth’s orbital motion around the sun is suddenly stopped. Find the time (in days) taken by the earth to fall into the sun.
Let g be the acceleration due to gravity at the earth’s surface and K be the rotational kinetic energy of the earth. Suppose the earth’s radius decreases by 2 % . Keeping mass of earth and angular momentum constant, then
16kg and 9kg are separated by25m. The velocity with which a body should be projected from the midpoint of the line joining the two masses so that it just escape is
Four masses ‘m’ each are orbiting in a circle of radius’r’ in the same direction under gravitational force. Velocity of each particle is
A body A of mass m is moving in a circular orbit of radius R about a planet . Another body B of mass m 2 collides with A with a velocity which is half v 2 the instantaneous velocity v of A. The collision is completely inelastic . Then the combine body:
The acceleration due to gravity on the earth’s surface at the poles is g and angular velocity of the earth about the axis passing through the pole is ω . An object is weighed at the equator and at a height h above the poles by using a spring balance. If the weights are found to be same, then h is : (h <<R, where R is the radius of the earth)
Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to R + 5 / 2 , then
A tunnel is dug along the diameter of the earth (radius R and mass M). There is a particle of mass ‘ m ‘ at the centre of the tunnel. The minimum velocity given to the particle so that it just reaches to the surface of the earth is
A projectile of mass m is fired from the surface of earth at an angle α = 60 ∘ from the vertical. The initial speed of the projectile is u = G M R where M and R are mass and radius of the earth. Find the height by the projectile in kilometer. Neglect air resistance and the rotation of earth. (Given R = 6400 km)
A uniform thin rod of mass m and length l is bent into a semicircle. Gravitational field intensity at the centre O is :
In a satellite if the time of revolution is T, then kinetic energy is proportional to
A satellite is revolving around the Earth in a circular orbit with a constant speed. If its speed is made 2 times by supplying energy from an external source then what will be the path of satellite after this?
A body is released from a point of distance R’ from the centre of earth. Its velocity at the time of striking the earth will be R ′ > R e
If a particle of mass z is projected with minimum velocity from the surface of a star with kinetic energy K 1 GMm / a and potential energy at surface of the star K 2 GMm / a towards the star of same mass m and radius a (K 1 and K 2 are constants) to reach the other star. Find the distance between the centre of the two stars
The sun’s mass is about 3.2 x 10 5 times the earth’s mass. The sun is about 400 times as far from the earth as the earth is from the moon. Assume that the sun-moon distance is constant and equal to sun-earth distance. The ratio of the magnitudes of the gravitational pull of the sun on the moon F ms and of the earth on the moon F em will be F ms F me = ‘ n ‘ . The value of n is
A ring of radius R = 4 m made of a highly dense material has a mass m 1 = 5 .4 × 10 9 kg distributed uniformly over its circumference. A highly dense particle of mass m 2 = 6 × 10 8 kg is placed on the axis of the ring at a distance 3 m from the center of the ring. Find the speed of the particle (in cm/s), when the particle is at the center of the ring. Except mutual gravitational interaction of the two, neglect all other forces.
A sky laboratory of mass 3 x 10 3 kg has to be lifted from one circular orbit of radius 2R into another circular orbit of radius 3R. Calculate the minimum energy (in x 10 10 J ) required if the radius of earth is R = 6 .4 × 10 6 m and g = 10 ms − 2
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
A solid sphere of mass M and a ring of mass m have their centres lying on the x-axis separated by a distance 2 2 r where r is the radius of the ring as shown in Fig. The gravitational force exerted by the sphere on the ring is
Find the percentage decrease in weight of a body, when taken 16 km below the surface of the earth. Take radius of the earth as 6400 km.
If the radius of Earth is 6000 km, what will be the weight of 120 kg body (in kg Wt.) if taken to a height of 2000 km above sea level (i.e., surface of the Earth).
A solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F 1 on a particle placed at P, distance 2R from the centre O of the sphere. A spherical cavity of radius R/2 is now made in the sphere as shown in figure. The sphere with cavity now applies an gravitational force F 2 on same particle placed at P. The ratio F 2 / F 1 will be
The distance of the centres of moon and earth is D . The mass of earth is 81 times the mass of the moon. At what distance from the centre of the earth, the gravitational force will be zero
Two sphere of mass m and M are situated in air and the gravitational force between them is F . The space around the masses is now filled with a liquid of specific gravity 3. The gravitational force will now be
The force of gravitation is
A body weighs 700 gm wt on the surface of the earth. How much will it weigh on the surface of a planet whose mass is 1 7 and radius is half that of the earth
The moon’s radius is 1/4 that of the earth and its mass is 1/80 times that of the earth. If g represents the acceleration due to gravity on the surface of the earth, that on the surface of the moon is
An object weights 72 N on earth. Its weight at a height of R /2 from earth is
What will be the acceleration due to gravity at height h if h >> R . Where R is radius of earth and g is acceleration due to gravity on the surface of earth
A body weight W newton at the surface of the earth. Its weight at a height equal to half the radius of the earth will be
The change in potential energy, when a body of mass m is raised to a height nR from the earth’s surface is ( R = Radius of earth)
Let g be the acceleration due to gravity at earth’s surface and K be the rotational kinetic energy of the earth. Suppose the earth’s radius decreases by 2% keeping all other quantities same, then
The masses and radii of the earth and moon are M 1 , R 1 and M 2 , R 2 respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is
If mass of earth is M , radius is R and gravitational constant is G , then work done to take 1 kg mass from earth surface to infinity will be
A body of mass m kg. starts falling from a point 2 R above the Earth’s surface. Its kinetic energy when it has fallen to a point ‘ R ’ above the Earth’s surface [ R -Radius of Earth, M -Mass of Earth, G- Gravitational Constant]
A body is projected vertically upwards from the surface of a planet of radius R with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is
Energy required to move a body of mass m from an orbit of radius 2 R to 3 R is
A particle falls towards earth from infinity. It’s velocity on reaching the earth would be
The angular velocity of rotation of star (of mass M and radius R ) at which the matter start to escape from its equator will be
The escape velocity from the surface of earth is V e . The escape velocity from the surface of a planet whose mass and radius are 3 times those of the earth will be
The escape velocity of a body on an imaginary planet which is thrice the radius of the earth and double the mass of the earth is ( v e is the escape velocity of earth)
Escape velocity on the surface of earth is 11.2 k m / s . Escape velocity from a planet whose mass is the same as that of earth and radius 1/4 that of earth is
Two particles of equal mass go round a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is
Suppose the gravitational force varies inversely as the n t h power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
In planetary motion the areal velocity of position vector of a planet depends on angular velocity ( ω ) and the distance of the planet from sun ( r ). If so the correct relation for areal velocity is
A tunnel is dug along a diameter of the earth. Find the force on a particle of mass M placed in the tunnel at a distance x from the center.
Two concentric spherical shells have masses M 1 , M 2 and radii R 1 , R 2 ( R 1 < R 2 ) , What is a force exerted by this system on a particle of mass m 1 if it is placed at a distance ( R 1 + R 2 ) / 2 from the centre?
R is the radius of the earth and ω is its angular velocity and g p is the value of g at the poles. The effective value of g at the latitude λ = 60 ° will be equal to
Assuming the earth to have a constant density, point out which of the following curves show the variation of acceleration due to gravity from the centre of earth to the points far away from the surface of earth
A body weight W newton at the surface of the earth. Its weight at a height equal to half the radius of the earth will be
A body of mass m rises to height h = R /5 from the earth’s surface, where R is earth’s radius. If g is acceleration due to gravity at earth’s surface, the increase in potential energy is
The gravitational field due to a mass distribution is E = K / x 3 in the x- direction. ( K is a constant). Taking the gravitational potential to be zero at infinity, its value at a distance x is
The ratio of the K.E. required to be given to the satellite to escape earth’s gravitational field to the K.E. required to be given so that the satellite moves in a circular orbit just above earth atmosphere is
Two identical satellites are at R and 7 R away from earth surface, the wrong statement is ( R = Radius of earth)
The orbital velocity of an artificial satellite in a circular orbit just above the earth’s surface is v . For a satellite orbiting at an altitude of half of the earth’s radius, the orbital velocity is
In a satellite if the time of revolution is T , then K.E. is proportional to
In the following four periods (i) Time of revolution of a satellite just above the earth’s surface ( T s t ) (ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth ( T m a ) (iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/kg ( T s p ) (iv) Period of an infinite length simple pendulum in the earth’s real gravitational field ( T i s )
If Gravitational constant is decreasing in time, what will remain unchanged in case of a satellite orbiting around earth
If g ∝ 1 R 3 (instead of 1 R 2 ) , then the relation between time period of a satellite near earth’s surface and radius R will be
A satellite of mass m is circulating around the earth with constant angular velocity. If radius of the orbit is R 0 and mass of the earth M , the angular momentum about the centre of the earth is
An astronaut inside an earth satellite, experiences weightlessness because
The tidal waves in the sea are primarily due to
A satellite of the earth is revolving in a circular orbit with a uniform speed v . If the gravitational force suddenly disappears, the satellite will
In order to find time, the astronaut orbiting in an earth satellite should use
If the angular speed of the earth is doubled, the value of acceleration due to gravity ( g ) at the north pole
At the surface of a certain planet, acceleration due to gravity is one-quarter of that on earth. If a brass ball is transported to this planet, then which one of the following statements is not correct
The weight of an object in the coal mine, sea level, at the top of the mountain are W 1 , W 2 a n d W 3 respectively, then
Spot the wrong statement :The acceleration due to gravity ‘ g ’ decreases if
A person will get less quantity of matter in kg –wt. at
A missile is launched with a velocity less than the escape velocity. The sum of its kinetic and potential energy is
The escape velocity of a particle of mass m varies as
For the moon to cease to remain the earth’s satellite, its orbital velocity has to increase by a factor of
The escape velocity for a body projected vertically upwards from the surface of earth is 11.2 km/s. If the body is projected at an angle of 45 o with the vertical, the escape velocity will be
The escape velocity for a rat of mass 1 kg from the earth surface is 11.2 k m s − 1 . The escape velocity for an elephant of mass 6000 kg would be
Two satellites of masses m 1 and m 2 ( m 1 > m 2 ) are revolving round the earth in circular orbits of radius r 1 and r 2 ( r 1 > r 2 ) respectively. Which of the following statements is true regarding their speeds v 1 a n d v 2 ?
A ball is dropped from a spacecraft revolving around the earth at a height of 120 km . What will happen to the ball?
If Gravitational constant is decreasing in time, what will change in case of a satellite orbiting around earth
A person sitting in a chair in a satellite feels weightless because
When a satellite going round the earth in a circular orbit of radius r and speed v looses some of its energy, then r and v change as
If satellite is shifted towards the earth. Then time period of satellite will be
If the mass of one particle is increased by 50% and the mass of another particle is decreased by50%, the force between them is
If the mass of the planet is 10% less than that of earth and radius of the planet is 20% greater than that of earth then the weight of 40kg person on that planet is
If the radius of earth decrease by 10% , the mass remaining unchanged , then the acceleration due to gravity
The height at which the value of acceleration due to gravity becomes 50% of that at the surface of the earth.(Radius of the earth =6400 km ) is
The angular velocity of the earth with which it has to rotate so that the acceleration due to gravity on 60° latitude becomes zero
A man weigh ‘W’ on the surface of the earth and his weight at a height of ‘R’ from surface of earth is (Radius of the earth)
Two satellites are revolving round the earth at different heights. The ratio of their orbital speeds is 2:1.If one of them is at a height of 100km, the height of the other satellite is(in km)
A person can jump to a height of 3m on the earth on a planet of density 1 3 rd that of the earth and radius 1 2 of the earth, he can jump to a height of
If g is acceleration due to gravity on the surface of the earth ,having radius R , the height at which the acceleration due to gravity reduces to g 2 is
Three identical particles each of mass ‘m’ are arranged at the corner of an equilateral triangle of side “a”. If they are to be in equilibrium, the speed with which they must revolve under the influence of one another’s gravity in a circular orbit circumscribing the triangle is
Three point masses each of mass ‘m’ rotate in a circle of radius r with constant angular velocity ω due to their mutual gravitational attraction. If at any instant, the masses are on the vertex of an equilateral triangle of side ‘a’, then the value of ω is
The period of revolution of a planet around the sun in circular orbit is same as that of period of similar planet revolving around a star of twice the radius of first orbit and ‘M’ is the mass of the sun and mass of star is
A body is projected with a velocity equal to 3/4th of the escape velocity from the surface of the earth. The height it reaches is (Radius of the earth is R)
Imagine a geo-stationery satellite of earth which is used as an inter continental telecast station. At what height will it have to be established
The work done to increase the radius of orbit of a satellite of mass ’m’ revolving around a planet of mass M from orbit of radius R in to another orbit of radius 3R is
At what height from the surface of earth , the total energy of satellite is equal to its potential energy at a height 2R from the surface of earth(R= radius of earth)
An artificial satellite is revolving around the earth in a circular orbit. Its velocity is one third of the escape velocity. Its height from the earth’s surface is (in km)
If an artificial satellite is moving in a circular orbit around earth with speed equal to one fourth of V e from earth, then height of the satellite above the surface of the earth is
By what percent ,the energy of the satellite has to be increased to shift it from an orbit of radius ’r’ to ‘3r’
The work done in bringing three particles each of mass 10gm from large distance to the vertices of an equilateral triangle of side 10cm is (approximately)
An artificial satellite revolves around earth in circular orbit of radius r with time period T. The satellite is made to stop in the orbit which makes it fall onto earth. Time of fall of the satellite on to earth is given by
If earth were to rotate on its own axis such that the weight of a person at the equator becomes half the weight at the poles, then its time period of rotation is (g=acceleration due to gravity near the poles and R is the radius of earth)
A satellite is moving with a constant speed’v’ in a circular orbit about the earth. An object of mass’m’ is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its ejection,the kinetic energy of the object is
A satellite of mass M is launched vertically upwards with initial speed u from the surface of the earth. After it reaches height R R = radius of the earth , it ejects a rocket of mass M 10 so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ( G is the gravitational constant; M e is the mass of the earth):
A box weighs 196N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to(Take g = 10 m s – 2 at the north pole and the radius of the earth = 6400km)
Consider two solid spheres of radii R 1 = 1 m , R 2 = 2 m and masses M 1 and M 2 , respectively. The gravitational field due to sphere 1 and 2 are shown. The value of M 1 M 2 is :
An asteroid is moving directly towards the centre of the earth. When at a distance of 10R(R is the radius of the earth) from the earth’s centre, it has a speed of 12 km/s. Neglecting the effect of earth’s atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is 11.2 km/s)? Give your answer to the nearest integer in kilometer/s——-
Planet A has mass M and radius R. Planet B has half the mass and half the radius of planet A. If the escape velocities from the planets A and B are u A a n d u B respectively , then u A u B = n 4 . The value of n is
The mass density of a spherical galaxy varies as K r over a large distance ‘r’ from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as
The height ‘h’ at which the weight of a body will be the same as that of at the same depth ‘h’ from the surface of the earth is (Radius of the earth is R and effect of the rotation of the earth is neglected):
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 R e . By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become 3 2 times larger. Due to this the farthest distance from the center of the earth that the satellite reaches is R. Value of R is :
The mass density of a planet of radius R varies with the distance r from its center as ρ ( r ) = ρ 0 1 – r 2 R 2 . Then the gravitational field is maximum at :
On the x-axis and at a distance x from the origin, the gravitational field due to a mass distribution is given by A x x 2 + a 2 3 2 in the x-direction. The magnitude of gravitational potential on the x-axis at a distance x, taking its value to be zero at infinity, is :
A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the orbit can be taken to be R itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is:
The value of the acceleration due to gravity is g 1 at a height h = R 2 ( R= radius of the earth) from the surface of the earth. It is again equal to g 1 at a depth d below the surface of the earth. The ratio d R equals
Three identical stars, each of mass M, form an equilateral triangle (stars are positioned at the corners) that rotates around the centre of the triangle. The system is isolated and edge length of triangle is ‘L’. The minimum amount of work done that is required to dismantle the system is (Assume L is much greater than size of stars)
Gravitational acceleration on the surface of the planet is 6 11 g , where g is the acceleration due to gravity on the surface of earth. The average mass density of the planet is 2/3 time that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms -1 , then find the escape speed on the surface of the planet (in kms -1 ).
What is gravitational potential energy of system consisting of two concentric spherical shells of mass, m, and 4 m and radii R and 2 R.
Gravitational acceleration on the surface of the planet is 8 11 g , where g is the acceleration due to gravity on the surface of earth. The average mass density of the planet is 2 times that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms -1 , then find the escape speed on the surface of the planet (in kms -1 ).
The ratio of Earth’s orbital angular momentum (about the sun) to its mass is 4.4 × 10 15 m 2 / s . The area enclosed by the earth’s orbit is N × 10 23 m 2 . Find the value of N ?
If the change in the value of g at a height h above the surface of the earth is the same as at a depth x below it, then (both x and h being much smaller than the radius of the earth)
The masses and radii of the earth and moon arc M 1 , R 1 and M 2 , R 2 , respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is
Two particles of equal mass go round a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is
At what depth below the surface of the earth, acceleration due to gravity is will be half its value 1600 km above the surface of the earth?
A body of mass m kg starts falling from a point 2R above the Earth’s surface. Its kinetic energy when it has fallen to a point R above the Earth’s surface [R: Radius of Earth, M:Mass of Earth, G: Gravitational Constant]
A missile is launched with a velocity less than the escape velocity. The sum of its kinetic and potential energy is
v e and v p denotes the escape velocity from the earth and another planet having twice the radius and the same mean density as the earth. Then
A satellite is moving around the earth with speed v in a circular orbit of radius r. If the orbit radius is decreased by 1%, its speed will
Two identical satellites are at R and 7 R away from earth surface, the wrong statement is (R = Radius of earth)
In the following four Periods, (i) Time of revolution of a satellite just above the earth’s surface T st (ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth T ma (iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/kg T sp (iv) Period of an infinite length simple pendulum in the earth’s real gravitational field T is
The magnitudes of the gravitational force at distances r 1 and r 2 from the centre of a uniform sphere of radius R and mass M are F 1 and F 2 , respectively. Then
A rocket of mass M is launched vertically from the surface of the earth with an initial speed V. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is
Two bodies of masses m 1 and m 2 are initially at rest at infinite distance apart They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is
A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius (1.01)R. The period of the second satellite is larger than that of the first one by approximately
A projectile is projected with velocity kv e in vertically upward direction from the ground into the space. (v e is escape velocity and k < 1). If air resistance is considered to be negligible, then the maximum height from the centre of earth to which it can go will be (R = radius of earth)
If the distance between the earth and the sun becomes half its present value, the number of days in a year would have been
Imagine a light planet revolving around a very massive star in a circular orbit of radius r with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to r 5/2 , then the square of the time period will be proportional to
A planet is revolving in an elliptical orbit around the Sun. Its closest distance from the Sun is r min and the farthest distance is r max . If the velocity of the planet at the distance of the closest approach is v 1 and that at the farthest distance from the Sun is v 2 , then v 1 /v 2
The earth moves around the Sun in an elliptical orbit as shown in the figure. The ratio OA/OB = x. The ratio of the speed of the earth at B to that at A is nearly
Suppose, the acceleration due to gravity at the Earth’s surface is 10 m/s 2 and at the surface of Mars it is 4.0 m/ s 2 . A 60-kg passenger goes from the Earth to the Mars in a spaceship moving with a constant velocity. Neglect all other objects in the sky. Which part of figure best represents the weight (net gravitational force) of the passenger as a function of time?
The gravitational force between two objects is proportional to 1/R (and not as 1/R 2 ) where R is separation between them, then a particle in circular orbit under such a force would have its orbital speed v proportional to
Two particles of equal mass go around a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is
The curves for potential energy (U) and kinetic energy (E k ) of a two particle system are shown in figure. At what points the system will be bound?
A sphere of mass M and radius R 2 has a concentric cavity of radius R 1 as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as ( 0 ≤ r ≤∝ )
If three uniform spheres, each having mass M and radius R, are kept in such away that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is
The correct graph representing the variation of total energy (E t ), kinetic energy (E k ) and potential energy (U) of a satellite with its distance from the centre of Earth is
The distance between the centres of the Moon and the earth is D. The mass of the earth is 81 times the mass of the Moon. At what distance from the centre of the Earth, the gravitational force will be zero?
As observed from the earth, the sun appears to move in an approximate circular orbit. For the motion of another planet like mercury as observed from the earth, this would
A solid sphere of radius R/2 is cut out of a solid sphere of radius R such that the spherical cavity so formed touches the surface on one side and the centre of the sphere on the other side, as shown. The initial mass of the solid sphere was M. lf a particle of mass m is placed at a distance 2.5R from the centre of the cavity, then what is the gravitational attraction on the mass m?
In our solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They
In planetary motion the areal velocity of position vector of a planet depends on angular velocity ( ω ) and the distance of the planet from sun (r). If so the correct relation for areal velocity is
Particles of masses 2M, m and M are respectively at points A, Band C with AB = 1 2 ( BC ) . m is much-much smaller than M and, at time t = 0, they are all at rest as given in figure. At subsequent times before any collision takes place.
The maximum and minimum distances of a comet from the sun are 8 × 10 12 m and 1.6 × 10 12 m. If its velocity when nearest to the sun is 60 m/s, what will be its velocity in m/s when it is farthest?
A planet moves around the sun. At a given point P, it is closest from the sun at a distance d 1 and has a speed v 1 . At another point Q, when it is farthest from the sun at a distance d 2 , its speed will be
Three identical point masses, each of mass 1 kg lie in the x-y plane at points (0, 0), (0, 0.2m) and (0.2m, 0). The net gravitational force on the mass at the origin is
A spherical shell is cut into two pieces along a chord as shown in the figure. P is a point on the plane of the chord. The gravitational field at P due to the upper part is I 1 and that due to the lower part is I 2 . What is the relation between them?
If g is same at a height h and at a depth d, then
If R is the radius of the earth and g the acceleration due to gravity on the earth’s surface, the mean density of the earth is
Figure shows two shells of masses m 1 and m 2 . The shells are concentric. At which point, a particle of mass m shall experience zero force?
The value of g (acceleration due to gravity) at earth’s surface is 10 ms -2 . Its value in ms -2 at the centre of the earth which is assumed to be a sphere of radius R metre and uniform mass density is
How many hours would make a day if the Earth were rotating at such a high speed that the weight of a body on the equator were zero?
If a man at the equator would weigh (3/5)th of his weight, The angular speed of the earth is
The distances from the centre of the Earth, where the weight of a body is zero and one-fourth that of the weight of the body on the surface of the Earth are (assume R is the radius of the earth)
If the radius of the earth decreases by 10%, the mass remaining unchanged, what will happen to the acceleration due to gravity?
What should be the angular velocity of rotation of the earth about its own axis so that the weight of a body at the equator reduces to 2/5 of its present value? (Take R as the radius of the earth)
If the mass of a planet is 10% less than that of the earth and the radius is 20% greater than that of the earth, the acceleration due to gravity on the planet will be
The value of g at a certain height h above the free surface of the earth is x/4 where x is the value of g at the surface of the earth. The height h is
The mass of the moon is 1/81 of the earth but the gravitational pull is 1/6 of the earth. It is due to the fact that
Acceleration due to gravity on moon is 1/6 of the acceleration due to gravity on earth. If the ratio of densities pf earth ρ e and moon ρ m is ρ e ρ m = 5 3 then radius of moon R m in terms of R e will be
The radii of two planets are respectively R 1 and R 2 and their densities are respectively ρ 1 and ρ 2 . The ratio of the accelerations due to gravity at their surfaces is
If W 1 , W 2 and W 3 represent the work done in moving a particle from A to B along three different paths 1 ,2 and 3, respectively, (as shown in the figure) in the gravitational field of a point mass m, find the correct relation between W 1 , W 2 and W 3 .
A space ship is launched into a circular orbit close to the surface of the earth. The additional velocity now imparted to the spaceship in the orbit to overcome the gravitational pull is
If g is acceleration due to gravity on the Earth’s surface, the gain in the potential energy of an object of mass m raised from the surface of Earth to a height equal to the radius R of the earth is
A skylab of mass m kg is first launched from the surface of the earth in a circular orbit of radius 2R (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius 3R. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit are
The gravitational potential due to earth at infinite distance from it is zero. Let the gravitational potential at a point P be − 5 J kg − 1 . Suppose, we arbitrarily assume the gravitational potential at infinity to be + 10 J kg − 1 , then the gravitational potential at P will be
Two bodies of masses M 1 and M 2 are placed at a distance R apart. Then at the position where the gravitational field due to them is zero, the gravitational potential is
In the solar system, the Sun is in the focus of the system for Sun-Earth binding system. Then the binding energy for the system will be [given that radius of the earth’s orbit round the Sun is 1.5 × 10 11 m and mass of the earth = 6 × 10 24 kg ]
The minimum energy required to launch a m kg satellite from the Earth’s surface in a circular orbit at altitude 2R, where R is the radius of earth is
In order to shift a body of mass m from a circular orbit of radius 3R to a higher orbit of radius 5R around the earth, the work done is
Assuming earth as a uniform sphere of radius R, if we project a body along the smooth diametrical chute from the centre of earth with a speed v such that it will just reach the earth’s surface then v is equal to
A body of mass m is situated at a distance 4R e . above the earth’s surface, where R e is the radius of earth. How much minimum energy be given to the body so that it may escape
The escape velocity from the centre of a uniform ring of mass M and radius R is
The potential energy of interaction between the semicircular ring of mass M and radius R, and the particle of mass M placed at the centre of curvature of the semicircular arc is
Two satellites A and B of masses m 1 and m 2 m 1 = 2 m 2 are moving in circular orbits of radii r 1 and r 2 r 1 = 4 r 2 , respectively, around the earth. If their periods are T A and T B , then the ratio T A /T B is
A satellite of mass m is revolving around the earth at height R (radius of the earth) from the earth’s surface. Its potential energy will be
A satellite moves around the earth in a circular orbit with speed v. If m is the mass of the satellite, its total energy is
Two satellites of masses m 1 and m 2 (m 1 > m 2 )are revolving around the earth in a circular orbit of radii r 1 and r 2 (r 1 > r 2 ),respectively. Which of the following statements is true regarding their speeds v 1 and v 2 ?
Two satellites A and B of the same mass are revolving around the earth in the concentric circular orbits such that the distance of satellite B from the centre of the earth is thrice as compared to the distance of the satellite A from the centre of the earth. The ratio of the centripetal force acting on B as compared to that on A is
A satellite is seen after each 8 hours over equator at a place on the earth when its sense of rotation is opposite to the earth. The time interval after which it can be seen at the same place when the sense of rotation of earth and satellite is same will be
A satellite is moved from one circular orbit around the earth to another of lesser radius. Which of the following statement is true?
The orbital velocity of an artificial satellite in a circular orbit just above the earth’s surface is v. For a satellite orbiting at an altitude of half of the earth’s radius, the orbital velocity is
In the following four periods (i) Time of revolution of a satellite just above the earth’s surface T s (ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth T ma (iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/kg T sp (iv) Period of an infinite length simple pendulum in the earth’s real gravitational field T is
A satellite can be in a geostationary orbit around earth in an orbit of radius r. If the angular velocity of earth about its axis doubles, & satellite can now be in a geostationary orbit around earth of radius
A satellite in equatorial plane is rotating in the direction of earth’s rotation with time interval between its two consecutive appearances over head of an observer as time period of rotation of the earth , T E . What is the time period of the satellite?
A satellite is orbiting with areal velocity v a . At what height from the surface of the earth, it is rotating, if the radius of earth is R?
A spherical planet has uniform density π 2 × 10 4 kg / m 3 . Find out the minimum period for a satellite in a circular orbit around it in seconds. (Use G = 20 3 × 10 − 11 N − m 2 kg 2 .
Two satellites of the same mass are launched in the same orbit around the earth so as to rotate opposite to each other. If they collide in elastically and stick together as wreckage, the total energy of the system just after collision is
A ball of mass m is fired vertically upwards from the surface of the earth with velocity nv e where v e is the escape velocity and n > 1. To what height will the ball rise? Neglecting, air resistance, take radius of the earth as R
The maximum vertical distance through which a fully dressed astronaut can jump on the earth is 0.5 m. If mean density of the Moon is two-thirds that of the Earth and radius is one quarter that of the Earth, the maximum vertical distance through which he can jump on the Moon and the ratio of the time of duration of the jump on the Moon to hold on the Earth are
A solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F 1 on a particle placed at a distance 2R from the centre of the sphere. A spherical cavity of radius R/2 is now made in the sphere as shown in the figure. The sphere with the cavity now applies a gravitational force F 2 on the same particle. The ratio F 1 /F 2 is
A uniform ring of mass m and radius r is placed directly above a uniform sphere of mass M and of equal radius. The centre of the ring is directly above the centre of the sphere at a distance r 3 as shown in the figure. The gravitational force exerted by the sphere on the ring will be
In a cosmic event, suppose a planet heavier than the earth with mass KM (K > 1) and radius K’R (K’ > 1) passes through a path near the earth (M and R are the mass and radius of earth).At what closest distance from surface of planet, we are in danger of being thrown into space:
A small body of superdense material, whose mass is twice the mass of the earth but whose size is very small compared to the size of the earth, starts from rest at a height H <<R above the earth’s surface, and reaches the earth’s surface in time t. Then t is equal to
Binary stars of comparable masses rotate under the influence of each other’s gravity at a distance 2 G / ω 2 1 / 3 where ω is the angular velocity of each of the system. If difference between the masses of two stars is 6 units. Find the ratio of the masses of smaller to bigger star.
A small mass 2kg moved slowly from the surface of earth to a height of 6.4 x 10 6 m above the earth. The work done [in Mega Joule] is .
A particle of mass m is projected from the surface of the earth with velocity 10 km/sec at an angle 60 o with horizontal. If V is the speed of the particle at the point of maximum height of the particle from the surface of the earth, then find the value of V which will be less than (a minimum value of velocity in km/sec) an integer from 0 to 9.
A satellite moves eastwards very near the surface of the Earth in equatorial plane with speed (v 0 ). Another satellite moves at the same height with the same speed in the equatorial plane but westwards. If R = radius of the Earth and ω be its angular speed of the Earth about its own axis. Then find the approximate difference in the two time period as observed on the Earth.
Two planets revolve with same angular velocity about a star. The radius of orbit of outer planet is twice the radius of orbit of the inner planet. If T is time period of the revolution of outer planet, find the time in which inner planet will fall into the star. If it was suddenly stopped.
A projectile of mas s m is fired from the surface of earth at an angle α = 60 ∘ from the vertical. The initial speed m of the projectile is u = GM R , where M and R are mass and radius of the earth. Find the height by the projectile in kilometer. Neglect air resistance and the rotation of earth. (Given R = 6400 km)
Find the percentage decrease in weight of a body, when taken 16 km below the surface of the earth. Take radius of the earth as 6400 km.
The radius of the earth is about 6400 km and that of mars is about 3200 km. The mass of the earth is about 10 times the mass of mars. An object weighs 80 N on the surface of earth. What will be its weight (in N) on the surface of mars?
The value of acceleration due to gravity at the surface of the earth is 9.8 ms − 2 and its mean radius is about 6.4 × 10 6 m . Assuming that we could get more soil from somewhere, estimate how thick (in km) would an added uniform outer layer on the earth have to have the value of acceleration due to gravity 10 ms − 2 exactly? Given the density of the earth’s soil, ρ = 5520 kgm − 3
A man can throw a ball at a speed on the earth which can cross a river of width 10 m. The man reaches on an imaginary planet whose mean density is twice that of the earth. Find out the maximum possible radius of the planet (in km) so that if the man throws the ball at the same speed it may escape from the planet. Given radius of the earth = 6.4 x 10 6 m.
A particle of mass 50 g is kept on the surface of a uniform sphere of mass 120 kg and radius 50 cm. Find the work to be done (in x 10 -10 joule) against the gravitational force between them to take the particle far away from the sphere. Take G = 6 .67 × 10 − 11 Nm 2 kg − 2 .
If the radius of the earth were to shrink by 2% its mass remaining the same, by how much percentage would the acceleration due to gravity on the earth’s surface would increase?
A planet revolves around the sun is an elliptical orbit. The maximum and minimum distances of the planet from the sun are a and b respectively. The time period of revolution of the planet is proportional to
A tunnel is dug along the diameter of the earth. An object is held in the tunnel at a distance x from the centre of the earth. The magnitude of the gravitational force on the object is proportional to
Which of the following astronomer first proposed that sun is static and earth rounds sun
A planet is revolving around the sun as shown in elliptical path The correct option is
The distance of a planet from the sun is 5 times the distance between the earth and the sun. The time period of the planet is
The radius of orbit of a planet is two times that of the earth. The time period of planet is
The orbital angular momentum of a satellite revolving at a distance r from the centre is L. If the distance is increased to 16r, then the new angular momentum will be
Kepler’s second law (law of areas) is nothing but a statement of
In planetary motion the areal velocity of position vector of a planet depends on angular velocity ω and the distance of the planet from sun (r). If so the correct relation for areal velocity is
The ratio of the distances of two planets from the sun is 1.38. The ratio of their period of revolution around the sun is
If a new planet is discovered rotating around Sun with the orbital radius double that of earth, then what will be its time period (in earth’s days)
In an elliptical orbit under gravitational force, in general
What does not change in the field of central force
The eccentricity of earth’s orbit is 0.0167. The ratio of its maximum speed in its orbit to its minimum speed is
Suppose the law of gravitational attraction suddenly changes and becomes an inverse cube law i.e. F ∝ 1 / r 3 , but still remaining a central force. Then
The mass of a planet that has a moon whose time period and orbital radius are T and R respectively can be written as
If orbital velocity of planet is given by v = G a M b R c , then
According to Kepler’s law the time period of a satellite varies with its radius as
Two satellite are revolving around the earth with velocities v 1 and v 2 and in radii r 1 and r 2 (r 1 > r 2 ) respectively. Then
The condition for a uniform spherical mass m of radius r to be a black hole is [G= gravitational constant and g= acceleration due to gravity]
Earth is revolving around the sun if the distance of the Earth from the Sun is reduced to 1/4 th of the present distance then the present day length reduced by
Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between planet and star is proportional to R − 5 2 , then T 2 is proportional to
A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of earth
The magnitudes of the gravitational force at distances r 1 and r 2 from the centre of a uniform sphere of radius R and mass M are F 1 and F 2 respectively. Then
A mass M is split into two parts, m and (M–m), which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts
If the radius of the earth were to shrink by 1% its mass remaining the same, the acceleration due to gravity on the earth’s surface would
Suppose the gravitational force varies inversely as the n th power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
The radius and mass of earth are increased by 0.5%. Which of the following statements are true at the surface of the earth
In order to make the effective acceleration due to gravity equal to zero at the equator, the angular velocity of rotation of the earth about its axis should be ( g = 10 ms − 2 and radius of earth is 6400 kms)
A simple pendulum has a time period T 1 when on the earth’s surface and T 2 when taken to a height R above the earth’s surface, where R is the radius of the earth. The value of T 2 / T 1 is
A body of mass m is taken from earth surface to the height h equal to radius of earth, the increase in potential energy will be
An artificial satellite moving in a circular orbit around the earth has a total (kinetic + potential) energy E 0 . Its potential energy is
A rocket of mass M is launched vertically from the surface of the earth with an initial speed V. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is
A solid sphere of uniform density and radius 4 units is located with its centre at the origin O of coordinates. Two spheres of equal radii 1 unit with their centres at A(– 2, 0, 0) and B(2, 0, 0) respectively are taken out of the solid leaving behind spherical cavities as shown in figure
A projectile is projected with velocity kv e in vertically upward direction from the ground into the space. (v e is escape velocity and k < 1). If air resistance is considered to be negligible then the maximum height from the centre of earth to which it can go, will be : (R = radius of earth)
If the distance between the earth and the sun becomes half its present value, the number of days in a year would have been
Two bodies of masses m 1 and m 2 are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is
Assuming the earth to have a constant density, point out which of the following curves show the variation of acceleration due to gravity from the centre of earth to the points far away from the surface of earth
A geostationary satellite orbits around the earth in a circular orbit of radius 36000 km. Then, the time period of a satellite orbiting a few hundred kilometres above the earth’s surface ( R Earth = 6400 km ) will approximately be
A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius (1.01)R. The period of the second satellite is larger than that of the first one by approximately
The diagram showing the variation of gravitational potential of earth with distance from the centre of earth is
Which one of the following graphs represents correctly the variation of the gravitational field (F) with the distance (r) from the centre of a spherical shell of mass M and radius a
A sphere of mass M and radius R 2 has a concentric cavity of radius R 1 as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as ( 0 ≤ r ≤ ∞ )
By which curve will the variation of gravitational potential of a hollow sphere of radius R with distance be depicted
Suppose, the acceleration due to gravity at the earth’s surface is 10 m/s 2 and at the surface of Mars it is 4.0 m/s 2 . A 60 kg passenger goes from the earth to the Mars in a spaceship moving with a constant velocity. Neglect all other objects in the sky. Which part of figure best represents the weight (net gravitational force)of the passenger as a function of time.
Which of the following graphs represents the motion of a planet moving about the sun
The curves for potential energy (U) and kinetic energy ( E k ) of a two particle system are shown in figure. At what points the system will be bound?
The correct graph representing the variation of total energy ( E t ) , kinetic energy ( E k ) and potential energy (U) of a satellite with its distance from the centre of earth is
A shell of mass M and radius R has a point mass m placed at a distance r from its centre. The gravitational potential energy U (r) vs r will be
Two planets A and B of masses m A and m B are considered fixed in space at separation d. The speed with which a body of mass m is to be projected from the midpoint of the line joining A and B so that the body will escape to infinity is nG m A + m B d . Find the value of n.
The escape velocity of 10g body from the Earth is 11.2 km s -1 . Ignoring air resistance, the escape velocity of 10kg of the iron ball from the Earth will be
A satellite revolving around earth in a circular orbit of radius 4R. Due to ejection of a rocket the speed of satellite is reduced to half as a result of which satellite starts falling toward earth. Find the angle at which it will hit the earth surface.
A rocket is launched from the surface of Earth with an initial speed of 10 kms -1 . The maximum distance of rocket from the surface of Earth is n × 10 4 km. The value of n is . (Radius of Earth = 6400 km and escape velocity from Earth’s surface = 11.2 km/s) Neglect atmospheric resistance.
The change in the value of g at a height h above the surface of earth is the same as at a depth d below the earth. When both d and h are much smaller than the radius of earth, then which one of the following is correct?
A mass m is placed at point P which lies on the axis of a ring of mass M and radius R at a distance R from its centre. The gravitational force on mass m is
Two stars of masses m 1 and m 2 are parts of a star system. The radii binary of their orbits are r 1 and r 2 respectively, measured from the centre of mass of the system. The magnitude of gravitational force m 2 exerts on m 1 , is
A point P is on the axis of a fixed ring of mass M and radius R, at a distance 2R from the centre O. A small particle starts from P and reaches O under the gravitational attraction only. Its speed at O will be
Two bodies of masses m 1 and m 2 are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is
A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work done against the gravitational force between them to take the particle far away from the sphere.
A particle of mass 50 g is kept on the surface of a uniform sphere of mass 120 kg and radius 50 cm. Find the work to be done in × 10 − 10 J against the gravitational force between them to take the particle far away from the sphere. Take G = 6 .67 × 10 − 11 Nm 2 kg − 2 .
A uniform ring of mass M and radius r is placed directly above a uniform sphere of mass 8 M and of same radius R. The centre of the ring is at a distance of d = 3 R from the centre of the sphere. The gravitational attraction between the sphere and the ring is
Imagine a light planet revolving around a very massive star in a circular orbit of radius r with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to R − 3 / 2 , then T 2 is proportional to
