Question

Derive a mathematical expression for acceleration due to gravity to show that
it does not depend on the mass of the object.​

Answer 1

Explanation:

The acceleration produced in a freely falling body by the gravitational pull of the earth is called the acceleration due to gravitation.

We know that,

Force = Mass × Acceleration

F=m×a

a=

m

F

…..1

Where F is the force on the object of mass m dropped from a distance r from the centre of earth of mass M.

So, force exerted by the earth on the object is

F=G

r

2

M×m

...2

M = Mass of Earth

m = mass of object

r = distance of object from centre of earth

Now, from equation 1 and 2,

a=G

r

2

×m

M×m

a=G

r

2

M

Now, from above,

a=g= Acceleration due to gravity

We also see that, although force is depending on the mass of the object,

F=G

r

2

M×m

But acceleration due to gravity is independent of the mass.

g=G

r

2

M

Factors on which g depends are:

(i) Value of gravitational constant (G)

(ii) Mass of Earth (M)

(iii) Radius of Earth (r)

As gravitational constant G and mass of earth M are always constant, so the value of acceleration due to gravity g is constant as long as the radius of earth remains constant.

At the surface of earth also, the value of g is not constant. (g∝

r

2

1

) .

At the poles, radius of earth is minimum, hence the g is maximum. Similarly at the equator, the radius of earth is maximum, and hence the value of g is minimum. Also as we go up from the surface of the earth, distance from the centre of the earth increases and hence value of g decreases.

The value of g also decreases as we go