JEE Advanced PYQ Gravitation – Acceleration Due to Gravity | Previous Year Questions

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Topic Question Set

Q 1 :

A planet of radius $R = \frac{1}{10} \times (radius of Earth)$ has the same mass density as Earth. Scientists dig a well of depth $\frac{R}{5}$ on it and lower a wire of the same length and a linear mass density $10^{- 3} {kg m}^{- 1}$ into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $= 6 \times 10^{6} m$ and the acceleration due to gravity on Earth is $10 {ms}^{- 2}$ ) [2014]

96 N
108 N
120 N
150 N

1

2

3

4

(2)

[IMAGE 273]

The mass of the wire $= 10^{- 3} \times 1.2 \times 10^{5} = 120 kg$

Acceleration due to gravity at the surface of the planet and earth

$g_{p} = \frac{4}{3} π ρ G R_{p}; g_{e} = \frac{4}{3} π ρ G R_{E}$

$∴$ $\frac{g_{p}}{g_{e}} = \frac{R_{p}}{R_{E}} = \frac{1}{10} \Rightarrow g_{p} = \frac{10}{10} = 1 {ms}^{- 2}$

Let $g_{p M}$ be the acceleration due to gravity at point M which is the midpoint of the wire and is at a depth of $\frac{R_{p}}{10} .$

$g_{p M} = g_{p} [1 - \frac{R_{p} / 10}{R_{p}}] = 1 [1 - 0.1] = 0.9 {ms}^{- 2}$

$∴$ $Force = mass of wire \times g_{p M} = 120 \times 0.9 = 108 N$

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