Work Done by a Force

Active learning builds physical intuition for work done by a force, letting students feel the difference between force, displacement, and their alignment. Through hands-on experiments and matching tasks, they connect abstract formulas to real motions and forces.

• Year 13
• Mathematics
• 4 activities
• 30 min–50 min

Learning Objectives

1. Calculate the work done by a constant force using the formula W = Fs cos θ.
2. Determine the work done by a variable force by integrating the force function with respect to displacement.
3. Analyze the effect of the angle between force and displacement on the total work done.
4. Construct calculations for work done against resistive forces like friction.
5. Explain the relationship between work done and energy transfer in mechanical systems.

Think-Pair-Share

⏱ 45 min·Small Groups

Pulley Experiment: Constant Force Work

Students attach weights to a pulley system and pull a mass horizontally over measured distances, recording force and displacement. They calculate work using W = F s and vary the angle by tilting the setup. Groups compare results and discuss cos θ effects.

Prepare & details

Explain the relationship between force, displacement, and work done.

Facilitation Tip: During the Pulley Experiment, remind students to zero the force sensor before each run and to measure the exact displacement from the pulley to the mass hanger.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 50 min·Pairs

Trolley Run: Work Against Friction

Release trolleys down inclines with varying surfaces; use motion sensors to log displacement and estimate friction forces. Students compute net work done and graph force vs. distance. Pairs verify calculations against energy loss.

Prepare & details

Analyze how the angle between force and displacement affects the work done.

Facilitation Tip: When running the Trolley Run, encourage multiple trials with different surface materials to show how friction changes the work values and energy loss.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 35 min·Whole Class

Graph Matching: Variable Forces

Provide printed force-displacement graphs; students match them to work values by shading areas and approximating integrals numerically. Extend to drawing their own curves for spring problems. Whole class shares and critiques methods.

Prepare & details

Construct a calculation for the work done by a force acting at an angle.

Facilitation Tip: In Graph Matching, ask students to sketch predicted force-displacement graphs before running the simulation to test their initial models.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 30 min·Small Groups

Card Sort: Angled Force Calculations

Distribute cards with scenarios, forces, angles, and displacements; students sort into work calculation sets. They solve matched sets and justify using vector diagrams. Rotate roles for peer teaching.

Prepare & details

Explain the relationship between force, displacement, and work done.

Facilitation Tip: For Card Sort, circulate to listen for groups that confuse cos θ with sin θ and prompt them to sketch the vectors before choosing equations.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Start with the Pulley Experiment to anchor the scalar nature of work and the role of θ. Use Card Sort to confront angle misconceptions early, before tackling variable forces. Emphasize that work is a scalar product, so direction matters but not the sign of individual components. Avoid rushing to the formula; insist on vector sketches and clear labels first. Research shows that students who draw free-body diagrams and displacement vectors before calculating retain concepts longer.

What to Expect

Students will confidently use W = F s cos θ for constant forces and W = ∫ F dx for variable forces. They will explain when work is zero and distinguish work by applied forces from work against resistance, supported by measured data and calculations.

Watch Out for These Misconceptions

Common Misconception

During Pulley Experiment, watch for students who multiply force by displacement without considering the angle θ.

What to Teach Instead

Have them measure θ with a protractor taped to the table and recalculate using W = F s cos θ, comparing predicted and measured values until the discrepancy is resolved.

Common Misconception

During Graph Matching, watch for students who use F_avg = (F_max + F_min)/2 instead of integrating.

What to Teach Instead

Ask them to shade the area under the curve with graph paper and compute the integral numerically, then compare to the shortcut result to reveal the error visually.

Common Misconception

During Trolley Run, watch for students who call the work done against friction the same as the work done by the applied force.

What to Teach Instead

Require them to write separate energy balances for the applied force and friction, using signs and vector directions to clarify why the two works have opposite signs.

Assessment Ideas

Quick Check

After Pulley Experiment and Card Sort, present a diagram of a box pulled at an angle across a floor. Ask students to calculate work done by the pulling force and work done against friction, explaining each step and referencing their experimental measurements of angle and friction.

Exit Ticket

During Trolley Run, give students a scenario: ‘A 5 kg object is lifted vertically by 2 meters.’ Ask them to: 1. Calculate work done against gravity. 2. State work done by the upward lifting force if it just overcomes gravity. 3. Explain why the angle matters in other scenarios using vector sketches.

Discussion Prompt

After Graph Matching and Pulley Experiment, pose the question: ‘If a force does zero work, does that mean the force is zero or the displacement is zero?’ Facilitate a class discussion where students justify answers using their pulley data for perpendicular forces and their graph integrals for zero displacement intervals.

Extensions & Scaffolding

• Challenge: Ask students to design a variable-force ramp for the trolley that keeps the total work constant while changing shape.
• Scaffolding: Provide pre-labeled graphs for the Graph Matching activity with key points marked to reduce cognitive load.
• Deeper exploration: Have students derive the work–energy theorem from their trolley data by calculating net work and change in kinetic energy.

Key Vocabulary

Work Done	The energy transferred when a force causes an object to move over a distance. It is calculated as the product of the force component in the direction of motion and the displacement.
Displacement	The change in position of an object, measured as a vector from its initial to its final position. It is distinct from distance traveled.
Force Vector	A representation of a force that includes both magnitude (strength) and direction. This is crucial for calculating work when force is not parallel to displacement.
Integration	A mathematical process used to find the area under a curve, which in this context represents the accumulation of work done by a variable force over a displacement.
Resistive Force	A force that opposes motion, such as friction or air resistance. Work done against these forces increases the energy dissipated, often as heat.