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A body of mass m rises to heighth=R5from the earth's surface, where R is the earth's radius. If g is the acceleration due to gravity at the earth's surface, the increase in potential energy is


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Solution

Step 1: Given data

  1. The radius of earth is R.
  2. Mass of the body is m.
  3. The height of the body from the earth's surface is h=R5 .

Step 2: Gravitational potential energy

  1. The gravitational potential energy at any point in a gravitational field of a system of masses is the amount of work done in bringing a unit mass from infinity to that point.
  2. Gravitational potential energy due to a point mass m at a distance r is, V=-rGMmx2dx=-GMmr , where G is the universal gravitational constant and M is the mass of the earth.

Step 3: Finding the change in gravitational potential energy

We know that the gravitational potential energy of a body of mass m placed on the earth's surface is

V=-GMmR ……………(1)

where M and R are the mass of the earth and the radius of the earth.

Now, gravitational potential energy at a height h=R5 from the earth's surface is,

V'=-GMmR+R5=-G5Mm6RorV'=-G5Mm6R..................(2)

So, the change in gravitational potential energy is

V=V'-V=-G5mM6R--GmMR.orV=-G5mM6R+GmMR=GmM6R.orV=GmM6R.

(From equations 1 and 2 )

Therefore, the increase in gravitational potential energy is GMm6R.


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