Question

How much deep inside the earth (radius \(R\)) should a man go, so that his weight becomes one-fourth of that on the earth’s surface?

• A. \( \frac{R}{2} \)
• B. \( \frac{3R}{4} \)
• C. \( \frac{R}{4} \)
• D. \( \frac{R}{3} \)

Hint:

Gravity decreases linearly inside Earth.

Answer 1

The Correct Option is B

To solve the problem of determining how deep inside the earth a man needs to go so that his weight becomes one-fourth of that on the earth’s surface, we can use the concept of gravitational force inside a spherical body.

The gravitational force at a distance \( r \) from the center of the Earth, but inside it, can be given by the formula:

\(F = \frac{G \cdot M(r)}{r^2}\), where \( M(r) \) is the mass contained within radius \( r \).

Inside the Earth, the gravitational force depends linearly on \( r \) because the shell of mass outside radius \( r \) does not contribute to the gravitational force at that point. Therefore, we can find \( M(r) \) using:

\(M(r) = M \cdot \left(\frac{r}{R}\right)^3\), where \( M \) is the total mass of the Earth and \( R \) is the Earth's radius.

The gravitational force \( F \) inside the Earth becomes:

\(F_{\text{inside}} = \frac{G \cdot M \cdot r}{R^3}\)

Now, the weight (gravitational force) at the surface of the Earth is:

\(F_{\text{surface}} = \frac{G \cdot M}{R^2}\)

We need the weight to become one-fourth of that on the Earth’s surface. Thus:

\(\frac{F_{\text{inside}}}{F_{\text{surface}}} = \frac{1}{4}\)

Substitute the expressions we derived for both the inside and surface force:

\(\frac{\frac{G \cdot M \cdot r}{R^3}}{\frac{G \cdot M}{R^2}} = \frac{1}{4}\)

Simplifying, we find:

\(\frac{r}{R} = \frac{1}{4}\)

Thus, \( r = \frac{R}{4} \).

This implies that the depth inside the Earth \( d \) where the man needs to go is:

\(d = R - r = R - \frac{R}{4} = \frac{3R}{4}\)

Therefore, the man should go to a depth of \( \frac{3R}{4} \) inside the Earth so that his weight becomes one-fourth of that on the Earth’s surface.

The correct option is \(\frac{3R}{4}\).