Work measures how much energy is transferred when a force moves an object through a distance. When the force is constant and along the direction of motion, work is simply W = Fd. Many real forces are not constant, such as a stretching spring or the force needed to pump water upward.
Calculus lets us add many tiny pieces of work to find the total work accurately.
For motion along a line, each small piece of work is approximately dW = F(x) dx, where F(x) is the force at position x. Adding all the pieces from x = a to x = b gives W = ∫_a^b F(x) dx, which is the signed area under the force versus position graph. In springs, Hooke's law gives F(x) = kx, so the work to stretch or compress a spring is an area under a line.
In pumping fluid problems, the integral adds the work needed to lift thin slices of liquid different distances.
Key Facts
- For constant force in the direction of motion, W = Fd.
- For variable force along a line, W = ∫_a^b F(x) dx.
- A small piece of work is dW = F(x) dx.
- On a force versus position graph, work equals the signed area under the curve.
- Hooke's law for an ideal spring is F(x) = kx.
- Work to stretch a spring from x = a to x = b is W = ∫_a^b kx dx = 1/2 k(b^2 - a^2).
Vocabulary
- Work
- Work is the energy transferred by a force acting through a displacement.
- Variable force
- A variable force is a force whose magnitude or direction changes with position, time, or another quantity.
- Force-position graph
- A force-position graph plots force F(x) on the vertical axis and position x on the horizontal axis.
- Hooke's law
- Hooke's law states that the restoring force of an ideal spring is proportional to its displacement from equilibrium.
- Riemann sum
- A Riemann sum approximates a total by adding many small products, such as force times a small displacement.
Common Mistakes to Avoid
- Using W = Fd for a changing force. This is wrong because Fd only works directly when the force is constant over the displacement.
- Forgetting the limits of integration. This is wrong because ∫ F(x) dx is only a general antiderivative until the starting and ending positions are specified.
- Using spring displacement from the wrong zero point. This is wrong because Hooke's law uses x measured from the spring's natural length, not from an arbitrary position.
- Treating pumping fluid as if every slice lifts the same distance. This is wrong because different layers of fluid may travel different distances to reach the outlet.
Practice Questions
- 1 A force varies with position as F(x) = 3x^2 newtons. Find the work done from x = 0 m to x = 4 m.
- 2 A spring has spring constant k = 200 N/m. How much work is required to stretch it from its natural length to 0.30 m?
- 3 A force-position graph is below the x-axis for part of an interval and above it for another part. Explain how signed work differs from total area on the graph.