Question

On a planet whose size is the same and mass four times as that of our Earth, find the amount of work done to lift $3kg$ mass vertically upwards through $3m$ distance on the planet. The value of $g$ on the surface of earth is $10m{{s}^{-2}}$.
$\text{A}\text{. }360J$
$\text{B}\text{. }160J$
$\text{C}\text{. }500J$
$\text{D}\text{. }460J$

Answer 1

Hint: Acceleration due to gravity on a planet depends upon its mass and radius. We will find the relation between acceleration due to gravity at Earth and the planet. We can then find the value of work done on the planet to lift the mass.

Formula used:
$g=\dfrac{GM}{{{R}^{2}}}$

Complete step by step answer:
When we lift an object, the work done against gravity or gravitational force is stored as potential energy in the object. An object’s gravitational potential is due to its position, relative position, being calculated from a reference point. To find gravitational potential, we must set a reference level at which potential energy is assumed to be zero. Usually, we take this level or point as ground.

Let’s assume the mass of Earth as $M$ and radius of Earth as $R$

Take mass of planet as $M'$ and radius of planet $R'$
We are given that $M'=4M$ and $R'=R$
Acceleration due to gravity on Earth $=g$
Acceleration due to gravity on planet $=g'$

For calculating the value of acceleration due to gravity we will use the formula,
$g=\dfrac{GM}{{{R}^{2}}}$

Finding $g'$,
$g'=\dfrac{GM'}{R{{'}^{2}}}$

Put $M'=4M$and $R'=R$

$g'=\dfrac{G\times 4M}{{{R}^{2}}}=\dfrac{4GM}{{{R}^{2}}}=4g$

Value of $g$ is given as $10m{{s}^{-2}}$, therefore,

$g'=40m{{s}^{-2}}$

Energy required to lift the body on Earth $=mgh$
Energy required to lift the body on planet $mg'h$
Therefore, \[\text{Energy required = 3}\times 40\times 3=360J\]

Hence, the correct option is A.

Additional information:
When the work is done on an object, positive work, it increases the potential of the object. When the work is done by the object, negative work, it decreases the potential energy of the object. We can also say, when work is done against gravity, it tends to decrease the kinetic energy of an object and is known as negative work. When the work is done by gravity, it tends to increase the kinetic energy of the object and is known as positive work. In case the kinetic energy of the object decreases, simultaneously, its potential energy increases and when the kinetic energy of the object increases, its potential energy decreases. This is due to the law of conservation of energy.

Note: While calculating the value of acceleration due to gravity, the mass in the formula is the mass of the planet, not the mass of the object. For any mass or size of object, acceleration due to gravity remains the same on a planet.