Work, Energy, and Simple Machines · Lesson 1 of 13
Work Done by a Constant Force
“Work gets a strict definition, so carrying three bags finally becomes measurable bragging.”
• Explain the scientific meaning of work. • Relate work to force and displacement in the direction of force. • Calculate work in joules using consistent units. • Identify the force and the object involved in a statement about work. • Interpret work as area under a force-displacement graph.
Imagine a five-kilogram bag of wheat resting on the floor. You lift it slowly onto a platform one metre above the floor. Your upward force produces an upward displacement, so something measurable has occurred. If you lift three identical bags to the same platform, the task demands three times as much work. If instead you lift one bag to a platform three metres high, the same force acts through three times the distance, and once again the work becomes three times as large. These comparisons reveal the two quantities hidden inside the scientific idea of work: force and displacement.
The everyday use of the word work is broad. Reading, thinking and holding a difficult pose may all feel like work. In mechanics, however, the word has a precise meaning. We first identify a force acting on an object and then ask whether the object is displaced in the direction of that force. A description is incomplete unless it identifies both the force or agency doing the work and the object on which the work is done.
The work done on an object by a constant force is the product of the force and the displacement of the object in the direction of that force.
| Symbol | Meaning | SI Unit |
|---|---|---|
| W | Work done by the stated force | joule (J) |
| F | Magnitude of the constant force | newton (N) |
| s | Displacement in the direction of the force | metre (m) |
How Force and Displacement Affect Work
Suppose a machine raises one bag slowly through one metre. Raising three identical bags together requires three times the upward force over the same displacement, so the work is tripled. Raising one bag through three metres uses the original force over three times the displacement, so the work is also tripled. Work therefore grows in direct proportion to force when displacement is fixed, and in direct proportion to displacement when force is fixed.
Only the displacement in the direction of the chosen force is used. In this first situation, force and displacement point in the same direction, so their magnitudes can be multiplied directly. Later we will distinguish cases in which the directions are opposite or perpendicular.
The Joule
One joule is the work done when a constant force of one newton displaces an object by one metre in the direction of the force.
Because one newton is equal to one kilogram metre per second squared, the joule can also be written entirely in base units.
Problem
A constant horizontal force of 35 N moves a cart 4 m in the direction of the force. Find the work done by this force.
- 1.Identify the agency and object: the stated horizontal force does work on the cart.
- 2.Write the given values in SI units: F = 35 N and s = 4 m.
- 3.The displacement is in the direction of the force, so use W = F × s.
- 4.Substitute: W = 35 N × 4 m.
- 5.Therefore, W = 140 J.
Problem
A person slowly lifts one 50 N bag through 2 m. How much work is done in lifting four identical bags one after another through the same height?
- 1.Work on one bag = force × vertical displacement.
- 2.W = 50 N × 2 m = 100 J for one bag.
- 3.Each of the four bags requires the same amount of work.
- 4.Total work = 4 × 100 J.
- 5.Therefore, the person does 400 J of work on the four bags.
Work from a Force-Displacement Graph
A force-displacement graph places force on the vertical axis and displacement in the direction of the force on the horizontal axis. For a constant force, the graph is a horizontal line. The region below that line forms a rectangle whose height is the force and whose width is the displacement. Its area is therefore force multiplied by displacement, exactly the expression for work.
Problem
A force-displacement graph shows a constant force of 10 N from zero displacement to 1 m. Find the work done.
- 1.The region under the graph is a rectangle.
- 2.Rectangle height = 10 N and width = 1 m.
- 3.Area = height × width = 10 N × 1 m.
- 4.The work done is 10 J.
When the force changes with displacement, the graph is no longer a single horizontal line. The total work is still obtained from the area under the force-displacement curve between the initial and final positions. The region may be divided into rectangles, triangles or other manageable shapes. This graphical method allows changing forces to be handled without pretending that one force value acts throughout.
Problem
A force increases uniformly from 0 N to 12 N while an object moves through 5 m. The force-displacement graph is a triangle. Find the work done.
- 1.The area under the graph represents work.
- 2.The triangular region has base 5 m and perpendicular height 12 N.
- 3.Use the triangle area: W = one-half × base × height.
- 4.W = one-half × 5 m × 12 N = 30 N m.
- 5.Since one newton metre is one joule, the work done is 30 J.
Do not say only that “work is done.” State which force does work and on which object. Different forces can act on the same object and may do different amounts of work.
Quiz
Which description best matches Work Done by a Constant Force?
Which description best matches Joule?
Which term matches this description: The work done on an object by a constant force is the product of the force and the displacement of the object in the direction of that force.
Which term matches this description: One joule is the work done when a constant force of one newton displaces an object by one metre in the direction of the force.
Which statement is a key takeaway from this lesson?
Practice Problems
- A 25 N force moves an object 6 m in its direction. Calculate the work done.
- A crane raises a 400 N load vertically through 3 m. Calculate the work done by the crane on the load.
- A constant 8 N force acts through 2.5 m. Represent the situation on a force-displacement graph and find its area.
- Explain why doubling force while halving displacement leaves the work unchanged.
Key Takeaways
• Work requires a stated force and a displacement produced in relation to that force. • For a constant force acting along the displacement, W = F × s. • The SI unit of work is the joule. • Work is represented by area under a force-displacement graph. • Larger force or larger displacement produces proportionally more work when the other quantity is unchanged.
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When Is Work Done Equal to Zero?