(i) According to Hooke's law the restoring spring force and displacement are linearly related as F = - k x, and are opposite in direction, the graph between' F and x is a straight line with dwelling only in the
second and fourth quadrant as. shown in Figure.
(ii) The elastic potential energy can be easily calculated by drawing a F- x graph. The shaded area (triangle) is the work done by the spring force.
(iii) Area (base)(height) = \(\frac{1}{2}\times(x)\times(kx)\).
=\(\frac{1}{2}kx^{2}\).

Force-displacement graph for a spring
(iv) Work done is stored as the potential energy of the spring.
\(\therefore U=\frac{1}{2}kx^{2}\)

It implies the following.
(i) If the work done by the force on the body is positive then its kinetic energy increases.
(ii) If the work done by the force on the-body is negative then its kinetic energy decreases.
(iii) If there is no work done by the force on the body then there is no change in its, kinetic energy, which means that the body has moved at constant speed provided its mass remains constant.

(i) The energy produced by mechanical means is called mechanical energy.
(ii) It is classified into 2 types : (1) Kinetic energy (2) Potential energy.
(iii) The energy possessed by a body due to its motion is called kinetic energy. The energy possessed by the body by virtue of its position is called potential energy. SI unit of energy: N m (or) joule (J).

(i) When the component of a variable force F acts on a body, the small work done (dW) by the force in producing a small displacement dr is given by the relation
dw=(f cosθ)dr
[F cos θ is the component of the variable force F]
where, F and θ are variables.
(ii) The total work. done for a displacement from initial position r_i to final position r_f is given by the relation,
\(W=\int _{ { r }_{ i } }^{ { r }_{ f } }{ dW } =\int _{ { r }_{ i } }^{ { r }_{ f } }{ F\cos\theta \ dr } \ \) ---- (1)
(iii) A graphical representation of the work done by a variable force is shown in Figure 1. The area under the graph is the work done by the variable force.

(i) If a force acting on the body in the opposite direction of displacement, the work done is negative.
(ii) For negative work (90° < θ <180°)
i.e. - 1 Example : In a football game, the goal keeper catches the ball coming towards him by applying a force such that the force is applied in a direction opposite to that of the motion of the ball till it comes to rest in his hands. During the time of applying the force, he does a negative work on the ball.