5.5 Work Done by a Constant and Variable Force

In physics, work is a measure of energy transfer that occurs when an object is moved over a distance by an external force. It is a crucial concept for understanding how energy is used and transformed in mechanical systems. The calculation of work depends on whether the force applied is constant or variable.

1. Work Done by a Constant Force

Definition

Work is done by a constant force when the force applied to an object does not change in magnitude or direction, and the object undergoes a displacement.

Formula: Work done ($W$) is defined as the dot product of the force vector $\vec{F}$ and the displacement vector $\vec{d}$: $W=\vec{F}\cdot \vec{d}=Fd\cos \theta$

Where:

• $F$ is the magnitude of the constant force (N)
• $d$ is the magnitude of the displacement (m)
• $\theta$ is the angle between the force vector and the displacement vector

Scalar and Vector Quantities→

Key Characteristics

• Scalar Quantity: Work is a scalar quantity — it has magnitude but no direction. It can be positive, negative, or zero.
• SI Unit: The Joule (J). One joule is the work done when a force of 1 N displaces an object by 1 m in the direction of the force: $1\text{ J}=1\text{ N}\cdot \text{m}$.
• Dimensions: $[M{L}^2{T}^{-2}]$

Derived Units→ Dimensions→

Conditions for Work Done

The sign of work depends on the angle $\theta$ between force and displacement:

Examples:

• A satellite in a circular orbit: gravity is always perpendicular to velocity, so $W_{gravity}=0$.
• Friction always does negative work since it opposes displacement.
• A person carrying a bucket horizontally: gravity acts downward ($\theta =90^{\circ }$), so gravity does zero work.

2. Work Done by a Variable Force

Definition

In many real-world scenarios, the force applied to an object changes as the object moves. Examples include:

• A spring force, which increases as the spring is stretched or compressed
• A rocket engine, whose thrust changes as fuel is consumed

For such cases, $W=Fd\cos \theta$ cannot be applied directly.

Calculation Methods

a) Graphical Method (Area Under F-d Graph)

The work done by a variable force equals the area under the Force-Displacement graph.

• Plot the component of force parallel to displacement ($F\cos \theta$) on the y-axis and displacement ($d$) on the x-axis.
• The area enclosed between the curve and the x-axis gives the total work done.

This method applies equally to Force-Extension graphs for elastic materials: the area under the $F$-$x$ graph gives the work done in stretching the material.

b) Calculus Method (Integral Form)

For a precise calculation:

1. Divide the total displacement into infinitesimally small intervals $\Delta d$.
2. Over each tiny interval, the force is approximately constant: $\Delta W=F\Delta d\cos \theta$.
3. Sum all contributions. As $\Delta d\to 0$, the sum becomes an integral:

$W={\int }_{d_i}^{d_f}F(d)\cos \theta dd$

This is the exact mathematical statement that the area under the $F$-$d$ curve equals the work done.

Why the Area Under the Force-Extension Graph Equals Work Done

For an elastic material (e.g., a spring) being stretched by a variable force $F(x)$:

• The work done over a tiny extension $\Delta x$ is $\Delta W=F\Delta x$.
• Total work: $W={\int }_0^{x_f}F(x)dx$ = area under the $F$-$x$ graph.
• For a spring obeying Hooke's Law ($F=kx$), this gives $W=\frac{1}{2}k{x}^2$, which is the elastic potential energy stored.

Summary Table