Q. 24.5( 25 Votes )

# Choose the correct alternative:

A. Acceleration due to gravity increases/decreases with increasing altitude.

B. Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).

C. Acceleration due to gravity is independent of mass of the earth/mass of the body.

D. The formula –G Mm(1/r_{2} – 1/r_{1}) is more/less accurate than the formula mg(r_{2} – r_{1}) for the difference of potential energy between two points r_{2} and r_{1} distance away from the centre of the earth.

Answer :

A. Acceleration due to gravity decreases with increasing altitude, as it varies inversely to the square of distance from centre of earth and is given by relation

Where, g is the acceleration due to gravity

G is universal gravitational Constant

M_{e} is mass of Earth

r is distance of the point from center of earth (point must be on or above surface of earth not inside)

**As can be seen in the figure**

so as the distance from center of earth r or Altitude increases, the acceleration due to gravity decreases

B. Acceleration due to gravity decreases with increasing depth, as though distance of center of earth from the point is decreasing but mass of earth is also decreasing as less section of earth’s mass will contribute to Gravity as can be seen in the figure

If at a depth d inside surface of earth, the acceleration due to gravity is given as

g’ = g(1-d/R)

where, g’ is acceleration due to gravity at a depth d inside surface of earth, R is the Radius of earth and g is acceleration due to gravity on surface of earth

so as we can see as the depth inside surface d increases, value of d/R increases and (1-d/R) decreases and becomes less than 1, and hence we get

g’ < g inside surface of earth

C. Acceleration due to gravity is independent of mass of body as it is given by the relation

Where, g is the acceleration due to gravity

G is universal gravitational Constant

Me is mass of Earth

R is the radius of earth

as we can see it does not include any term of mass of body, so acceleration of gravity has same value for all bodies and is independent of mass of other body

D. The formula –G Mm(1/r_{2} – 1/r_{1}) is more accurate than the formula mg(r_{2} – r_{1}) for the difference of potential energy between two points r_{2} and r_{1} distance away from the centre of the earth, as acceleration due to gravity varies with distance from centre from earth

Potential energy assuming acceleration due to gravity to be constant is given as

V = mgr

Where V is the gravitational Potential Energy of body of mass m at a distance r from centre of earth, g is acceleration due to gravity

So at distance r_{1} from centre of earth gravitational potential energy will be

V_{1} = mgr_{1}

at distance r_{2} from centre of earth gravitational potential energy will be

V_{2} = mgr_{2}

So difference in potential energy is

V_{2} – V_{1} = mgr_{2} – mgr_{1} = mg(r_{2} - r_{1})

But as we know Acceleration due to gravity decreases with increasing altitude, as it varies inversely to the square of distance from centre of earth and is given by relation

Where, g is the acceleration due to gravity

G is universal gravitational Constant

M_{e} is mass of Earth

r is distance of the point from center of earth

so g has different value at both the points i.e. the difference in potential energy is not accurate

The accurate relation for Gravitational Potential energy of a body of mass m at any point above surface of earth is given by relation

V = -GMm/r

Where V is the gravitational Potential Energy of body of mass m at a distance r from centre of earth and M is the mass of earth,

So at distance r_{1} from centre of earth gravitational potential energy will be

V_{1} = -GMm/r_{1}

at distance r_{2} from centre of earth gravitational potential energy will be

V_{2} = -GMm/r_{2}

So difference in potential energy is

V_{2} – V_{1} = -GMm/r_{2} – (-GMm/r_{1})

= –G Mm(1/r_{2} – 1/r_{1})

This is more accurate formula for change in gravitational potential energy

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