Gravitation

### A satellite is in an elliptic orbit around the earth with an aphelion of 6R and perihelion of 2R where R = 6400 km is the radius of the earth. Find eccentrically of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred to a circular orbit of radius 6R?

Solution: Radius of perigee is given as $r_{p}=2R$ Radius of apogee is given as $r_{a}=6R$ And we know, $r_{p}=a(1-e)=2R$ and, $r_{a}=a(1+e)=6R$ From the above equations, we get $e = 1/2$ From the...

### Earth’s orbit is 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 throughout the year. Assume that earth’s spin axis is normal to its orbital plane and find out the length of the shortest and the longest day. A day should be taken from noon to noon. Does this explain the variation of the length of the day during the year?

Solution: Velocity of the earth at perigee is given as $v_{p}$ Velocity of the earth at apogee is given as $v_{a}$ Angular velocity of the earth at perihelion is given as $\omega_{p}$ Angular...

### A satellite is to be placed in equatorial geostationary orbit around the earth for communicationa) calculate height of such a satelliteb) find out the minimum number of satellites that are needed to cover entire earth so that at least one satellites is visible from any point on the equator

a) Mass of the earth is given as $M=6\times 10^{24}kg$ Radius of the earth is given as $R=6.4 \times 10^{3}m$ Time period is given as $24.36 \times 10^{2}s$ $G=6.67 \times 10^{-11}Nm^{2}kg^{-1}$...

### 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.

$AE = AG + EG$ $AG + AG = 2AG$ $= 2l cos 30^{o}$ $AE = AC = \sqrt3l$ $AD = 2l$ As a result, $\frac{Gm^{2}}{l^{2}}$ is the force on A due to B along B to A $\frac{Gm^{2}}{3l^{2}}$ is the force on A...

### 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 ω, kinetic energy K, gravitational potential energy U, total energy E, and angular momentum l. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.

When a body moves around a star in equilibrium, the gravitational attraction produces a centripetal force. Consider a body of mass $m$ revolving in a circular path of radius $r$ around the star S of...

### 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. If the mass is removed further away such that OP becomes 2h, by what factor the force of gravitation will decrease, if h = r?

Solution: Let the radius of the ring be r Let the mass of the ring be m When small element dM is considered as the mass, the gravitation force becomes, $dF=\frac{G(dM)m}{x^{2}}$ where...

### 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?

Potential Energy of the body on the earth is given by $-GMm/R$ When the body is closer to the equator, then PE becomes $-GMm/2R$ Gain in PE is $1/2 mgR$

### Shown are several curves. Explain with reason, which ones amongst them can be possible trajectories traced by a projectile

Solution: Amongst the given figures, (c) show the focus of trajectory.

### Show the nature of the following graph for a satellite orbiting the earth.TE vs orbital radius R

Total energy of the satellite is $-GMm/2R$

### Show the nature of the following graph for a satellite orbiting the earth.a) KE vs orbital radius Rb) PE vs orbital radius R

a) $\mathrm{K}=1 / 2 \mathrm{mv}^{2}=(1 / 2 \mathrm{~m})(\mathrm{GM} / \mathrm{R})$ b) PE of satellite is $U = -GMm/R = -2K$

### Two identical heavy spheres are separated by a distance 10 times their radius. Will an object placed at the midpoint of the line joining their centres be in stable equilibrium or unstable equilibrium? Give a reason for your answer.

M = mass of the spheres R = radius of the spheres. P = midpoint of A and B. The magnitude of the force is expressed as, $F_{1}=F_{2}=\frac{GMm}{5R^{2}}$ As the resultant force exerted on the object...

### Mean solar day is the time interval between two successive noon when the sun passes through zenith point. The sidereal day is the time interval between two successive transits of a distant star through the zenith point. By drawing the appropriate diagram showing earth’s spin and orbital motion, show that mean solar day is four minutes longer than the sidereal day. In other words, distant stars would rise 4 minutes early every successive day.

The polar axis of the earth and its movement are E and E’ respectively. Translational motion is P’ After every 24 hours, earth's orbit is approximately advanced by $1^{o}$ As a result, time taken...

### What is the angle between the equatorial plane and the orbital plane ofa) polar satellite?b) geostationary satellite?

a) The equatorial plane and the orbital plane of a polar satellite form a $90^{o}$ angle. b) A geostationary satellite's equatorial plane and orbital plane are at an angle of $0^{o}$.

### Out of aphelion and perihelion, where is the speed of the earth more and why?

According to Kepler’s second law, a real velocity is constant and is given as: $r_{p}\times v_{p}=r_{A}\times v_{A}$ $\frac{r_{A}}{r_{p}}=\frac{v_{p}}{v_{A}}$ $r_{A}>r_{p}$ and...

### The gravitational force between a hollow spherical shell and a point mass is F. Show the nature of F vs r graph where r is the distance of the point from the centre of the hollow spherical shell of uniform density.

R is the spherical shell's radius, while r is the distance between m and M. The point's mass is m, while the hollow spherical shell's mass is M. Now, $F=\frac{GMm}{r^{2}}$ When F = 0,...

### 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 astronaut will experience variation when the size of the space station orbiting the earth is large, and this is due to acceleration due to gravity.

### We can shield a charge from electric fields by putting it side 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?

No. As gravitation is independent of the medium, a body can be shielded from the gravitational pull of adjacent matter.

### Is it possible for a body to have inertia but no weight?

Yes. It is possible for a body to have inertia but no weight as inertia is associated with the mass of the body. Satellite revolving around the earth is an example of a body with inertia and has no...

### How is the gravitational force between two point masses affected when they are dipped in the water keeping the separation between them the same?

The gravitational force between point masses is unaffected by the medium and remains constant regardless of their surroundings. As a result, even if the two-point masses are submerged in water, the...

### What is the direction of areal velocity of the earth around the sun?

The direction of the earth's areal velocity around the sun is determined by the product of r and v.

### Give one example each of central force and non-central force.

Example of Central force: Electrostatic force acting on the point charge Example of Non-central force: Nuclear force between the atoms

### Molecules in the air in the atmosphere are attracted by the gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree.

The gravitational force of the earth attracts molecules in the atmosphere, yet they do not descend into the earth since they are in random motion, whereas an apple moves downhill.

### The centre of mass of an extended body on the surface of the earth and its centre of gravitya) are always at the same point for any size of the bodyb) are always at the same point only for spherical bodiesc) can never be at the same pointd) is close to each other for objects, say of sizes less than 100 me) both can change if the object is taken deep inside the earth

The correct option is d) is close to each other for objects, say of sizes less than 100 m

### Supposing Newton’s law of gravitation for gravitation forces F1 and F2 between two masses m1 and m2 at positions r1 and r2 read Exemplar Solutions Physics Class 11 Chapter 8 – 21 where Mo is a constant of the dimension of mass, r12 = r1 – r2 and n is a number. In such a case,a) the acceleration due to gravity on earth will be different for different objectb) none of the three laws of Kepler will be validc) only the third law will become invalidd) for n negative, an object lighter than water will sink in water

The correct options are a) the acceleration due to gravity on earth will be different for different object c) only the third law will become invalid d) for n negative, an object lighter than water...

### There have been suggestions that the value of the gravitational constant G becomes smaller when considered over a very large time period in the future. If that happens for our earth,a) nothing will changeb) we will become hotter after billions of yearsc) we will be going around but not strictly in closed orbitsd) after a sufficiently long time we will leave the solar system

The correct options are c) we will be going around but not strictly in closed orbits d) after a sufficiently long time we will leave the solar system

### If the sun and the planets carried huge amounts of opposite charges,a) all three of Kepler’s laws would still be validb) only the third law will be validc) the second law will not changed) the first law will still be valid

The correct options are a) all three of Kepler’s laws would still be valid c) the second law will not change d) the first law will still be valid

### If the mass of sun were ten times smaller and gravitational constant G were ten times larger in magnitudesa) walking on ground would become more difficultb) the acceleration due to gravity on earth will not changec) raindrops will fall much fasterd) aeroplanes will have to travel much faster

The correct options are a) walking on ground would become more difficult c) raindrops will fall much faster d) aeroplanes will have to travel much faster

### If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube-lawa) planets will not have elliptic orbitsb) circular orbits of planets is not possiblec) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolicd) there will be no gravitational force inside a spherical shell of uniform density

The correct options are a) planets will not have elliptic orbits c) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic Explanation: The planets...

### Which of the following options is correct?a) acceleration due to gravity decreases with increasing altitudeb) acceleration due to gravity increases with increasing depthc) acceleration due to gravity increases with increasing latituded) acceleration due to gravity is independent of the mass of the earth

The correct options are a) acceleration due to gravity decreases with increasing altitude c) acceleration due to gravity increases with increasing latitude

### 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. 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 Solution: The correct option is c) m will move towards 2M

### Choose the wrong option.a) inertial mass is a measure of the 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 massb) that the gravitational mass and inertial mass are equal is an experimental resultc) that the acceleration due to gravity on earth is the same for all bodies is due to the equality of gravitational mass and inertial massd) gravitational mass of a particle-like proton can depend on the presence of neighbouring heavy objects but the inertial mass cannot

The correct option is d) gravitational mass of a particle-like proton can depend on the presence of neighbouring heavy objects but the inertial mass cannot

### In our solar system, the inter-planetary region has chunks of matter called asteroids. Theya) will not move around the sun since they have very small masses compared to the sunb) will move in an irregular way because of their small masses and will drift away outer spacec) will move around the sun in closed orbits but not obey Kepler’s lawsd) will move in orbits like planets and obey Kepler’s laws

The correct option is d) will move in orbits like planets and obey Kepler’s laws

### Both earth and moon are subject to the gravitational force of the sun. as observed from the sun, the orbit of the moona) will be elliptical will not be strictly elliptical because the total gravitational force on it is not centralc) is not elliptical but will necessarily be a closed curved) deviates considerably from being elliptical due to the influence of planets other than earth

The correct option is b) will not be strictly elliptical because the total gravitational force on it is not central

### Satellites orbiting the earth have a finite life and sometimes debris of satellites fall to the earth. This is becausea) the solar cells and batteries in satellites run outb) the laws of gravitation predict a trajectory spiralling inwardsc) of viscous forces causing the speed of the satellite and hence height to gradually decreased) of collisions with other satellites

The correct option is c) of viscous forces causing the speed of the satellite and hence height to gradually decrease

### Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the cm causing translation and a net torque at the cm causing translation and a net torque at the cm causing rotation around an axis through the cm. For the earth-sun systema) the torque is zerob) the torque causes the earth to spinc) the rigid body result is not applicable since the earth is not even approximately a rigid bodyd) the torque causes the earth to move around the sun

The correct option is a) the torque is zero

### As observed from earth, the sun appears to move in an approximately circular orbit. For the motion of another planet like mercury as observed from earth, this woulda) be similarly trueb) not be true because the force between earth and mercury is not inverse square lawc) not be true because the major gravitational force on mercury is due to sund) not be true because mercury is influenced by forces other than gravitational forces

The correct option is c) not be true because the major gravitational force on mercury is due to sun

### The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravitya) will be directed towards the centre but not the same everywhereb) will have the same value everywhere but not directed towards the centrec) will be same everywhere in magnitude directed towards the centred) cannot be zero at any point

The correct option is d) cannot be zero at any point

### lo, one of the satellites of Jupiter, has an orbital period of days and the radius of the orbit is Show that the mass of J upiter is about one-thousandth that of the sun.

Orbital period of $I0$, is given as $T_{10}=1.769$ days $=1.769 \times 24 \times 60 \times 60 \mathrm{~s}$ Orbital radius of $I0$, is given as $R_{10}=4.22 \times 10^{8} \mathrm{~m}$ Mass of jupiter...

### 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?

Time taken by the earth for one complete revolution is represented by $\mathrm{T}_{\mathrm{E}}$ having value $1$ Year Radius of Earth's orbit is represented by $\mathrm{R}_{\mathrm{E}}$ having value...

### Choose the correct alternative:(a) Acceleration due to gravity is independent of the mass of the earth/mass of the body.(b) The formula is more/less accurate than the formula for the difference of potential energy between two points and distance away from the centre of the earth.

(a) Acceleration due to gravity is given by the formula: $g=G{{M}_{e}} /{{ {R}_{e}}^{2}}$ is​​ Hence, it is independent of mass of body, but is dependent on mass of earth. (b) Gravitational...

### 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).

(a) According to the formula g′=g(1-R2h), acceleration decreases. where h is the height and R is the earth's radius (b) According to the formula g′=g(1−Rd​),, acceleration decreases. where d is the...