Gravitation

Mass

Mass is the measurement of inertia and inertia is the property of any object which opposes the change in state of the object. It is inertia because of which an object in rest has tendency to remain in rest and an object in motion has tendency to remain in motion.

Inertia depends upon the mass of an object. Object having greater mass has greater inertia and vice versa. Mass of an object remains constant everywhere, i.e. mass will remain same whether that object is at the moon, at the earth or anywhere in the universe.


Weight:

Earth attracts every object towards it. We know that force is the product of mass and acceleration due to gravity.

This means, `F = m xx g`  ----(i)

The force by which earth attracts an object towards it is called the weight of the object, which is the product of mass (m) of the object and acceleration due to gravity (g).

Weight is denoted by ‘W’.

Therefore, by substituting in the expression ‘F = mg’ we get,

`W = m xx g`   ----(ii)

Since weight is the force which is acting vertically downwards, therefore, weight has both magnitude and direction and hence it is a vector quantity.

Since the value of ‘g’ is always constant at a given place,

Therefore, expression ‘W = m x g’ can be written as follows:

`W prop m`  ------(iii)

This means weight of any object is directly proportional to its mass, i.e. weight will increase with the increase of mass and decrease with decrease in mass.

This is the cause that weight of any object is the measure of its mass.


The unit of weight

Since, weight of an object is equal to the force by which an object is attracted towards earth, therefore, unit of weight is same as the unit of force.

Therefore, Unit of weight is ‘newton (N)’.

Weight of an Object on the Surface of Moon

Since, weight of an object on the earth is the force by which earth attracts that very object towards it. In similar way, weight of an object on the surface of moon or any other planet is the force by which moon or any other planet will attract the object towards it.

We know that,

The Mass of Earth = 5.98 × 10 24 kg

Radius of earth = 6.37 × 106 m

Mass of moon = 7.36 × 1022 kg

Radius of moon = 1.76 × 106 m


Since, mass of the moon is less than that of earth, therefore, moon will exert less force of attraction on any object; in comparison to the earth.

Let mass of an object is ‘m’
The weight of the object on earth is We
The weight of the object on moon is Wm
Mass of the earth is M
Mass of the moon is Mm
Radius of earth is R
Radius of moon is Rm
Acceleration due to gravity on earth is ‘g’
Acceleration due to gravity on moon is ‘gm’.

Therefore,

Weight of the object on earth We = m × g

By substituting the value of ‘g’ from the expression of Universal Law of Gravitation we get

`W_e=m*(GM)/R^2`

`=>W_e=(mxxGxx5.98xx10^(24)kg)/(6.37xx10^6m)^2`

`=>W_e=(mxxGxx5.98xx10^(24)kg)/(6.37xx6.37xx10^6xx10^6m^2)`

`=>W_e=(mxxGxx5.98xx10^(24)kg)/(40.57xx10^(12)m^2)`

`=>W_e=(mxxGxx5.98xx10^(12)kg)/(40.57m^2)`

`=>W_e=mxxGxx0.1474xx10^(12)kgm^(-2)`

`=>W_e=mxxGxx1.474xx10^(11)kgm^(-2)`

Weight of the object on moon `W_m=mxxg_m`

By the expression of universal law of gravitation,

`W_m=m(GM_m)/(R_m^2)`

`=>W_m=(mxxGxx7.36xx10^(22)kg)/((1.74xx10^6m)^2)`

`W_m=(mxxGxx7.36xx10^(22)kg)/(1.74xx1.74xx10^6xx10^6m^2)`

`=>W_m=(mxxGxx7.36xx10^(22)kg)/(3.0276xx10^(12)m^2)`

`=>W_m=(mxxGxx7.36xx10^(10)kg)/(3.0276m^2)`

`=>W_m=mxxGxx2.4309xx10^(10)kg\ m^(-2)`

Now, `W_m/W_e` `=(mxxGxx2.4309xx10^(10)kgm^(-2))/(mxxGxx1.474xx10^(11)kg\ m^(-2))`

`=>W_m/W_e=(2.4309)/(1.474xx10)`

`=>W_m/W_e=(2.4309)/(14.74)`

`=>W_m/W_e==1/(6.030)~~ 1/6`


Therefore, Weight of an object on moon / Weight of an object on earth = 1/6

Or, Weight of an object on the moon = 1/6th of the weight of the object on earth.




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