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Calculate The Work Done by Graph Integral

Reviewed by Calculator Editorial Team

Calculating work done by a graph integral involves determining the area under a force vs. displacement curve. This method is essential in physics for analyzing variable forces and energy transfer. This guide explains the concept, provides a calculation formula, and includes an interactive calculator to compute work from force-displacement graphs.

What is Work Done by Graph Integral?

Work done by a variable force is calculated by finding the area under the force-displacement curve on a graph. This method is used when the force changes as the object moves, making a simple F × d calculation insufficient. The integral of force with respect to displacement gives the total work done.

In physics, work is defined as the product of force and displacement when the force is constant. However, when force varies with displacement, we use calculus to find the exact work done. The graph of force vs. displacement becomes a curve, and the area under this curve represents the work done.

Work done by a variable force is always positive when the force and displacement are in the same direction. If they are in opposite directions, the work is negative.

The Formula

The work done (W) by a variable force is given by the integral of force (F) with respect to displacement (d):

W = ∫ F(d) dd

Where:

  • W = Work done (in joules, J)
  • F(d) = Force as a function of displacement (in newtons, N)
  • d = Displacement (in meters, m)

For a graph, you can approximate the area under the curve using geometric shapes or numerical integration methods like the trapezoidal rule or Simpson's rule.

How to Calculate Work from a Force-Displacement Graph

  1. Plot the force (F) on the y-axis and displacement (d) on the x-axis.
  2. Identify the curve representing F(d).
  3. Divide the area under the curve into simple shapes (rectangles, trapezoids, triangles) or use numerical methods.
  4. Calculate the area of each shape and sum them to get the total work done.
  5. For exact calculation, use calculus to find the integral of F(d) with respect to d.

If the force-displacement graph is a straight line, you can use the simple formula W = ½ × F × d, where F is the average force.

Worked Example

Suppose a force F = 2d² N acts on an object as it moves from d = 0 to d = 3 m. Calculate the work done.

Using the integral formula:

W = ∫ (2d²) dd from 0 to 3

= [ (2/3)d³ ] from 0 to 3

= (2/3)(3³) - (2/3)(0³)

= (2/3)(27) - 0

= 18 J

The work done is 18 joules.

FAQ

What units are used for work in physics?
Work is measured in joules (J) in the International System of Units (SI).
Can work be negative?
Yes, work is negative when force and displacement are in opposite directions.
How do I calculate work for a non-linear force-displacement graph?
Use calculus to find the integral of force with respect to displacement or approximate the area under the curve using geometric shapes.
What if the force-displacement graph is not smooth?
Use numerical integration methods like the trapezoidal rule or Simpson's rule to estimate the area.