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The correct OPTION is (B) v(k^2-1)^1/2

Easy explanation: From the law of conservation of energy, we have;

(m x (kv) ^2)/2 – (G x M x m)/R = (m x p^2)/2

m = MASS of object

M = Mass of earth

R = Radius of the earth

p = Velocity of the object at infinity

(kv)^2/2 –(g x R) = p^2/2

(kv)^2/2 – v^2/2 = p^2/2; v = (2 x g x R) ^½

(k^2-1)v^2 = p^2

Therefore; p = v(k^2-1) ^1/2.

The correct choice is (c) they are CLOSER to the surface of the earth and can cover VAST areas very QUICKLY

To explain: SINCE polar satellites orbit close to the surface of the earth, it is easier to click images of the planet’s surface with greater detail. Their high orbital velocity enables them to view most of the surface strip-by-stip very quickly with which high-resolution images can be constructed.

Right answer is (b) False

The EXPLANATION is: The ESCAPE VELOCITY of a planet is INDEPENDENT of the radius of orbit of its parent star. It only depends on the radius of the planet and the mass of the planet.

The correct OPTION is (a) True

For explanation I would say: A GEOSTATIONARY satellite cannot orbit in any PLANE other than the equatorial plane because then it would have to be at rest, which cannot happen because the gravitational pull of the EARTH would eventually crash the satellite on the surface.

Right answer is (c) the normal force is zero

Explanation: Since the person sitting in the chair in the satellite is essentially in FREEFALL ALONG with the satellite, he does not experience any reaction force. The lack of reaction force (or) normal force MAKES a person FEEL WEIGHTLESS.

Correct choice is (b) They are falling TOWARDS the earth

To explain: The parabolic trajectory of the free fall of the international space station is at such high velocity that it essentially ORBITS the PLANET. The weightlessness experienced by the ASTRONAUTS in the ISS is the consequence of this FREEFALL.

Correct CHOICE is (b) it has INERTIA

To explain: SINCE the object has mass, it always has inertia. However, weight force is zero and hence the experienced acceleration is zero because it is EXPERIENCING weightlessness.

Correct option is (a) True

To ELABORATE: A geosynchronous satellite is a satellite with a time PERIOD of 24 hours. It may or may not lie in the equatorial plane and HENCE it may not always appear fixed in the sky. However, it will return to the same position after every 24 hours.

Therefore, a geosynchronous satellite orbiting the EARTH in the equatorial plane is a geostationary satellite.

Every geostationary satellite is a geosynchronous satellite but not vice versa.

Correct answer is (b) False

For EXPLANATION: POLAR satellites orbit close to the earth at distances of about 500 – 800 km above the SURFACE and have time periods of about 90 – 100 MINUTES.

Right choice is (a) Orbital speeds of A and B are equal

The explanation is: SINCE orbital velocities are independent of the mass of SATELLITES and only depends on the radius of orbits, they are equal for both A and B.

The kinetic energy of a satellite depends on the mass of the satellite. HENCE, it differs for satellite A and B.

Right option is (b) 2.5 x 10^16 J

Easy EXPLANATION: Mass of 50 cubic km of water = 5X 10^13 kg; DENSITY of water = 1 g/cubic cm

Work done = m x g x h

 = (5x 10^13 x 10 x 50) JOULES

 = 2.5 x 10^16 J.

The CORRECT answer is (b) False

For explanation: The TIME PERIOD of the moon around the earth cannot be 24 hours because then it would be a geostationary satellite. However, we know it is not true. The time period of the moon around the earth is found to be APPROXIMATELY 27.3 earth DAYS.

Correct answer is (a) 0.07 times

The explanation is: Escape VELOCITY (V) = [(2*G*M)/R]^1/2

Mass of moon = (DENSITY) x (Volume)

 = (Density) x (Volume of the earth)/50.65;

[Volume is directly proportional to the cube of the radius]

Vmoon/vearth = [(2*G*Mmoon)/Rmoon^1/2/[(2*G*Mearth)/Rearth]^1/2

 = [1/(50.65 x 3.7)] ^½

 = 0.07

Therefore; Vmoon = 0.07 (vearth).

The correct OPTION is (d) [(T^2*G*M)/(4*pi^2)]^1/3

To explain I would say: The time period of a satellite is given by;

T = 2 X pi x (R+h) ^3/2/ (G x M)^1/2

Solving for ORBITAL radius “(R+h)”, we get;

(R+h) = [(T^2 x G x M)/(4 x pi^2)]^1/3.

Right choice is (a) True

Best explanation: The KINETIC ENERGY of a satellite is;

KE = (1/2) x (G x m x M)/(R+H)

h: Height above the surface of the planet

For an ELLIPTICAL ORBIT, “h” changes with time. Hence, the kinetic energy also varies with time.

The correct answer is (a) True

The best I can explain: Perigee is the point in the orbit CLOSEST to the surface of the planet. HENCE, it has a maximum VELOCITY. Therefore, at perigee, the KINETIC energy is maximum and potential energy is minimum.

The correct choice is (b) -[(G*M*m)/2*(R+h)]

BEST explanation: TOTAL ENERGY (E) of a satellite is the sum of its potential energy (PE) and kinetic energy (KE).

PE = – (G x M x m)/(R+h)

KE = (G x M x m)/(2 x (R+h))

E = PE + KE

 = -(G x M x m)/(R+h) + (G x M x m)/(2 x (R+h))

 = -(G x M x m)/(2 x (R+h)).

Right choice is (c) [3/(2*D*G)]^1/2

For explanation I would say: The time period of a satellite flying very close to the surface of the earth is;

T = 2 X pi x [R^3 / (G X M)]^1/2;[Height is negligible compared to the RADIUS of the earth].

Mass (M) = Density (D) x Volume

M = D x (4/3 x pi x R^3)

Substituting the RELATION of mass into the time period, we GET;

T = [3/(2 x D x G)]^1/2

Right option is (d) 3

Best explanation: The time periods (T) of a SATELLITE revolving AROUND the earth of radius R and at height h is:

T = 2 x pi x [(R + h)^3 / (G X M)]^1/2

Hence, the time PERIOD is directly proportional to (R + h)^3/2

From this, we have the ratio of the time periods as 1.03.

1.03 x 100 = 103. Hence, there is a 3% CHANGE in the time period.

The correct answer is (b) False

The BEST I can explain: The expression for gravitational POTENTIAL energy is “-(G*M*m)/r”.

“-(G*M)/r” is the expression for ‘gravitational potential’.

Gravitational potential at a POINT can be defined as the work done in BRINGING a UNIT mass from infinity to that point.

The correct CHOICE is (d) cannot be zero at any point

The explanation is: Since we assumed the earth to have a non-uniform density, the acceleration DUE to gravity will not be directed TOWARDS the GEOMETRIC centre. FURTHERMORE, for a perfect sphere, the acceleration due to gravity will be equal at all points on the surface and will be non-zero.

The correct answer is (a) 0 g-FORCE

Best explanation: Since a body at the CENTRE of the earth WOULD experience equal gravitational force from all sides, the vector addition of all of these forces amount to zero. Therefore, the force experienced by an object at the centre of the earth is zero.

Correct option is (b) 6.67 X 10^-11 N m^2/kg^2

The best explanation: 6.022 x 10^23 is the Avogadro number.

6.67 x 10^-11 N m^2/kg^2 is the value of universal gravitational constant.

1.602 x 10^-19 C is the CHARGE of a proton.

9.81 m/s^2 is the acceleration DUE to gravity on the surface of the EARTH.

The correct answer is (a) True

For explanation I would say: The weight of an object is zero when the NET gravitational force ACTING on the object is zero. However, the mass of an object can NEVER be zero SINCE mass is a property of matter.

The correct answer is (d) The gravitational constant is INDEPENDENT of the medium

For explanation: SINCE the gravitational constant is an empirical constant, it does not vary with the medium. HENCE, the value of the gravitational constant is the same in any PART of the known UNIVERSE.

Correct answer is (a) True

To elaborate: The amount of WORK done on the BODY by the force of gravity is INDEPENDENT of the path. Hence, it is a CONSERVATIVE force.

Right ANSWER is (b) Isaac Newton

Easiest explanation: The UNIVERSAL law of GRAVITATION is a part of Isaac Newton’s WORK “Philosophiæ Naturalis Principia Mathematica (the Principia)”, FIRST published on 5 July 1687.

Correct answer is (c) Henry Cavendish

To EXPLAIN: The value of gravitational CONSTANT was FIRST experimentally determined by Henry Cavendish in the year 1798. It is ALSO known as the Cavendish Gravitational Constant.

The CORRECT answer is (d) Zero

For explanation: Gravitational POTENTIAL ENERGY (U) = -(G*M*m)/r

At infinity, r -> infinity

Therefore, U -> 0.

Right choice is (a) True

The EXPLANATION: The point inside the spherical SHELL EXPERIENCES gravitational pull by all points of point of the shell. However, the net gravitational force is zero DUE to vector addition. Hence, the net acceleration due to GRAVITY is also zero.

The CORRECT choice is (a) the heliocentric theory

To EXPLAIN: Kepler’s LAWS of planetary motion improved the heliocentric theory of Nicolaus COPERNICUS, replacing its circular orbits with epicycles with ELLIPTICAL orbits.

Correct ANSWER is (a) our SUN

The explanation is: The Kepler’s laws of planetary motion were published by Johannes Kepler between 1609 and 1619. They are three scientific laws DESCRIBING the motion of planets around the Sun.

The CORRECT option is (d) Remains UNCHANGED

To elaborate: The value of acceleration due to GRAVITY does not depend on the speed of revolution but only the DISTANCE between both the centres of gravity. Hence, the magnitude and direction would remain unchanged.

Correct choice is (C) 2h = d

Easy explanation: ACCELERATION due to gravity above the surface of the EARTH;

g’ = [(G*M1)/R^2] X [1 – (2h)/R)]

Acceleration due to gravity below the surface of the earth;

g’’ = (G x M1/R^3) x (R – d)

g’ = g’’

Therefore;

[(G*M1)/R^2] x [1 – (2h)/R)] = (G x M1/R^3) x (R – d)

2h = d.

The correct answer is (a) True

To ELABORATE: SINCE the earth is not a PERFECT sphere and has many irregularities, the acceleration due to GRAVITY is different at different POINTS on the earth’s surface.

The correct OPTION is (a) 1.6675 N

The explanation is: From NEWTON’s law of gravitation, we have;

F = (G*M1*M2*)/R^2

G = 6.67 x 10^-11 N m^2/kg^2

M1 = 10kg

M2 = 1000kg

R = 200m

F = (6.67 x 10^-11 x 10 x 1000) / 200^2

= 1.6675 N.

The CORRECT ANSWER is (a) True

For explanation: If the ECCENTRICITY is taken as zero, then the trajectory would become CIRCULAR. So, the sun would be at the centre of the circle since both foci would also lie at the centre.

The correct CHOICE is (c) 3T

For explanation: TIME period of the simple pendulum on the SURFACE of the earth;

T = (2 x pi) x (l / g)^1/2

l = Length of the simple pendulum

The time period of the simple pendulum at a certain height above the earth’s surface;

T’ = (2 x pi) x (l / g’)^1/2

g’ = ACCELERATION due to gravity at a certain height above the surface of the earth

g’ for a height of 2R (R = Radius of the earth);

g’ = (G*M1)/(R + 2R)^2

 = (G*M1)/(3R)^2

 = (1/9) x g

T’ = (2 x pi) x (l / (1/9)g) ^½

 = 3 x [(2 x pi) x (l / g)^½]

 = 3T.

The correct CHOICE is (d) g = [(G*M1)/R^2] x [1 – (2h)/R)]

The best I can explain: If “r” is the distance between the CENTRE of the earth and the object at a height “h” above the earth’s surface, then;

R = R + h

And; g = (G*M1)/r^2

 = (G*M1)/(R + h)^2

 = (G*M1)/R^2 x (1 + h/R)^-2

Since h < < R;

Using binomial expansion and NEGLECTING higher order terms, we can write the above expression as;

g = [(G*M1)/R^2] x [1 – (2h)/R)].

Correct option is (c) g = (G*M1)/(R + h)^2

For explanation: If “r” is the distance between the centre of the earth and the OBJECT at a HEIGHT “h” above the earth’s surface, then;

R = R + h

And; g = (G*M1)/r^2

 = (G*M1)/(R + h)^2.

Correct answer is (a) True

To explain: From Kepler’s LAW of area, we know that a planet sweeps equal areas at equal intervals of TIME. Perigee is the closest point to the SUN and HENCE, the velocity of the planet is GREATEST at the perigee because, only then, can the planet sweep an area equal to that when it was farther from the sun for the same interval of time.

Correct answer is (c) 72 HOURS

Easiest EXPLANATION: According to Kepler’s law of AREA, a line segment joining a planet and the Sun sweeps out EQUAL areas during equal intervals of time.

If it takes 36 hours to sweep an area of 1 million square KM, it would take (36 + 36) hours to sweep an area of (1 + 1) million square km.

Therefore, the answer is 72 hours.

Right option is (a) Lead

Explanation: Henry Cavendish USED lead SPHERES in his experiment because lead was EASILY available at the time. Iron was also abundantly available. However, Henry Cavendish chose lead because it is denser than iron and the MASS of lead is about 44% higher than that of iron for the same size of the sphere – this would lead to a larger gravitational force which is easier to MEASURE.