A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let r be the distance of the body from the centre of the star and let its linear velocity be v, angular velocity to, kinetic energy K, gravitational potential energy U, total energy E and angular momentum /. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.
In equilibrium, the gravitation pull provides the necessary centripetal force. The situation is shown in the diagram, where a body of mass m is revolving around a star of mass M.
Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface?
In the following two exercises, choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig.)
(i) a, (ii) b, (iii) c, (iv) 0.
We can shield a charge from electric fields by putting it inside a hollow conductor. Can we shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?
Molecules in air in the atmosphere are attracted by gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree.
There have been suggestions that the value of the gravitational constant G becomes smaller when considered over very large time period (in billions of years) in the future. If that happens, for our earth,
(a) nothing will change
(b) we will become hotter after billions of years
(c) we will be going around but not strictly in closed orbits
(d) after sufficiently long time we will leave the solar system
Mean solar day is the time interval between two successive noon when sun passes through zenith point (meridian).
Sidereal day is the time interval between two successive transit of a distant star through the zenith point (meridian).
By drawing appropriate diagram showing the earth's spin and orbital motion, show that mean solar day is 4 minutes longer than the sidereal day. In other words, distant stars would rise 4 minutes early every successive day.
Does the escape speed of a body from the Earth depend on (a) the mass of the body, (b) the location from where it is projected, (c) the direction of projection, (d) the height of the location from where the body is launched?
For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow (i) d, (ii), e, (iii) f (iv) g.
An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity?
The centre of mass of an extended body on the surface of the earth and its centre of gravity
(a) are always at the same point for any size of the body
(b) are always at the same point only for spherical bodies
(c) can never be at the same point
(d) is close to each other for objects, say of sizes less than 100 m
(e) both can change if the object is taken deep inside the earth
Shown are several curves [Fig. (a), (b), (c), (d), (e) and (f)]. Explain with reason, which ones amongst them can be possible trajectories traced by a projectile (neglect air friction).
A comet orbits the Sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed (b) angular speed (c) angular momentum (d) kinetic energy (e) potential energy (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
Choose the correct alternative:
(a) Acceleration due to gravity increases/decreases with increasing altitude.
(b) Acceleration due to gravity increases/decreases with increasing depth (assume the Earth to be a sphere of uniform density).
(c) Acceleration due to gravity is independent of the mass of the Earth/mass of the body.
(d) The formula – GMm (1/r2-1/r1) is more/less accurate than the formula mg (r2 – r1) for the difference of potential energy between two points r2 and r1 distance away from the centre of the Earth.
Let us assume that our galaxy consists of 2.5 x 1011 stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution? Take the diameter of the Milky way to be 105 ly.
If the mass of the sun were ten times smaller and gravitational constant G were ten times larger in magnitude. sun Then,
(a) walking on ground would become more difficult
(b) the acceleration due to gravity on the earth will not change
(c) raindrops will fall much faster
(d) airplanes will have to travel much faster
How is the gravitational force between two point masses affected when they are dipped in water keeping the separation between them the same?
Two identical heavy spheres are separated by a distance 10 times their radius. Will an object placed at the mid-point of the line joining their centres be in stable equilibrium or unstable equilibrium? Give reason for your answer.
What is the angle between the equatorial plane and the orbital plane of
(a) polar satellite?
(b) geostationary satellite?
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 x 1024 kg, radius = 6400 km.
Answer the following:
(a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?
(b) An astronaut inside a small spaceship orbiting around the Earth cannot detect gravity. If the space station orbiting around the Earth has a large size, can he hope to detect gravity?
(c) If you compare the gravitational force on the Earth due to the Sun to that due to the Moon, you would find that the Sun’s pull is greater than the Moon’s pull. (You can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the Moon’s pull is greater than the tidal effect of Sun. Why?
Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 x 108 m. Show that the mass of Jupiter is about one-thousandth that of the Sun.
Supposing Newton's law of gravitation for gravitational forces F1 and F2 between two masses m1 and m2 at positions r1, and r2 read
(a) the acceleration due to gravity on the earth will be different for different objects
(b) none of the three laws of Kepler will be valid
(c) only the third law will become invalid
(d) for n negative, an object lighter than water will sink in water
If mean angular velocity co corresponds to 1 ° per day, then p = 1.034 ° per day and a = 0.967 ° per day.
Since, 361 ° = 24 mean solar day we get (360 + 1.034) which corresponds to 24 h, 8.14″ (8.1″ longer) and 360.967 °, corresponds to 23 h 59 min 52″ (7.9″ smaller).
This does not explain the actual variation of the length of the day during the year.
[G = 6.67 x 10-11 SI unit and M= 6 x 1024 kg]
Suppose there existed a planet that went around the Sun twice as fast as the Earth. What would be its orbital size as compared to that of the Earth?
A body weighs 63 N on the surface of the Earth. What is the gravitational force on it due to the Earth at a height equal to half the radius of the Earth?
A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 x 1024 kg; radius of the earth = 6.4 x 106 m; G = 6.67 x 10-11 N m2 kg-2.
Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem.
A Saturn year is 29.5 times the Earth year. How far is the Saturn from the Sun if the Earth is 1.50 x 108 km away from the Sun?
Two stars each of one solar mass (=2 x 1030 kg) are approaching each other for a head on collision.When they are at a distance 109 km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
Two heavy spheres each of mass 100 kg and radius 0.10 mare placed 1.0 m apart on ahorizontal table. What is the gravitational field and potential at the mid point of the line joining the centres of the spheres ? Is an object placed at that point in equilibrium ? If so, is the equilibrium stable or unstable?
Particles of masses 2M, m and M are respectively at points A, B and 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.
(a) m will remain at rest
(b) m will move towards M
(c) m will move towards 2M
(d) m will have oscillatory motion
Choose the correct alternative:
(a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
(b) The energy required to launch an orbiting satellite out of Earth’s gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of Earth’s influence.
A star 2.5 times the mass of the sun and collapsed to a size of 12 km rotates with a speed of 1.2 rev. per second. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (mass of the sun = 2 x 1030 kg).
How will you ‘weigh the sun’, that is, estimate its mass? The mean orbital radius of the earth around the sun is 1.5 x 108 km.
If the sun and the planets carried huge amounts of opposite charges,
(a) all three of Kepler's laws would still be valid
(b) only the third law will be valid
(c) the second law will not change
(d) the first law will still be valid
The escape speed of a projectile on the Earth’s surface is 11.2 km s-1. A body is projected out with thrice this speed. What is the speed of the body far away from the Earth? Ignore the presence of the Sun and other planets.
If the law of gravitation, instead of being inverse square law, becomes an inverse cube law
(a) planets will not have elliptic orbits
(b) circular orbits of planets is not possible
(c) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic
(d) there will be no gravitational force inside a spherical shell of uniform density
A mass m is placed at P a distance h along the normal through the centre O of a thin circular ring of mass M and radius r (figure).
If the mass is moved further away such that OP becomes 2h, by what factor the force of gravitation will decrease, if h = r?
A rocket is fired vertically’ from the surface of Mars with a speed of 2 km s-1. If 20% of its initial energy is lost due to Martian atmospheric resistance, how far will the rocket go from the surface of mars before returning to it? Mass of Mars = 6.4 x 1023 kg; radius of Mars = 3395 km; G = 6.67 x 10-11 N m2 kg-2
A spaceship is stationed on Mars. How much energy must be expended on the spaceship to rocket it out of the solar system? Mass of the spaceship = 1000 kg, Mass of the Sun = 2 x 1030 kg. Mass of the Mars = 6.4 x 1023 kg, Radius of Mars = 3395 km. Radius of the orbit of Mars = 2.28 x 1011 m, G = 6.67 x 10-11 N m2 kg-2.
An object of mass m is raised from the surface of the earth to a height equal to the radius of the earth, that is, taken from a distance R to 2R from the centre of the earth. What is the gain in its potential energy?
Choose the wrong option.
(a) Inertial mass is a measure of difficulty of accelerating a body by an external force whereas the gravitational mass is relevant in determining the gravitational force on it by an external mass.
(b) That the gravitational mass and inertial mass are equal is an experimental result.
(c) That the acceleration due to gravity on the earth is the same for all bodies is due to the equality of gravitational mass and inertial mass.
(d) Gravitational mass of a particle like proton can depend on the presence of neighbouring heavy objects but the inertial mass cannot.
Earth's orbits an ellipse with eccentricity 0.0167. Thus, the earth's distance from the sun and speed as it moves around the sun varies from day-to-day. This means that the length of the solar day is not constant through the year. Assume that the earth's spin axis is normal to its orbital plane and find out the length of the shortest and the longer, day. A day should be taken from noon to noon. Does this explain variation of length of the day during the year?
A rocket is fired vertically with a speed of 5 km s-1 from the earth’s surface. How far from the earth does the rocket go before returning to the earth? Mass of the earth = 6.0 x 1024 kg; mean radius of the earth = 6.4 x 106 m; G = 6.67 x 10-11 N m2 kg-2.
Which of the following are true?
(a) A polar satellite goes around the earth's pole in north-south direction
(b) A geostationary satellite goes around the earth in east-west direction
(c) A geostationary satellite goes around the earth in west-east direction
(d) A polar satellite goes around the earth in east-west direction
Six point masses of mass m each are at the vertices of a regular hexagon of side l. Calculate the force on any of the masses.