Question

Determine the magnitude of free fall acceleration at an altitude of 500km. By what percentage is the weight of a body reduced at this altitude? (Assume the values of G, mass of the earth and radius of the earth)
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11th

Physics

Gravitation

Variation in Value of Acceleration due to Gravity

At a height h above the sur...

PHYSICS

At a height h above the surface of earth, the change in value of g is same as that at a depth x below the surface on the earth. Both x and h are very small in comparison to the radius of the earth. Then,

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ANSWER

Let, g

h

be the acceleration due to gravity at height h.

g

x

be the acceleration due to gravity at depth x.

The acceleration due to gravity on the surface of Earth is

g=

R

2

GM

where,R is the radius o Earth,M is mass of Earth and G is gravitational constant.

∴g∝

R

2

1

⇒g

h

∝

(R+h)

2

1

Therefore

g

g

h

=

(R+h)

2

R

2

g

g

h

=

(1+

R

h

)

2

1

g

g

h

=(1+

R

h

)

−2

g

g

h

=(1−

R

2h

)

g

h

=g−

R

2gh

g−g

h

=

R

2gh

Also, the acceleration due to gravity at depth x,

g

x

=

3

4

Gρ(R−x)

⇒g

x

∝(R−x)

Therefore

g

g

x

=

R

R−x

g

g

x

=1−

R

x

g

x

=g−

R

gx

g−g

x

=

R

gx

If the change in the value of g at height h above earth surface is the same as that at depth x i.e.

g−g

h

=g−g

x

R

2gh

=

R

gx

∴x=2h