A block of mass 2 kg is released from rest on a smooth...

A block of mass $2 \mathrm{~kg}$ is released from rest on a smooth inclined plane of inclination $30^{\circ}$ and connected by a massless spring of force constant $1000 \mathrm{~N} / \mathrm{m}$ as shown in Fig. $4.25$. Initially, the spring is in its natural length. An external variable force also acts on the block down the inclined plane. Block comes to rest for a moment after travelling a distance of $20 \mathrm{~cm}$ along the inclined plane. From initial to this moment, Column-I gives work done by various forces and Column-II gives their values. (A) Work done by gravity (1) Zero (B) Work done by spring (2) $18 \mathrm{~J}$ (C) Work done by external force (3) $-20 \mathrm{~J}$ (D) Work done by normal force (4) $2 \mathrm{~J}$ (5) $20 \mathrm{~J}$

A block of mass $2 \mathrm{~kg}$ is released from rest on a smooth inclined plane of inclination $30^{\circ}$ and connected by a massless spring of force constant $1000 \mathrm{~N} / \mathrm{m}$ as shown in Fig. $4.25$. Initially, the spring is in its natural length. An external variable force also acts on the block down the inclined plane. Block comes to rest for a moment
after travelling a distance of $20 \mathrm{~cm}$ along the inclined plane. From initial to this moment, Column-I gives work done by various forces and Column-II gives their values.
(A) Work done by gravity
(1) Zero
(B) Work done by spring
(2) $18 \mathrm{~J}$
(C) Work done by external force
(3) $-20 \mathrm{~J}$
(D) Work done by normal force
(4) $2 \mathrm{~J}$
(5) $20 \mathrm{~J}$

Key Concepts

Work Done by a Force

This concept refers to the energy transferred to or from an object by a force acting over a displacement. It is defined as the product of the magnitude of the force, the displacement of the object, and the cosine of the angle between the force vector and the displacement vector. Understanding this concept is crucial for analyzing how different forces contribute to the motion and energy changes in a system.

Work-Energy Principle

The work-energy principle states that the net work done on an object equals the change in its kinetic energy. This principle is fundamental in solving dynamics problems because it allows us to account for the energy contributions from all forces acting on the object without having to analyze the detailed path of the motion, focusing instead on the initial and final energy states.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. When an object moves along an incline, its height relative to a reference level changes, which in turn changes its gravitational potential energy. This concept is key to understanding how gravity performs work on an object moving on an inclined plane.

Elastic Potential Energy

Elastic potential energy is the energy stored in elastic materials like springs when they are deformed. According to Hooke’s law, the force exerted by a spring is proportional to its displacement from its equilibrium position, and the potential energy stored is given by one-half the product of the spring constant and the square of the displacement. This concept is fundamental when analyzing systems involving springs and their associated work contributions.

Normal Force in Work Calculations

The normal force is the force perpendicular to the surface of contact between two bodies. In many cases, such as motion along a frictionless inclined plane, the normal force does no work because it acts perpendicular to the direction of displacement. Recognizing when forces do or do not do work is critical in setting up accurate energy balances in mechanical systems.

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