Derive an expression for acceleration due to gravity on a planet of mass M and radius R
Answers
accen due to gravity is shown as g
mass of planet = M
radius =R
.. .
according to universal law of gravitation
f = GMm/r2 ............. eqn I
....
when a object fall vertically on toward s the earth
then,
f= mg........ eqn II
..
using eqn I and II,
we get,,
= mg = GMm/r2
= g= GM/r2
... this expression show the accen due to gravity..
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=mF …..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=Gr2M×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=Gr2×mM×m
a=Gr2M
Now, from above,
a=g= Acceleration due to gravity
We also see that, although force is depending on the mass of the object,
F=Gr2M×m
But acceleration due to gravity is independent of the mass.
g=Gr2M
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∝r21) .
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 inside the surface of earth and g is zero at the centre of earth, as at the centre of the earth the object has mass around it, so net force cancels and thus net acceleration becomes zero.