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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 -

R/3

R/2

R/4

R/5

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Explanation

$Total energy on the surface = total enrgy on height h - \frac{GMm}{R} + \frac{1}{2} m \frac{2 GM}{4 R} = - \frac{GMm}{(R + h)} - \frac{1}{R} + \frac{1}{4 R} = - \frac{1}{(R + h)} R + h = \frac{4 R}{3} h = \frac{R}{3}$

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A satellite is launched into a circular orbit of radius ‘R’ around earth while a second satellite is launched into an orbit of radius 1.02 R. The percentage difference in the time periods of the two satellites is

0.7

1.0

1.5

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Explanation

$(d) T = \frac{2 {πr}^{\frac{3}{2}}}{\sqrt{GM}} \frac{∆ T}{T} \times 100 = \frac{3}{2} \times \frac{∆ r}{r} \times 100 = \frac{3}{2} \times \frac{0.02 R}{R} \times 100 = 3 %$

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If the radius of the earth shrinks by 1.5% (mass remaining same), then the value of acceleration due to gravity changes -

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Explanation

Percentage change in g = 2(percentage change in R)

= $2 \times 1.5 = 3 %$

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Distance of geostationary satellite from the surface of the earth $r a d i u s (R_{e} = 6400 k m)$ in terms of $R_{e}$ is

$13.76 R_{e}$

$10.76 R_{e}$

$6.56 R_{e}$

$2.56 R_{e}$

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Explanation

(c)The distance of a geostationary satellite from centre of earth, r=36000 + 6400 km=42400 km

Hence, $d = \frac{42400}{6400} R_{e} = 6.56 R_{e}$

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Energy required to move a body of mass m from an orbit of radius 2R to 3R is

$G M m / 12 R^{2}$

$G M m / 3 R^{2}$

$G M m / 8 R$

$G M m / 6 R$

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Explanation

(d) Change in potential energy in displacing a body from $r_{1}$ to $r_{2}$ is given by

$∆ U = G M m [\frac{1}{r_{1}} - \frac{1}{r_{2}}] = G M m (\frac{1}{2 R} - \frac{1}{3 R}) = \frac{G M m}{6 R}$

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If radius of the earth contracts by 2% and its mass remains the same, then weight of the body at the earth surface :

Will decrease

Will increase

Will remain the same

None of these

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Explanation

(b) $g \propto \frac{1}{R^{2}}$ . If radius of earth decreases by 2% then g will increase by 4% i.e. weight of the body at earth surface will increase by 4%.

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The kinetic energy needed to project a body of mass m from the earth surface (radius R) to infinity is -

mgR/2

2 mgR

mgR

mgR/4

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Explanation

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A satellite is to revolve round the earth in a circle of radius 8000 km. The speed at which this satellite be projected into an orbit, will be

$3 k m / s$

16 km/s

7.15 km/s

8 km/s

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Explanation

(c) $v_{0} = \sqrt{\frac{G M}{r}} = \sqrt{\frac{g R^{2}}{r}} = \sqrt{\frac{10 \times {(64 \times 10^{5})}^{2}}{8000 \times 10^{3}}} = 71.5 \times 10^{2} m / s = 7.15 k m / s$

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If mass of a body is M on the earth surface, then the mass of the same body on the moon surface is

M/6

Zero

None of these

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Explanation

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Radius of orbit of satellite of earth is R. Its kinetic energy is proportional to -

$\frac{1}{R}$

$\frac{1}{\sqrt{R}}$

$\frac{1}{R^{3 / 2}}$

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Explanation

(a) $K . E = \frac{G M m}{2 R}$

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