Work, Energy, and Simple Machines · Lesson 2 of 13
When Is Work Done Equal to Zero?
“A wall can exhaust your muscles while stubbornly receiving zero mechanical work.”
• Identify situations in which work done by a force is zero. • Distinguish positive, negative and zero work from directions. • Explain why tiredness does not prove that work was done on an object. • Assign the correct sign to displacement relative to a chosen force. • Calculate signed work in familiar situations.
Stand in front of a rigid wall and push as hard as you can. Your muscles consume energy and you soon feel tired, but the wall does not move. Now walk across a room while holding a box at a constant height. Your arms apply an upward force to support the box, yet the box moves horizontally. These situations show that the presence of force and effort alone does not guarantee that a particular force has done mechanical work on an object.
The work done by a force is zero when the force is zero, when the object has no displacement, or when the displacement has no component in the direction of the force.
No Force or No Displacement
From W = F × s, work is zero if F is zero. Work is also zero if s is zero, no matter how large the applied force may be. A wall that remains fixed has no displacement, so your force does no work on the wall. Your tiredness has a different cause: muscles repeatedly contract and use the internal chemical energy of the body while maintaining the push.
Force Perpendicular to Displacement
When a person carries a box horizontally at constant height, the supporting force exerted by the person is upward while the box displacement is horizontal. These directions are perpendicular. There is no displacement in the direction of the supporting force, so that force does zero work on the box. This statement concerns the upward force only; other forces must be analysed separately.
Positive and Negative Work Done
A force does positive work when the displacement of the object is in the same direction as that force.
A person pushing a wheelchair forward applies a forward force while the wheelchair moves forward. The work done by the person on the wheelchair is positive. Positive work transfers energy to the object or system and can increase its kinetic energy.
A force does negative work when the displacement of the object is opposite to that force.
A goalkeeper stopping a ball applies a force opposite to the ball’s displacement. The goalkeeper therefore does negative work on the ball. Negative work removes mechanical energy from the object under consideration. When signs are used with W = F × s, choose the force direction as positive; displacement opposite to it is then negative.
| Relative Direction | Sign of Work | Example |
|---|---|---|
| Same direction | Positive | Pushing a wheelchair forward |
| Opposite directions | Negative | Brakes slowing a bicycle |
| Perpendicular directions | Zero | Supporting a box carried horizontally |
| No displacement | Zero | Pushing a rigid wall |
Problem
A girl lifts a dumbbell slowly and then lowers it slowly. Identify the sign of the work done by her force during each part.
- 1.Her hands exert an upward force on the dumbbell during both parts.
- 2.While lifting, the dumbbell displacement is upward, in the same direction as her force.
- 3.She therefore does positive work on the dumbbell while lifting it.
- 4.While lowering, the dumbbell displacement is downward but her supporting force remains upward.
- 5.She therefore does negative work on the dumbbell while lowering it.
Problem
A goalkeeper applies a force of 200 N while her hands and the ball move backward through 15 cm opposite to that force. Calculate the work done by the goalkeeper on the ball.
- 1.Choose the direction of the goalkeeper’s force as positive.
- 2.The displacement is opposite to the force, so s = -15 cm.
- 3.Convert centimetres to metres: s = -15 ÷ 100 = -0.15 m.
- 4.Use W = F × s.
- 5.W = 200 N × (-0.15 m) = -30 J.
- 6.The negative result agrees with the fact that the goalkeeper removes kinetic energy from the ball.
Problem
A box slides 5 m to the right while friction of 12 N acts to the left. Find the work done by friction.
- 1.Take the direction of friction as the reference direction.
- 2.The displacement is opposite to friction, so s = -5 m.
- 3.Use W = F × s = 12 N × (-5 m).
- 4.The work done by friction is -60 J.
- 5.The negative work shows that friction reduces the box’s mechanical energy.
Two Objects Can Do Different Work on Each Other
During an interaction, two objects apply equal and opposite forces to each other. Their displacements need not point the same way relative to those forces. A moving ball can do positive work on a goalkeeper’s hands as it pushes them backward, while the goalkeeper simultaneously does negative work on the ball. The pair of forces is equal and opposite, but the work statements refer to different objects.
Analysing Each Force Separately
An object may experience several forces at once, and each force can do a different kind of work. A box sliding across a rough floor may receive positive work from a forward pull, negative work from friction, and zero work from its weight and the upward support of the floor because those vertical forces are perpendicular to horizontal displacement. The overall change in the box’s motion depends on the combined work, but every individual work statement must still be tied to one force.
Problem
A box moves 4 m horizontally. A person pulls forward with 30 N while friction acts backward with 10 N. The weight and supporting force are vertical. Find the work done by each horizontal force and identify the work done by each vertical force.
- 1.Work by the pull = 30 N × 4 m = 120 J because force and displacement agree.
- 2.Work by friction = 10 N × (-4 m) = -40 J because they are opposite.
- 3.Weight is perpendicular to horizontal displacement, so gravity does zero work.
- 4.The supporting force is also perpendicular to displacement, so it does zero work.
- 5.The combined work is 120 J - 40 J = 80 J, showing a net transfer of energy to the box.
Do not decide the sign from words such as lifting, falling or stopping alone. Draw or state the direction of the chosen force and the direction of displacement, then assign the sign.
Quiz
Which description best matches Zero Work?
Which description best matches Positive Work?
Which term matches this description: The work done by a force is zero when the force is zero, when the object has no displacement, or when the displacement has no component in the direction of the force.
Which term matches this description: A force does positive work when the displacement of the object is in the same direction as that force.
Which statement is a key takeaway from this lesson?
Practice Problems
- State whether the work done by the supporting force is positive, negative or zero when a tray is carried horizontally.
- A force of 40 N acts opposite to a displacement of 3 m. Calculate its work.
- Identify the sign of work done by gravity on a ball during its upward and downward motion.
- A weightlifter holds a barbell steady. Explain the work done on the barbell by the weightlifter during this interval.
Key Takeaways
• A force does no work if there is no displacement in its direction. • Work is positive when force and displacement agree in direction. • Work is negative when they are opposite. • A perpendicular force does zero work. • Feeling tired does not by itself show that work was done on the external object.