Hello everyone, here we are going to discuss the effect of height on acceleration due to gravity. We have derived the formula for acceleration due to gravity on the surface of earth, in that we concluded that the acceleration due to gravity depends upon mass and distance of object from the centre of planet.
Derivation of formula for acceleration due to gravity at height ‘h’
Let’s find the formula for the acceleration due to gravity at height ‘h’ above the surface of earth.
Consider a body of mass ‘m’ at height ‘h’ above the surface of earth of mass ‘M’ and radius ‘R’. The distance of object from the centre of earth will be, r= R+h
If ‘g’ is acceleration due to gravity on the surface of earth then, we have
Let gh be the acceleration due to gravity at altitude h above the earth surface given by
dividing equation (2) by (1) we get
Equation (3) and (4) gives the different forms of acceleration due to gravity at height ‘h’ above the surface of earth.
Points to remember:
- We can conclude that the value of acceleration due to gravity decreases as body moves above the surface earth.
- Note that for small values of ‘h’, ‘g’ won’t get changed by larger extent.
- Value of ‘g’ becomes almost zero when we are altitude of 36000 km above the surface of earth.
Numerical based in the acceleration due to gravity at height ‘h’ above earth:
Ex: Find the height at which the acceleration due to gravity becomes 1/16th of acceleration due to gravity on the surface of earth. (R=6400 km)
Solution:
Here, gh = (1/16) g.
The value for acceleration due to gravity at height ‘h’ is given as,
Taking square root on both sides we get,
∴1/4 = R/(R+h)
∴R+h = 4 R
∴ h = 3 R
∴ h = 3×6400
∴h = 19200 km