Question

Assuming the earth to be a sphere of uniform density, the ratio of acceleration due to gravity on the earth's surface to its value at halfway towards the centre of the earth, will be

• A. \(2 : 1\)
• B. \(2 : 3\)
• C. \(1 : 1\)
• D. \(1 : 2\)

Hint:

Inside the earth (uniform density), gravity decreases linearly with distance from the centre.

Answer 1

The Correct Option is A

Step 1: Relation of gravity inside the earth.
For a sphere of uniform density, acceleration due to gravity at a distance \(r\) from the centre is directly proportional to \(r\):
\[ g(r) \propto r \]
Step 2: Gravity at the earth’s surface.
At the surface, \(r = R\), so
\[ g_{\text{surface}} \propto R \]
Step 3: Gravity at halfway to the centre.
Halfway towards the centre means \(r = \frac{R}{2}\).
\[ g_{\text{half}} \propto \frac{R}{2} \]
Step 4: Ratio calculation.
\[ g_{\text{surface}} : g_{\text{half}} = R : \frac{R}{2} = 2 : 1 \]
Step 5: Conclusion.
The required ratio is \(2 : 1\).