Class 11 > Unit # 05: Work, Energy & Power > Work done by Variable Force

Work Done by Variable Force — Unit 5 Work, Power & Energy

Welcome to Talha's Physics Academy. In this lecture, we explore how to calculate the work done by a force that changes in magnitude or direction over a displacement, moving beyond standard constant-force formulas.

Video Lecture

1. Introduction: Constant vs. Variable Force

The standard formula for work assumes a constant force acting in the same direction as the displacement:

$$W = F \cdot s$$

However, when a force is variable (its magnitude changes continuously throughout the motion), this direct formula cannot be applied directly. To solve this, we rely on graphical partitioning and calculus.

2. Graphical Method & Partitioning

To find the total work done by a variable force:

• The total displacement is divided into many small, individual intervals (\(\Delta x_1, \Delta x_2, \dots, \Delta x_n\)).
• Within each small interval, the force is assumed to remain approximately constant (\(f_1, f_2, \dots, f_n\)).
• The work done over each small strip is calculated individually:
  $$w_1 = f_1 \Delta x_1, \quad w_2 = f_2 \Delta x_2, \quad \dots \quad w_n = f_n \Delta x_n$$

3. Total Work via Summation

The total work (\(W\)) is the sum of all individual work increments across the entire displacement:

$$W = \sum_{i=1}^{n} w_i = \sum_{i=1}^{n} f_i \Delta x_i$$

4. Using Calculus for Exact Results

To get an exact value, we take the limit as the number of intervals approaches infinity (\(n \to \infty\)) and the width of each interval approaches zero (\(\Delta x \to 0\)). This summation transitions into a definite integral:

$$W = \int_{x_1}^{x_2} F \, dx$$

This integral represents the area under the force-displacement curve, serving as the definitive method for calculating work done by a variable force in physics.