Physics- Chapter 3: Gravitation

Universal Law of Gravitation(ULG)

"Every object in the universe attracts every other object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining the centers of two objects."

Equations of motion for a free falling body

(u=0, a=g, s=h) v= u + at s= ut + 1/2at^2 => v= gt => h=1/2 gt^2 v^2= u^2 + 2as => v^2=2gh

Equations of motion for a body thrown vertically up

(v=0, a=-g, s=h) v= u + at s= ut + 1/2gt^2 0= u - gt => h= ut - 1/2gt^2 => u=gt v^2= u^2 + 2as 0= u^2 - 2gh => u^2=2gh

Expression for g

Consider an object A of m at a distance d from Earth. Let M be mass of the Earth and R be the radius of the Earth. Let h be the height of the object from the surface of the Earth. According to Newton's 2nd Law, F=mg ---- 1 According to ULG, F=GMm/d^2 ---- 2 From equations 1 and 2, we get mg=GMm/d^2 g=GM/d^2 This shows that acceleration due to gravity(g) is independent of the falling mass(m). For objects closer to the Earth's surface, h<<R. So, we get g=GM/R^2

Universal Law of Gravitation Equation

F=GMm/d^2

Difference between G and g

Gravitational Constant(G) - Its value is 6.67 x 10^-11 Nm^2/kg^2. - It is a scalar quantity. - Its value remains constant always and everywhere. - Its unit is Nm^2/kg^2. Gravitational acceleration(g) - Its average value is 9.8 m/s^2. - It is a vector quantity. - Its value varies at various places. - Its unit is m/s^2 (or) N/kg.

Proof for ULG

Let two objects A and B of masses M and m lie at a distance d from each other. Let F be the force of attraction between them. According to the ULG, F∝ Mm and F∝ 1/d^2. So, we get F∝ Mm/d^2 (or) F∝ GMm/d^2 where G is the constant of proportionality and is called the universal gravitational constant. Rearranging, G=Fd^2/Mm => Nm^2/kg^2 (or) Nm^2 kg^-2

Calculate the value of G

The values of G, M, R are: G= 6.7×10^-11 Nm^2 kg^-2 M= 6×10^24 kg R= 6.4×10^6 m Substituting the values, g= GM/R^2 = (6.7×10^-11 Nm^2 kg^-2)(6×10^24 kg)/(6.4×10^6 m)^2 = 9.8 ms^-2 ∴ g= 9.8 m/s^2

What is acceleration due to gravity?

When an object falls towards the earth, it accelerates due to the gravitational force of earth. This is called acceleration due to gravity and it is represented by g.

What is free fall?

Whenever an object falls towards the earth due to the earth's gravitational force or force of gravity, we say that the object is in free fall.