One end of a light spring of spring constant k is fixed to a...

One end of a light spring of spring constant $k$ is fixed to a wall and the other end is tied to a block placed on a smooth horizontal surface. In a displacement, the work done by the spring is $\frac{1}{2} k x^{2}$. The possible cases are: (A) The spring was initially compressed by a distance $x$ and was finally in its natural length. (B) It was initially stretched by a distance $x$ and finally was in its natural length. (C) It was initially in its natural length and finally in a compressed position. (D) It was initially in its natural length and finally in a stretched position.

One end of a light spring of spring constant $k$ is fixed to a wall and the other end is tied to a block placed on a smooth horizontal surface. In a displacement, the work done by the spring is $\frac{1}{2} k x^{2}$. The possible cases are:
(A) The spring was initially compressed by a distance $x$ and was finally in its natural length.
(B) It was initially stretched by a distance $x$ and finally was in its natural length.
(C) It was initially in its natural length and finally in a compressed position.
(D) It was initially in its natural length and finally in a stretched position.

Key Concepts

Hooke's Law

Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium (natural) position and acts in the opposite direction. This fundamental principle explains how a spring behaves when it is either compressed or stretched and forms the basis for analyzing forces in oscillatory and static systems involving springs.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is either compressed or stretched. This energy, which is proportional to the square of the displacement from the spring's natural length, is calculated as ½kx². It plays a crucial role in understanding energy transformations in systems where springs are involved.

Work Done by a Spring Force

The work done by the spring is related to the change in its elastic potential energy. When a spring moves from a position of deformation (compressed or stretched) back to its equilibrium state, the work done by the spring force is positive and equals ½kx². Conversely, moving from equilibrium to a deformed state involves the spring force doing negative work, which is essential for analyzing energy conservation in such systems.

Sign Convention in Work Calculations

In the context of springs, the sign of the work done depends on the direction of displacement relative to the spring force. When the spring force and the displacement are in the same direction, the work done by the spring is positive, as seen when the spring returns to its natural length from a deformed state. Conversely, if the displacement is in the direction in which the spring is being deformed, the work done by the spring is negative. This understanding is crucial for properly applying the work-energy principle in spring-related problems.

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