A particle of mass \(10 \mathrm{~g}\) is kept on the surface of a uniform sphere of mass \(100 \mathrm{~kg}\) and radius \(10 \mathrm{~cm}\). Find the work to be done against the gravitational force between them to take the particle is away from the sphere \(\left(\mathrm{G}=6.67 \times 10^{-11} \mathrm{SI}\right.\) unit \()\) (A) \(6.67 \times 10^{-9} \mathrm{~J}\) (B) \(6.67 \times 10^{-10} \mathrm{~J}\) (C) \(13.34 \times 10^{-10} \mathrm{~J}\) (D) \(3.33 \times 10^{-10} \mathrm{~J}\)

Short Answer

Expert verified
The work to be done against the gravitational force between the particle and the sphere is \(6.67 \times 10^{-10} \mathrm{J}\). The correct answer is (B).

Step by step solution

01

Identify the given information

We are given the following: - Mass of particle (m1) = 10 g = 0.01 kg (converted to kilograms) - Mass of sphere (m2) = 100 kg - Radius of sphere (r) = 10 cm = 0.1 m (converted to meters) - Gravitational constant (G) = 6.67 × 10^-11 N m²/kg²
02

Calculate the gravitational force

We will calculate the gravitational force between the particle and the sphere using the formula F = G * (m1 * m2) / r^2: F = (6.67 × 10^-11 N m²/kg²) * (0.01 kg * 100 kg) / (0.1 m)^2
03

Simplify the expression

Next, we will simplify the expression: F = (6.67 × 10^-11 N m²/kg²) * (1 kg) / 0.01 m² F = 6.67 × 10^-9 N
04

Calculate the work done against gravitational force

Since work is done against the force, the work is equal to the force multiplied by the radius of the sphere (the distance over which the force is applied): W = F * r W = (6.67 × 10^-9 N) * (0.1 m)
05

Simplify the expression and find the answer

Now, we will simplify the expression to obtain the answer: W = 6.67 × 10^-10 J So the correct answer is (B) 6.67 × 10^-10 J.

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