Question 430: Gravitation

Question

If the Earth is assumed to be a sphere of uniform mass density and its weight on the surface is 200 N , then the weight of the body at a depth halfway to the Earth's center will be

Accepted Answer

We are given that the weight of the body on the surface of the Earth is $200 \, \mathrm{N}$.
At a depth $h$ below the surface of a uniformly dense spherical Earth, the acceleration due to gravity decreases according to the formula:
$$ g' = g \left(1 - \frac{h}{R}ight) $$
Here, $R$ is the radius of the Earth and $h$ is the depth from the surface.
Since the body is placed halfway to the Earth's center, we have:
$$ \frac{h}{R} = \frac{1}{2} $$
Substitute this value in the formula for $g'$:
$$ g' = g \left(1 - \frac{1}{2}ight) = \frac{g}{2} $$
The new weight of the body at this depth will be:
$$ 	ext{Weight at depth} = m g' = \frac{m g}{2} $$
We know that $m g = 200 \, \mathrm{N}$. Therefore,
$$ m g' = \frac{200}{2} = 100 \, \mathrm{N} $$
Thus, the weight of the body at a depth halfway to the Earth's center is $100 \, \mathrm{N}$.

Answer

50 N

Answer

25 N

Answer

75 N

Answer

100 N

Gravitation

If the Earth is assumed to be a sphere of uniform mass density and its weight on the surface is 200 N , then the weight of the body at a depth halfway to the Earth's center will be

Choose the correct answer: