## GRAVITATIONAL FORCE OF EARTH

- If we release a small stone without pushing it from a height, it accelerates towards earth.
- The stone is when accelerated towards earth, means some force is acting on it. The force which pulls the objects towards the centre of the earth is known as gravitational force of the earth.
- Here, stone also attracts earth. It means every object in universe attracts every other object.

## NEWTON'S UNIVERSAL LAW OF GRAVITATION

Sir Isaac Newton in 1687 proposed a law about the force of attraction between the two objects in the universe which is known as Newton’s law of gravitation. According to Universal law of Gravitation, every mass in this universe attracts every other mass with a force which is directly proportional to the product of two masses and inversely proportional to the square of the distance between them.

- Let masses (M) and (m) of two objects are distance (d) apart, then force of attraction (F) between them

F ∝ M×m

F ∝ 1/d2

F ∝ Mm/d2

F = (GMm)/d2

where,

G is a constant and is known as Gravitational constant. Value of G = 6.67×10-11 Nm2/kg2

G is called universal gravitational constant. - If unit of F is in Newton, m is in kg, d is in metre, then unit of G can be calculated as : G = (F×d2)/Mm, therefor unit will be Nm2/kg2

## RELATION BETWEEN NEWTON'S THIRD LAW OF MOTION AND NEWTON'S LAW OF GRAVITATION

- According to Newton’s third law of motion, “Every object exerts equal and opposite force on other object but in opposite direction.”
- According to Newton’s law of gravitation, “Every mass in the universe attracts the every other mass.”

→ In case of freely falling stone and earth, stone is attracted towards earth means earth attracts the stone. - But according to Newton’s third law of motion, the stone should also attract the earth. It is true that stone also attracts the earth with the same force F = m × a but due to very less mass of the stone, the acceleration (a) in its velocity is 9.8 m/s2 and acceleration (a) of earth towards stone is 1.65×10-24 m/s2 which is negligible and we cannot feel it.

## IMPORTANCE OF UNIVERSAL LAW OF GRAVITATION

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

(ii) The motion of moon around the earth.

(iii) The motion of earth around the sun.

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

## FREE FALL OF AN OBJECT AND ACCELERATION (g)

- When an object is thrown upward, it reaches certain height, then it starts falling down towards earth. It is because the earth’s gravitational force exerts on it.

- This fall under the influence of earth is called ‘free fall of an object’.
- During this free fall direction do not change but velocity continuously changes which is called acceleration due to gravity. It is denoted by ‘g’.
- Its unit is same as acceleration m/s .

## GRAVITATIONAL ACCELERATION AND ITS VALUE AT THE SURFACE OF THE EARTH

- The uniform acceleration produced in a freely falling object due to the gravitational force of earth, is called acceleration due to gravity.
- It is represented by ‘g’ and it always acts towards the centre of the earth. Value of ‘g’ on the surface of earth
- The force acting on an object is F = (GMem)/R
- where,Me = Mass of earth, m = Mass of an object, R = Radius of earth
- If acceleration due to gravity is ‘g’ due to force F then, F = m×g

## RELATIONSHIP AND DIFFERENCE BETWEEN 'G' AND 'g'

- G = Gravitational constant
- g = Acceleration due to gravity g = GM/R2
- Difference between G (Gravitational constant) and g (Acceleration due to gravity)
**Gravitation Constant (G)**Its value is 6.67×10 Nm /kg . Its value remains constant always and everywhere.

**Gravitational acceleration (g)**Its value is 9.8 m/s . Its value varies at various places. Its unit is Nm /kg . Its unit is m/s . It is a scalar quantity. It is a vector quantity.

**Example:**If two stones of 150 gm and 500 gm are dropped from a height, which stone will reach the surface of earth first and why ? Explain your answer.**Solution**→ It was Galileo, who first time demonstrated and depicted that the acceleration of an object falling freely towards earth does not depend on the mass of the

object.→ It can be verified by universal law of gravitation. Let an object of mass m, is allowed to fall from a distance of R, from the centre of the earth. Then, the gravitational force, F = (GMem)/R2 (Me= Mass of the earth) The force acting on the stone is F = m×a

∴ m × a = (GMem)/R2

⇒ a = GMe/R2→ So, acceleration in an object falling freely towards earth depends on the mass of earth and height of the object from the centre of the earth. So, stones of mass 150 gm and 500 gm will reach the earth surface together.

## EQUATION OF MOTION WHEN AN OBJECT IS FALLING FREELY TOWARDS EARTH OR THROWN VERTICALLY UPWARDS

**Case 1:**When an object is falling towards earth with initial velocity (u)

Velocity (v) after t seconds, v = u + ght

Height covered in t seconds, h = ut + 1⁄2gt2

Relation between v and u when t is not given: v2 = u2 + 2gh**Case 2:**When object is falling from rest position means initial velocity u=0

Velocity (v) after t seconds, v = gt

Height covered in t seconds, h = 1⁄2gt2

Relation between v and u when t is not given: v2 = 2gh**Case 3:**When an object is thrown vertically upwards with initial velocity u, the gravitational acceleration will be negative (-g)

Velocity (v) after t seconds, v = u − gt

Height covered in t seconds, h = ut − 1⁄2gt2

Relation between v and u when t is given: v2 = u2 − 2gh

## MASS

- The mass of a body is the quantity of matter contained in it. Mass is a scalar quantity which has only magnitude but no direction.
- SI unit of mass is kilogram which is written in short form as kg.
- Mass of a body is constant and does not change from place to place.
- Mass of a body is usually denoted by the small ‘m’.
- Mass of a body cannot be zero.

## WEIGHT

- The force with which an object is attracted towards the centre of the earth, is called the weight of the object.

Force = m × a

In case of earth, a = g

∴ F = m× g

But the force of attraction of earth on an object is called its weight (W). ∴W = m × g - Weight is the force and its SI unit is Newton (N). It depends on ‘g’ and is a vector quantity.

### Relation between 1 kg wt and express it into Newton

- We know that
- W = m × g

If mass (m) = 1 kg, g = 9.8 m/s2, then W = 1 kg × 9.8 m/s2

⇒ 1 kg wt = 9.8 N - The gravitational force of earth that acts on an object of mass 1 kg is called as 1 kg wt.

## DIFFERENCE BETWEEN MASS AND WEIGHT

## FACTORS AFFECTING THE VALUE OF 'g'

- Earth is not a perfect sphere. The radius of the earth increases when we go from pole to equator.
- The value of gravitational acceleration ‘g’ decreases with increasing height.
- The weight of an object on the moon is one-sixth of the weight on earth. Let the mass of an object be m, its weight on earth means the force by which earth attracts it towards the centre.

## THRUST AND PRESSURE

- The force acting on an object perpendicular to the surface is called thrust.
- The effect of thrust per unit area is called pressure.
- Pressure (P) = Force/Area = F/A
- SI unit is N/m or Nm .
- SI unit of pressure is Pascal denoted by ‘Pa’.
- Pressure depends on two factors :(i) Force applied(ii) Area of surface over which force acts.
- Examples of Pressure: (i)The base of high buildings is made wider so that weight of walls act over a large surface area and pressure is less. (ii)School bags are having broad strap so that the weight of school bags fall over a larger area of the shoulder and produce less pressure and becomes less painful. (iii)The blades of knives are made sharp so very small surface area and on applying force, it produces large pressure and cuts the object easily.(iv)All liquids and gases are fluids and they exert pressure in all directions.

## BUOYANCY

- The upward force experienced by an object when it is immersed into a fluid is called force of buoyancy.
- It acts in upward direction and it depends on the density of the fluid.
- When force of gravitational attraction of the earth on the surface of the object < buoyant force exerted by fluid on the surface of the object, object floats in the fluid.
- When force of gravitational attraction of the earth on the surface of the object > buoyant force exerted by fluid on the surface of the object, the object sinks in the fluid.
- This is the reason, why a pin sinks and boat/ship floats on the surface of water. (Archimedes’ principle)

## DENSITY

- The mass per unit volume is called density of an object.
- If M is the mass and V is the volume, then • Density (d) = Mass(M)/Volume(V)
- SI unit = kg/m3

## ARCHIMEDES' PRINCIPLE

### APPLICATIONS OF ARCHIMEDES' PRINCIPLE

- It is used in determining the relative density of substances.
- It is used in designing ships and submarines.
- Hydrometers and lactometers are made on this principle.
- It is because this ship made of iron and steel floats in water whereas a small piece of iron sinks in it.

## RELATIVE DENSITY

- The ratio of the density of a substance to that of the density of water is called relative density.
- Relative density = Density of a substance/Density of water.
- It has no unit as it is a ratio.