Question

A particle of mass 'm' is raised from the surface of earth to a height h = 2R. Find work done by some external agent in the process. Here, R is the radius of earth and g the acceleration due to gravity on earth's surface.

Answer 1

Work done by the process =  $\frac{2}{3}$ mghR    

Given:

A particle of mass 'm' is raised from the surface of earth to a height h = 2R.

R is the radius of earth

g = acceleration due to gravity on earth's surface.

To find:

Work done by some external agent in the process.

Formula used:

Energy = Potential energy + kinetic energy

Potential energy = $\frac{GMm}{R}$

g = acceleration due to gravity = $\frac{GM}{R^2}$

Where      M = mass of earth.

                 m = mass of object.

                g = acceleration due to gravity.

                 R = Radius of earth.

Explanation:

As no information is given to the kinetic energy so we can assume that kinetic energy is zero.

work done by some external agent in the process = Change in Energy.

Initial Energy =  $\frac{GMm}{R}$

Final Energy = $\frac{GMm}{R+h}$        

We can take the mass from earth surface to Height.

Change in Energy = Final Energy - Initial Energy  

Change in Energy = $\frac{GMm}{R+h}$ - $\frac{GMm}{R}$

Change in Energy = $\frac{GM}{R^2}$. ($\frac{mh}{1+\frac{h}{R} }$)

Given that H = 2R

                     = mgh . $(\frac{1}{1+\frac{h}{R} })$

                     = $\frac{2}{3}$ mghR

Work done by the process =  $\frac{2}{3}$ mghR              

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