Sample Problems

Problem 1: The mass of the earth is 6 × 10²⁴ kg and that of the moon is 7.4 × 10²² kg. If the distance between the earth and the moon is 3.84 × 10⁵ km, calculate the force exerted by the earth on the moon. 

Solution: 

Given that,

mass of the earth, m₁ is 6 × 10²⁴ kg.

mass of the moon, m₂ is 7.4 × 10²² kg. 

distance between earth and moon, d is 3.84 × 10⁵ km.

formula to calculate the force exerted by the earth on the moon is:

[Tex]\begin{aligned}F&=\dfrac{Gm_1m_2}{d^2}\\&=6.67\times10^{-11}\times\dfrac{6\times10^{24}\times7.4\times10^{22}}{({3.84\times10^8})^2}\\&=2.02\times10^{20} \text{ N}\end{aligned}[/Tex]

Hence, the force exerted by the earth on the moon is equal to 2.02 × 10²⁰ N.

Problem 2: Your teacher is teaching you the law of gravitation 5 m far from you, if his Mass is 60 kg then what is the value of gravitational force exerted by you on him? (Hint: Take your weight by yourself).

Solution: 

Given that,

mass of the teacher, M is 60 kg.

mass of the person, m is 70 kg.

distance between them, d is 5 m.

Formula to calculate the gravitational force exerted by the person on the teacher is:

[Tex]\begin{aligned}F&=\dfrac{GMm}{d^2}\\&=6.67\times10^{-11}\times\dfrac{60\text{ kg}\times70\text{ kg}}{(5\text{ m})^2}\\&=1.1\times 10^{-9}\text{ N}\end{aligned}[/Tex]

Hence, the gravitational force exerted by the person on the teacher is 1.1 × 10^-9 N.

Problem 3: A satellite A has twice the mass of satellite B. The satellite A orbit the planet with the half-radius of B satellite. How many times the force on A satellite in comparison to B satellite by earth?

Solution: 

Consider the mass of B satellite as m and the mass of A satellite as 2m.

Now, to calculate gravitational force for both satellite divide both forces to obtain an equation as:

[Tex]F_{A}= G \times \dfrac{M_{earth}\times 2m}{R^2}\\~\\ F_{B}= G \times \dfrac{M_{earth}\times m}{(2R)^2}\\~\\[/Tex]

Now divide the force on A by Force on B as:

[Tex]F_{A} = 8\times F_{B}[/Tex]

Satellites orbiting earth

Hence, the force on satellite A is 8 times the force on satellite B.

Problem 4: The Newton’s law of gravitation applies to:

a.  Small bodies only      

b.  Plants only      

c.  All bodies irrespective of their size      

d.  For solar system

Solution: 

The Newton’s law of gravitation is applicable to all bodies irrespective of their size.

Hence, Option c. is correct.

Problem 5: Consider two bodies in contact such that the distance between them is six times greater than the usual distance. Calculate how much the force changed.

Solution:

The force between two bodies, according to Newton’s law of universal gravitation, is inversely proportional to the square of the distance between them. So, if the distance between the bodies increases by a factor of 6, the force will decrease by a factor of 6² = 36

In other words, the force will decrease by a factor of 36.

Whether or not that apple actually landed on Isaac Newton’s head, as some stories would have it, this equation given by universal law of gravitation describes why everyone stays rooted to the ground, what locks the Earth in orbit around the sun, and to send men to the moon. It summarizes the idea that all the particles of matter in the universe attract each other through the force of gravity, Newton’s law tells how strong that attraction is. Let’s discuss how this law come to light and derived.