Work Done by a Constant Force

Introduction

In our daily life, we frequently push, pull, lift, or drag objects. Whenever a force acts on an object and causes it to move, work is said to be done. The concept of work helps us understand how energy is transferred from one object to another.

The simplest situation occurs when the force acting on an object remains constant in both magnitude and direction during the motion. Such a force is known as a constant force. The work done by a constant force can be calculated using a simple mathematical relation.

Definition

The work done by a constant force is defined as the scalar (dot) product of the force vector \(\vec{F}\) and the displacement vector \(\vec{s}\).

\(W=\vec{F}\cdot\vec{s}\)

If the angle between the force vector \(\vec{F}\) and the displacement vector \(\vec{s}\) is \(\theta\), then

\(W=Fs\cos\theta\)

where,

  • \(W\) = Work done
  • \(\vec{F}\) = Force vector acting on the object
  • \(\vec{s}\) = Displacement vector of the object
  • F = Magnitude of force
  • s = Magnitude of displacement
  • \(\theta\) = Angle between force and displacement vectors

Figure

\( \vec{F} \)
\( \theta \)
\( \vec{s} \)
Initial Position Final Position

Explanation

The above figure shows a block placed on a horizontal surface. A constant force \(\vec{F}\) acts on the block at an angle \(\theta\) with the horizontal. As a result, the block moves from its initial position to its final position and undergoes a displacement \(\vec{s}\).

Only the component of force along the direction of displacement contributes to the work done. The component of force in the direction of displacement is

\(F\cos\theta\)

Therefore, the work done by the constant force becomes

\(W=(F\cos\theta)s\)

or

\(W=Fs\cos\theta\)

This equation shows that work depends not only on the magnitudes of force and displacement but also on the angle between them.

SI Unit of Work

The SI unit of work is the joule (J).

One joule is defined as the work done when a force of one newton produces a displacement of one metre in the direction of the force.

\(1\,J=1\,N\times1\,m\)

Special Cases

1. Force and Displacement in the Same Direction

When the force acts in the same direction as displacement,

\(\theta=0^\circ\)

\(W=Fs\cos0^\circ=Fs\)

In this case, the work done is maximum and positive.

2. Force Perpendicular to Displacement

When the force is perpendicular to the displacement,

\(\theta=90^\circ\)

\(W=Fs\cos90^\circ=0\)

Hence, no work is done by the force.

3. Force Opposite to Displacement

When the force acts opposite to the displacement,

\(\theta=180^\circ\)

\(W=Fs\cos180^\circ=-Fs\)

The work done is negative because the force opposes the motion.

Importance

  • Understanding work done by a constant force helps us explain how energy is transferred from one object to another. Whenever work is done, energy is either transferred or transformed from one form to another.
  • The concept forms the basis of the Work-Energy Theorem, which relates the work done on an object to the change in its kinetic energy.
  • The concept is widely used in solving numerical problems involving force, displacement, energy, and motion in mechanics.
  • Work done by forces plays an important role in engineering, transportation, construction, and machine design where forces continuously act on moving objects.
  • A proper understanding of work done by a force provides a strong foundation for studying advanced topics such as energy, power, momentum, and dynamics.

Examples

  • A person pushing a box across a floor. Since the applied force produces displacement, positive work is done by the person.
  • A worker pulling a cart on a road using a rope inclined at an angle to the horizontal.
  • Dragging a suitcase while walking through a railway station or airport.
  • Lifting a bucket of water from a well against the force of gravity.
  • A crane lifting heavy construction materials to a certain height.

Important Points

  • Work is done only when a force acting on an object produces displacement. If there is no displacement, no work is done.
  • Work is a scalar quantity because it is obtained from the scalar (dot) product of two vectors.
  • Only the component of force along the direction of displacement contributes to the work done.
  • The mathematical expression for work done by a constant force is \(W=Fs\cos\theta\).
  • The work done can be positive, negative, or zero depending on the angle between the force and the displacement.
  • When force and displacement are in the same direction, the work done is positive.
  • When force acts opposite to the displacement, the work done is negative.
  • When force is perpendicular to displacement, the work done is zero.
  • The SI unit of work is the joule (J).
  • One joule is defined as the work done when a force of one newton produces a displacement of one metre in the direction of the force.

Conclusion

Work done by a constant force is one of the fundamental concepts of mechanics. It explains how forces transfer energy and produce motion. The relation \(W=Fs\cos\theta\) shows that the work done depends not only on the magnitudes of force and displacement but also on the angle between them. Understanding this concept is essential for studying energy, power, and many other topics in Physics.

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