Question

What is the work done against gravity when a body is moved horizontally along a frictionless surface?

Answer 1

In order to solve this question we need to understand work done and the force of gravity. Work done is defined as the magnitude of force causing a body to be displaced by one unit in its direction. If the force and displacement make some angle with each other, say $\theta$ then work is mathematically defined as $W = \vec F.\vec d$ or $W = Fd\cos \theta$ and the force of gravity is defined as the force with which we push the earth or the earth attracts us. Gravitational force is one of the basic forces in nature which is attractive in nature. Also the work done is scalar in nature.

Complete step by step answer:
Consider a body of mass “m”. Let the body be displaced horizontally by distance “d”.So the force of gravity on the mass “m” is given by,
$F = mg$
Here, $g$ is acceleration due to gravity and this force is in a downward direction.
Also the floor is frictionless, so force is acting on the body horizontally.So the work done by force of gravity in displacing a body horizontally is given as,
$W = \vec F.\vec d$
Or removing dot product we get,
$W = Fd\cos \theta$
Here the force of gravity makes an angle $\theta = \dfrac{\pi }{2}$.
Putting value of $\theta$ and Force of gravity in work equation we get,
$W = mgd\cos (\dfrac{\pi }{2})$
$\therefore W = 0$ Since $\cos (\dfrac{\pi }{2}) = 0$

So there is no work done by gravity force in displacing the body horizontally.

Note: It should be remembered that work can be positive, negative or zero depending on the angle between force applied and displacement of body. Work is positive if the force applied is in the direction of displacement, work done is negative if the force applied is in the opposite direction of displacement, Work done is zero if the applied force is perpendicular to displacement.