Question

A heavenly body has a mass equal to half of the mass of the earth and its radius half as that of the earth. If a stone weighs 100 N on the surface of the earth, find the weight of the stone on the heavenly body.​

Answer 1

Answer:-

Let the mass of the stone be m.

Given:-

Weight of the stone on earth = 100 N.

We know,

Weight = mg

where,

• m = mass of the object
• g = acceleration due to gravity.

Also,

g = GM/r²

Where,

• G is gravitational constant
• M is mass of the surface where the object is placed
• r is the radius of the surface

So,

Let the mass of earth be M kg and its radius be r km.

Hence,

⟹ m * GM/r² = 100 N

⟹ GMm/r² = 100 N -- equation (1)

It is also given that,

A heavenly body has mass & radius equal to half of the mass & radius of earth.

So,

• Mass of the heavenly body = M/2 kg.

• Radius of the heavenly body = r/2 km

Now,

⟹ Weight of stone on the heavenly body = m * G (M/2) / (r/2)²

⟹ Weight of stone on the heavenly body = (GMm/2) * 4/r²

⟹ Weight of stone on the heavenly body = (GMm/r²) * 4/2

Substitute the value of GMm/r² from equation (1).

⟹ Weight of stone on the heavenly body = 100 * 2

∴ Weight of stone on the heavenly body = 200 N.

Answer 2

Answer :-

• Weight of the stone on the heavenly body is 200 N.

Explanation :-

Given :

• A heavenly body has a mass equal to half of the mass of the earth.
• Its radius if half as that of the earth.
• If a stone weight 100 N on the surface of the earth.

To find :

• Weight of the stone on the heavenly body.

Solution :

Using Newton's Law of Universal Gravitation :

• $\bold{F=G\frac{m_1m_2}{d^2} }$

Where we have :

• F = Force
• m$_1$ = Mass of object 1 [Stone]
• m$_2$ = Mass of object 2 [Heavenly body]
• d = Distance between center of masses.

According to the question :

• m$_2$ = $\bold{\frac{1}{2} }$ m$_1$

Hence, the force of gravity will be reduced to 50 N from 100 N.

Also, new d is half the old d, so the value of F increases by a factor of 4, that is $\bold{(\frac{1}{2})^2}$.

So, the force of the heavenly body acting on the stone is $\bold{\frac{4}{1}\times50 }$ = 200 N