CBSE Class 11-science Answered

Q. Derive an expression of work done in making a body of mass 'm' free from earth's gravitational field. Using the result, obtain an expression of escape velocity. Also show escape velocity depends only on the density of the planet of a given size.

Asked by neerajprataphazarika | 24 Sep, 2018, 10:50: PM

Expert Answer

If the position of the body of mass m changes on account of forces acting on it,

the change in potential energy is just the amount of workdone on the body by the force.

We know the force of gravitation on a body of mass m, directed towards the centre of earth is

begin mathsize 12px style F space equals space G fraction numerator M cross times m over denominator r squared end fraction space space space..................... left parenthesis 1 right parenthesis end style

where G is gravitational constant, M is mass of earth , m is mass of the body and r is the distance from centre of eart to the body.

Workdone in lifting the body of mass m from earth's surface (r = R ) to infinite distance (r = ∞ ) .

begin mathsize 12px style W space equals space integral subscript R superscript infinity G fraction numerator M cross times m over denominator r squared end fraction d r space equals space G fraction numerator M cross times m over denominator R end fraction end style

..........................................(2)

Escape Speed:-

Let us assume,the body is projected with speed V_iat earth surface so that it escapes the gravitational field.

We consider that potential energy at earth's surface is zero and when the body reaches a point beyond the gravitational field,

all its kinetic energy is converted to potential energy at infinite point ( same as workdone to move it to infinite distance).

Hence from eqn.(2), we have,

begin mathsize 12px style 1 half m cross times V subscript i superscript 2 space equals space G fraction numerator M cross times m over denominator R end fraction V subscript i superscript 2 space equals space 2 cross times fraction numerator G cross times M over denominator R end fraction space equals space 2 cross times fraction numerator begin display style G cross times M end style over denominator begin display style R squared end style end fraction cross times R space equals space 2 cross times g cross times R E s c a p e space s p e e d space V space equals space square root of 2 cross times g cross times R end root space m divided by s end style

Planet of given size means, volume or radius is constant.

We can see that escape speed is directly proportional to square root of mass of planet.

Since volume is constant we can say that escape speed is directly proportional to square root of density

because other factors G and R are constant

Answered by Thiyagarajan K | 26 Sep, 2018, 03:51: PM

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CBSE Class 11-science Answered

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