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.
Understanding Calculus: Work as an Integral
The integral is needed because the force can change from one position to the next. Imagine pulling a rubber band. At first it is easy to stretch, but each extra bit of stretch needs more pull.
Divide the motion into many very short distances. Over one tiny distance, the force changes so little that it can be treated as constant. The work for that piece is the force near that location multiplied by the tiny distance.
Adding every piece gives a close estimate. Making the pieces thinner makes the estimate approach the exact total.
The sign of work carries important physical meaning. A force pointing in the same direction as the motion does positive work. It transfers energy into the moving object.
A force pointing opposite the motion does negative work. It removes energy from the object. Friction usually does negative work when a box slides across a floor.
Gravity does negative work while an object is lifted, since gravity pulls downward while the object moves upward. On a force versus position graph, regions above the horizontal axis count positively and regions below it count negatively. Students should not add all visible areas as positive amounts unless the problem specifically asks for total area rather than net work.
Units provide a useful error check. Force is measured in newtons and distance is measured in metres. Their product is a joule, the unit of work and energy.
A force graph therefore has newtons on its vertical scale and metres on its horizontal scale. Its area has units of newton metres, which is one joule. This check matters in applications with changing dimensions.
For example, a spring constant has units of newtons per metre. Multiplying that constant by a stretch gives a force. Integrating that force over a distance then gives energy in joules.
Spring questions require careful attention to what the position variable means. It usually measures displacement from the spring's natural length, not the spring's total length. The force exerted by the spring points back toward that natural length.
The force applied by a person who stretches the spring slowly points the other way. These forces have equal size during slow motion but opposite signs. A problem asking for work done on the spring usually wants positive energy stored in it.
A problem asking for work done by the spring can have the opposite sign. The work from zero stretch to a chosen stretch is related to the triangular region under a straight force graph.
In fluid pumping and lifting problems, the moving object is not a single point. Split the fluid into thin horizontal layers. Find the weight of one layer from its volume, density, and gravity.
Then determine how far that particular layer must rise. Layers near the top may travel only a short distance, while lower layers travel farther. Their work contributions are different, so one distance cannot represent the whole tank.
The same idea appears when lifting a hanging cable, draining a container, or pulling a chain onto a platform. A clear diagram, a defined position variable, correct limits, and a distance expression for each slice are usually more important than performing the final integration.
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.