Question 348: Gravitation

Question

The depth from the surface of the earth of radius $$\mathrm{R}$$, at which acceleration due to gravity will be $$60 \%$$ of the value on the earth surface is

Accepted Answer

To determine the depth from the surface of the Earth at which the acceleration due to gravity is 60% of its value on the surface, we can use the concept of the gravitational force inside the Earth.
Let $$g_0$$ be the acceleration due to gravity at the surface of the Earth, and let the radius of the Earth be $$R$$. When at a depth $$d$$ from the surface, the gravity is determined by the distance from the center of the Earth, which is $$R - d$$.
The gravity inside the Earth at a depth $$d$$ is given by the formula:
$$ g_d = g_0 \left(1 - \frac{d}{R}\right) $$
Given that at depth $$d$$, the gravity is 60% of $$g_0$$, we get:
$$ g_d = 0.60 \cdot g_0  $$
Substitute this into the formula:
$$ 0.60 \cdot g_0 = g_0 \left(1 - \frac{d}{R}\right) $$
Divide both sides by $$g_0$$:
$$ 0.60 = 1 - \frac{d}{R} $$
Rearrange the equation to solve for $$d$$:
$$ \frac{d}{R} = 1 - 0.60 = 0.40 $$
Hence:
$$ d = 0.40 R $$
Thus, the depth $$d$$ is:
$$ d = 0.40 R $$
To find the depth from the surface, use:
$$ d = R - \frac{d}{R} $$
Rearranging for $$d$$ we get:
$$ d = 0.40 R $$
Therefore, the depth from the surface is:
$$ d = 0.40 R  $$
Which corresponds to the option:
Option B
$$\frac{2 R}{5}$$

Answer

$$\frac{2 R}{3}$$

Answer

$$\frac{2 R}{5}$$

Answer

$$\frac{3 R}{5}$$

Answer

$$\frac{5 R}{3}$$

Gravitation

The depth from the surface of the earth of radius $\mathrm{R}, at which acceleration due to gravity will be 60 \%$ of the value on the earth surface is

Choose the correct answer: