GRE Physics Companion: Work Done by a Force

The fastest way to solve most work questions is to begin with the dot product

(1)

The angle is always the angle between the force and the displacement. Do not automatically use the angle drawn relative to the horizontal or vertical unless that is also the angle between force and displacement.

1 Fast triage

High-value shortcuts are

(2)

(3)

and

(4)

2 Cosine controls the sign

This makes angle questions extremely fast. If a force becomes more nearly parallel to the displacement while F and Δr stay fixed, its work increases. If it becomes more nearly perpendicular, its work approaches zero.

3 Component shortcut

When vectors are given in components, skip angle reconstruction and use

(5)

This avoids unnecessary inverse trigonometric calculations.

4 Worked GRE example 1: rank forces by work

Three forces of equal magnitude F act through the same displacement d. Their angles between force and displacement are 0^∘, 60^∘, and 120^∘. Rank their works from greatest to least.

Because

(6)

we have

(7)

Therefore

(8)

The negative result is smaller than zero, not merely smaller in magnitude.

5 Worked GRE example 2: component dot product

A force and displacement are

(9)

(10)

Then

(11)

Thus

(12)

No angle calculation is needed.

6 GRE practice questions

1. A constant force of magnitude F acts through displacement d. If the force is perpendicular to the displacement, what is the work?
2. A force of magnitude F acts at 60^∘ to a displacement d. Express the work in terms of F and d.
3. The angle between force and displacement changes from 60^∘ to 120^∘ while F and d remain fixed. How does the work change?
4. A particle moves in a circle under a force that is always directed toward the center. What work does that force do over one revolution?
5. A constant 5 N force acts in the +x direction while a particle moves 3 m in the −x direction. Find the work.
6. A constant force acts on a particle. The magnitude of the force doubles and the displacement magnitude triples, while the angle between force and displacement remains unchanged. By what factor does the work change?

7 Answers

1. Zero.
2. Fd∕2.
3. It changes from +Fd∕2 to −Fd∕2.
4. Zero, because the radial force is perpendicular to each tangential displacement.
5. −15 J.
6. Factor of 6.

8 Final checklist

• Work is a scalar.
• Use the angle between force and displacement.
• Parallel gives maximum positive work.
• Perpendicular gives zero work.
• Antiparallel gives maximum negative work.
• Use components directly when they are supplied.
• Separate work by one force from net work by all forces.
• Area under an F_x versus x graph is signed work.

References