NEET · Physics · STD 11 - 7. gravitation

The amount of work done to raise a mass ' \(m\) ' from the surface of the Earth to a height equal to the radius of the Earth ' \(R\) ', will be :

A \(2 m g R\)
B \(m g \frac{R}{4}\)
C mgR
D \(m g \frac{R}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(m g \frac{R}{2}\)

Step-by-step Solution

Detailed explanation

(D) \(m g \frac{R}{2}\)
Initial potential energy at the surface of the Earth is \(U_i=-\frac{G M m}{R}\)
Final potential energy at height \(h=R\) is \(U_f=-\frac{G M m}{R+R}=-\frac{G M m}{2 R}\)
Work done \(W=\Delta U=U_f-U_i\)
\(W=-\frac{G M m}{2 R}-\left(-\frac{G M m}{R}\right)=\frac{G M m}{2 R}\)
Using the relation \(g=\frac{G M}{R^2}\), we get \(G M=g R^2\)
Substituting this in the expression for work done:
\(W=\frac{g R^2 m}{2 R}=m g \frac{R}{2}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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