Gravitation — Top 20 Questions

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A wire of uniform mass density (mass \(M\), length \(L\)) is bent into a semicircular arc. A particle of mass \(m\) is at centre. Gravitational force on particle:

A\(\dfrac{GM m\pi^2}{L^2}\)

B\(\dfrac{GM m\pi}{2L^2}\)

D\(\dfrac{2GM m\pi}{L^2}\)

✅ Correct: D

\(dF = \dfrac{Gm\,dm}{r^2}\). Radius \(r = L/\pi\). By symmetry only the component toward centre survives: \(F = \frac{2GMm}{r^2/\pi} \cdot \frac{1}{\pi} \cdot ... = \mathbf{\dfrac{2GMm\pi}{L^2}}\).

If \(G\) is gravitational constant and \(u\) is energy density (energy per volume), \(\sqrt{uG}\) has dimensions of:

AGravitational potential

BPressure gradient per unit mass

CEnergy per unit mass

DForce per unit mass

✅ Correct: D

\([u] = ML^{-1}T^{-2}\), \([G] = M^{-1}L^3T^{-2}\). \([uG] = L^2T^{-4}\). \([\sqrt{uG}] = LT^{-2} = \) acceleration \(= \mathbf{\text{force per unit mass}}\).

Acceleration due to gravity at height \(h \ll R\) above Earth's surface is:

A\(g' = g\left(1 - \dfrac{2h}{R}\right)\)

B\(g' = g\left(1 - \dfrac{2h^2}{R^2}\right)\)

C\(g' = g\left(1 - \dfrac{h^2}{2R^2}\right)\)

D\(g' = g\left(1 - \dfrac{h}{2R}\right)\)

✅ Correct: A

\(g' = \dfrac{GM}{(R+h)^2} = \dfrac{GM}{R^2}\left(1+\dfrac{h}{R}\right)^{-2} \approx g\left(1 - \dfrac{2h}{R}\right)\) for \(h \ll R\).

Force between planet and star \(\propto R^{-3/2}\). Time period \(T\) and orbit radius \(R\) relation:

A\(T^2 \propto R^{7/2}\)

B\(T^2 \propto R^3\)

C\(T^2 \propto R^{5/2}\)

D\(T^2 \propto R^2\)

✅ Correct: C

\(F = \dfrac{mv^2}{R} \propto R^{-3/2}\) → \(v^2 \propto R^{-1/2}\) → \(v \propto R^{-1/4}\). \(T = \dfrac{2\pi R}{v} \propto R^{5/4}\) → \(T^2 \propto \mathbf{R^{5/2}}\).

Simple pendulum at height \(h = R\) (Earth radius) above surface. Length \(l = 4\) m, \(g = \pi^2\) m/s². Time period (s):

A4 s

B8 s

C12 s

D6 s

✅ Correct: B

At \(h = R\): \(g' = g/4 = \pi^2/4\). \(T = 2\pi\sqrt{l/g'} = 2\pi\sqrt{4/(\pi^2/4)} = 2\pi\sqrt{16/\pi^2} = 2\pi \cdot 4/\pi = \mathbf{8\text{ s}}\).

Two identical particles (mass \(m\)) revolve in circular orbit of radius \(a\) under mutual gravity. Angular speed:

A\(\sqrt{\dfrac{Gm}{2a^3}}\)

B\(\sqrt{\dfrac{Gm}{4a^3}}\)

C\(\sqrt{\dfrac{Gm}{a^3}}\)

D\(\sqrt{\dfrac{4Gm}{a^3}}\)

✅ Correct: B

Separation between them \(= 2a\). \(F = \dfrac{Gm^2}{(2a)^2} = m\omega^2 a\) → \(\omega^2 = \dfrac{Gm}{4a^3}\) → \(\omega = \mathbf{\sqrt{\dfrac{Gm}{4a^3}}}\).

Escape velocity from Earth's surface is \(v_e\). Escape velocity from a planet with mass \(4M\) and radius \(2R\):

A\(v_e\)

B\(\sqrt{2}\,v_e\)

C\(2v_e\)

D\(4v_e\)

✅ Correct: B

\(v_e = \sqrt{2GM/R}\). New: \(v' = \sqrt{2G(4M)/(2R)} = \sqrt{4GM/R} = \sqrt{2}\sqrt{2GM/R} = \mathbf{\sqrt{2}\,v_e}\).

Orbital velocity of satellite just above Earth's surface (radius \(R\), \(g = 9.8\) m/s²):

A7.9 km/s

B11.2 km/s

C9.8 km/s

D4.9 km/s

✅ Correct: A

\(v_o = \sqrt{gR} = \sqrt{9.8 \times 6.4\times10^6} \approx \sqrt{62.72\times10^6} \approx \mathbf{7.9\text{ km/s}}\).

Gravitational PE of a body (mass \(m\)) at height \(h = 2R\) above Earth's surface (mass \(M\), radius \(R\)):

A\(-\dfrac{GMm}{3R}\)

B\(-\dfrac{GMm}{2R}\)

C\(-\dfrac{GMm}{R}\)

D\(-\dfrac{2GMm}{3R}\)

✅ Correct: A

Distance from centre \(= R + 2R = 3R\). \(U = -\dfrac{GMm}{3R}\). \(\mathbf{-\dfrac{GMm}{3R}}\).

Q10

Time period of satellite at height \(h = R\) is \(T\). Time period at Earth's surface would be:

A\(T/2\sqrt{2}\)

B\(T/2\)

C\(T\sqrt{2}\)

D\(2\sqrt{2}T\)

✅ Correct: A

\(T \propto r^{3/2}\). At surface \(r_1 = R\), at height \(h=R\): \(r_2 = 2R\). \(\dfrac{T_1}{T} = \left(\dfrac{R}{2R}\right)^{3/2} = \dfrac{1}{2\sqrt{2}}\) → \(T_1 = \mathbf{T/2\sqrt{2}}\).

Q11

Acceleration due to gravity at depth \(d = R/2\) below Earth's surface:

A\(g/4\)

B\(g/2\)

C\(3g/4\)

D\(g/3\)

✅ Correct: B

\(g_d = g\left(1 - \dfrac{d}{R}\right) = g\left(1 - \dfrac{1}{2}\right) = \mathbf{g/2}\).

Q12

Gravitational field is zero at a point between Earth (mass \(M\)) and Moon (mass \(m\)) separated by distance \(d\). Distance from Earth:

A\(\dfrac{d}{1+\sqrt{m/M}}\)

B\(\dfrac{d\sqrt{M}}{\sqrt{M}+\sqrt{m}}\)

C\(\dfrac{d}{1+\sqrt{M/m}}\)

D\(d\sqrt{M/(M+m)}\)

✅ Correct: B

\(\dfrac{GM}{x^2} = \dfrac{Gm}{(d-x)^2}\) → \(\dfrac{x}{d-x} = \sqrt{\dfrac{M}{m}}\) → \(x = \dfrac{d\sqrt{M}}{\sqrt{M}+\sqrt{m}}\).

Q13

A tunnel is dug through Earth's centre. Time to cross Earth for an object dropped in (\(g = 10\) m/s², \(R = 6400\) km):

A2530 s

B5060 s

C1265 s

D3580 s

✅ Correct: A

Motion is SHM with \(T = 2\pi\sqrt{R/g} = 2\pi\sqrt{6.4\times10^6/10} = 2\pi\times800 \approx 5027\) s. Time to cross = \(T/2 \approx \mathbf{2530\text{ s}}\).

Q14

Period of geostationary satellite is:

A12 hours

B24 hours

C48 hours

D6 hours

✅ Correct: B

A geostationary satellite has period equal to Earth's rotation: \(\mathbf{24\text{ hours}}\). It remains stationary relative to Earth's surface.

Q15

Binding energy of a satellite (mass \(m\)) in circular orbit at radius \(r\):

A\(\dfrac{GMm}{r}\)

B\(\dfrac{GMm}{2r}\)

C\(-\dfrac{GMm}{2r}\)

✅ Correct: B

Total energy \(E = -\dfrac{GMm}{2r}\). Binding energy \(= -E = \mathbf{\dfrac{GMm}{2r}}\) (energy needed to escape orbit).

Q16

Two spheres (same density) have radii \(R_1\) and \(R_2\). Ratio of gravitational fields at their surfaces:

A\(R_1^2 : R_2^2\)

B\(R_2 : R_1\)

C\(R_1 : R_2\)

D\(R_1^3 : R_2^3\)

✅ Correct: C

\(g = \dfrac{GM}{R^2} = \dfrac{4\pi G\rho R^3/3}{R^2} = \dfrac{4\pi G\rho R}{3}\). Same density → \(g \propto R\) → ratio \(= \mathbf{R_1:R_2}\).

Q17

A satellite is moved from circular orbit radius \(R\) to \(2R\). Change in KE:

AIncreases by \(GMm/4R\)

BDecreases by \(GMm/4R\)

CIncreases by \(GMm/2R\)

DDecreases by \(GMm/2R\)

✅ Correct: B

\(KE = GMm/2r\). \(\Delta KE = \dfrac{GMm}{2R} - \dfrac{GMm}{4R} = \dfrac{GMm}{4R}\). KE decreases (r increases). \(\mathbf{\text{Decreases by }GMm/4R}\).

Q18

Weight of body at pole vs equator differs because of:

AVariation in \(g\) with latitude only

BCentrifugal effect only

CBoth variation in \(g\) with latitude and centrifugal effect

DNeither

✅ Correct: C

Both effects contribute: Earth is oblate (larger \(R\) at equator → smaller \(g\)) AND centrifugal effect at equator reduces apparent weight. \(\mathbf{\text{Both}}\).

Q19

Kepler's second law states that the area swept by planet-sun line per unit time is:

AProportional to distance from Sun

BConstant

CProportional to orbital speed

DInversely proportional to distance

✅ Correct: B

Kepler's 2nd law: equal areas in equal times → area swept per unit time is constant. This follows from conservation of angular momentum (no torque from gravity about Sun).

Q20

Gravitational potential at centre of a solid sphere (mass \(M\), radius \(R\)):

A\(-\dfrac{GM}{R}\)

B\(-\dfrac{3GM}{2R}\)

C\(-\dfrac{2GM}{R}\)

✅ Correct: B

\(V_\text{centre} = -\dfrac{3GM}{2R}\) (more negative than surface value \(-GM/R\)). \(\mathbf{-\dfrac{3GM}{2R}}\).

Roshan

Expert · 5 Years Experience

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