**Gravitation Notes – Chapter 10**

**Gravitation**

Gravitation is defined as the non-contract force of attraction between any two bodies in the universe. The earth attracts (or pulls) all objects lying on or near its surface towards its centre. The force with which the earth pulls the objects towards its centre is called the gravitational force the earth or gravity of the earth.

**Universal law of gravitation**

Universal law of gravitation was given by Isaac Newton. According to this law, the attractive force between any two objects in the universe is directly proportional to the product of their masses and inversely proportional to the square of distance between them. The direction of the force is along the line joining the centres of two objects. Consider two bodies A and B having masses m_{1} and m_{2}, whose centres are at a distance of d from each from each other.

Then, the force between two bodies is directly proportional to the product of their masses, i.e.

F ∝ m_{1}m_{2} …(i)

And the force between two bodies is inversely proportional to the square of the distance between them, i.e.

Where, G = 6.67 10^{-11}N-m^{2}/kg^{2 }is the universal gravitational constant.

**Importance of Universal law of gravitation**

The universal law of gravitation successfully explained several phenomena given as below:

(i) The force that binds us to the earth.

(ii) The motion of the moon around the earth.

(iii) The motion of planets around the sun.

(iv) The tides due to the moon and the sun.

(v) The flow of water in rivers is also due to gravitational force of the earth on water.

**Motion of moon around earth and centripetal force.**

The motion of the moon around the earth is due to centripetal force provided by the force of attraction of the earth. The force that keeps a body moving along the circular path is acting towards the centre and responsible for the change in direction of velocity or acceleration is called centripetal (centre seeking) force.

Us,

**Kepler’s laws of planetary motion**

**Kepler’s first law**

It states that the path of any planet in an orbit around the sun follows the shape of an ellipse with the sun at one of the foci.

**Kepler’s second law**

It states that an imaginary line from the sun to the planet sweeps out equal areas in equal intervals of time.

Thus, if the time of travelling of a planet from A to B and from C to D is same, then the areas AOB and COD are equal.

**Kepler’s third law**

It states that the cube of the mean distance of a planet from the sun is directly proportional to the square of its orbital period T. it is expressed as

r^{3 }∝ T^{2}, r^{3 }= K T^{2}

Where, T = time period of the planet (around the sun)

r = radius as mean distance of the planet from the sun

and K = Kepler constant

**Calculation of value of g**

To calculate the value of g, we should put the values of G, M and R in above formula, i.e. g=GM/R^{2}.

∵ Mass of the earth, M= 6 10^{24} kg

Radius of the earth, R = 6.4 10^{6} m

Universal gravitational constant, G = 6.67 10^{-11} N-m^{2}/kg^{2}

g= 9.8 m/s^{2}

**Motion of objects under the influence of gravitational force of the earth**

General equations Equations for body under

of motion free fall

1. v = u+ at v = u + gt [∵ a = g]

2. s = ut + 1/2 at^{2} h = ut + 1/2gt^{2} [∵ a = g]

3. v^{2 }= u^{2 }+ 2as v^{2 }= u^{2 }+ 2gh [∵ a= g]

**Mass **

Mass is the total content of the body which measures the inertia of a body. It is a scalar quantity and its SI unit is kilogram.

**Weight **

The weight of an object is the force with which it is attracted towards the earth.

Weight of an object, w = mg

Where, m = mass, g = acceleration due to gravity

Here, M = mass of the earth and R = radius of the earth

Weight of an object on the moon

W_{m} = 1/6 x w_{e}

**Thrust and pressure**

Thrust is the force acting on an object perpendicular to its surface. The effect of thrust depends on the area on the area on which it acts.

**Pressure of fluids **

All liquids and gases are combinedly called fluids.

**Buoyancy**

The tendency of a liquid to exert an upward force on an object immersed in it is called buoyancy.

Bouyant Force is an upward force which acts on an object when it is immersed in a liquid. It is also called upthrust.

**Factors affecting buoyant force**

(i) Density of the fluid

The liquid having higher density exerts more upward buoyant force on an object than another liquid of lower density.

(ii) Volume of object immersed in the liquid

As the volume of solid object immersed inside the liquid increases, the upward buoyant force also increases.

**Floating or sinking of objects in liquid**

(i) If the buoyant force or upthrust exerted by the liquid is less than the weight of the object, the object will sink in the liquid.

(ii) If the buoyant force is equal to the weight of the object, the object will float in the liquid.

(iii) If the buoyant force is more than the weight of the object, the object will rise in the liquid and then float.

**Density **

**Archimedes’ Principle**

“When an object is fully or partially immersed in a liquid, it experiences a buoyant force or upthrust, which equal to the weight of liquid displaced by the object”.

i.e. Buoyant force or upthrust acting on an object = Weight of liquid displaced by an object

**Applications of Archimedes’ principle**

- Designing ships and submarines
- Lactometer (a device used to determine the purity of milk).
- Hydrometer (a device used for determining the density of liquid)
- Determining the relative density of a substance.

**Relative density**