Intensity of Gravitational Field | JEE Main previous year questions with Solutions

Q 11 :

At a certain depth “ $d$ ” below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height 3R above earth surface, where R is radius of earth (Take R = 6400 km). The depth $d$ is equal to [2023]

5260 km
640 km
2560 km
4800 km

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(4)

$\frac{G M}{R^{2}} [1 - \frac{d}{R}] = \frac{4 \times G M}{{(4 R)}^{2}}$

$1 - \frac{d}{R} = \frac{1}{4} \Rightarrow \frac{d}{R} = \frac{3}{4} \Rightarrow d = \frac{3}{4} R$

$d = 4800 km$

Q 12 :

A body of weight $W$ is projected vertically upwards from the earth's surface to reach a height above the earth which is equal to nine times the radius of the earth. The weight of the body at that height will be [2023]

$\frac{W}{91}$
$\frac{W}{100}$
$\frac{W}{9}$
$\frac{W}{3}$

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(2)

[IMAGE 70]

$g' = \frac{G M}{{(10 R)}^{2}} = (\frac{g}{100})$

$W' = (\frac{W}{100})$

Q 13 :

Given below are two statements: [2023]

Statement I: Acceleration due to gravity is different at different places on the surface of earth.

Statement II: Acceleration due to gravity increases as we go down below the earth’s surface.

In the light of the above statements, choose the correct answer from the options given below:

Both Statement I and Statement II are true
Both Statement I and Statement II are false
Statement I is true but Statement II is false
Statement I is false but Statement II is true

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(3)

$g_{eff} = g - ω^{2} R_{e} \sin^{2} θ$

$g_{eff} = g (1 - \frac{d}{R_{e}})$

Q 14 :

The weight of a body on the surface of the earth is 100 N. The gravitational force on it when taken at a height from the surface of earth, equal to one-fourth the radius of the earth, is: [2023]

100 N
64 N
50 N
25 N

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(2)

$\Rightarrow g' = \frac{g R^{2}}{r^{2}} = \frac{g R^{2}}{{(R + \frac{R}{4})}^{2}} = \frac{16 g}{25}$

$∴$ $Weight = \frac{16}{25} \times 100 = 64 N$

Q 15 :

The weight of a body on the earth is 400 N. Then the weight of the body when taken to a depth half of the radius of the earth will be: [2023]

300 N
100 N
Zero
200 N

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(4)

$W = m g = 400 N$

$At depth d, gravity g' = g (1 - \frac{d}{R})$

$For d = \frac{R}{2}, g' = g (1 - \frac{d}{2 R}) = \frac{g}{2}$

$W' = m g' = \frac{m g}{2} = 200 N$

Q 16 :

The acceleration due to gravity at height $h$ above the earth, if $h ≪ R$ (radius of earth), is given by [2023]

$g' = g (1 - \frac{2 h}{R})$
$g' = g (1 - \frac{h}{2 R})$
$g' = g (1 - \frac{2 h^{2}}{R^{2}})$
$g' = g (1 - \frac{h^{2}}{2 R^{2}})$

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(1)

$For a point outside the surface of Earth, g = \frac{G M}{r^{2}}$

[IMAGE 71]

$r = distance from centre of Earth$

$\Rightarrow g (h) = \frac{G M}{{(R + h)}^{2}} = \frac{G M}{R^{2} {(1 + \frac{h}{R})}^{2}}$

$\Rightarrow g (h) = \frac{G M}{R^{2}} {(1 + \frac{h}{R})}^{- 2}$

$If h ≪ R, {(1 + \frac{h}{R})}^{- 2} \approx 1 - \frac{2 h}{R}$

$\Rightarrow g (h) = \frac{G M}{R^{2}} (1 - \frac{2 h}{R})$

$\Rightarrow g (h) = g_{surface} (1 - \frac{2 h}{R}), \frac{G M}{R^{2}} = g_{surface}$

Q 17 :

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $d = \frac{R}{2}$ from the surface of the earth, if its weight on the surface of the earth is 200 N, will be (Given R = radius of earth) [2023]

400 N
500 N
300 N
100 N

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(4)

$M = \frac{W}{g} = \frac{200}{10} = 20 kg$

$Acceleration due to gravity at a depth$

$g' = g (1 - \frac{d}{R})$

$d = depth from surface, d = \frac{R}{2}$

$g' = g (1 - \frac{R / 2}{R}) = \frac{g}{2} = {5 m/s}^{2}$

$Weight = m \times g' = 20 \times 5 = 100 N$

Q 18 :

Given below are two statements: [2023]

Statement I: Rotation of the earth shows effect on the value of acceleration due to gravity (g).

Statement II: The effect of rotation of the earth on the value of 'g' at the equator is minimum and that at the pole is maximum.

In the light of the above statements, choose the correct answer from the options given below:

Statement I is false but Statement II is true.
Statement I is true but Statement II is false.
Both Statement I and Statement II are true.
Both Statement I and Statement II are false.

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(2)

$Statement I is true due to centrifugal force.$

$Statement II is incorrect.$

$At pole: g = g_{s} (no effect)$

$At equator: g = g_{s} - r ω^{2} \cos^{2} λ = g_{s} - r ω^{2}$

$∴$ $\cos^{2} λ is maximum at λ = 0 ° (i.e., at equator)$

$Effect is maximum at equator.$

Q 19 :

Two planets A and B of radii R and 1.5 R have densities $ρ$ and $ρ / 2$ respectively. The ratio of acceleration due to gravity at the surface of B to A is [2023]

2 : 3
2 : 1
3 : 4
4 : 3

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(3)

$g = \frac{G M}{R^{2}} = \frac{4}{3} π G ρ R$

$∴$ $\frac{g_{2}}{g_{1}} = \frac{ρ_{2}}{ρ_{1}} \times \frac{R_{2}}{R_{1}} = \frac{1}{2} \times 1.5 = \frac{3}{4}$

Topic Question Set

Q 11 :

At a certain depth “ $d$ ” below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height 3R above earth surface, where R is radius of earth (Take R = 6400 km). The depth $d$ is equal to [2023]

Q 12 :

A body of weight $W$ is projected vertically upwards from the earth's surface to reach a height above the earth which is equal to nine times the radius of the earth. The weight of the body at that height will be [2023]

Q 14 :

The weight of a body on the surface of the earth is 100 N. The gravitational force on it when taken at a height from the surface of earth, equal to one-fourth the radius of the earth, is: [2023]

Q 15 :

The weight of a body on the earth is 400 N. Then the weight of the body when taken to a depth half of the radius of the earth will be: [2023]

Q 16 :

The acceleration due to gravity at height $h$ above the earth, if $h ≪ R$ (radius of earth), is given by [2023]

Q 17 :

Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $d = \frac{R}{2}$ from the surface of the earth, if its weight on the surface of the earth is 200 N, will be (Given R = radius of earth) [2023]

Q 19 :

Two planets A and B of radii R and 1.5 R have densities $ρ$ and $ρ / 2$ respectively. The ratio of acceleration due to gravity at the surface of B to A is [2023]

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