Notifications

Log in to see your notifications.

Work done by forces is taught in the note for Applications of Vectors. Read it below and look for the section on this topic.

Applications of Vectors

14 min read

Motion of a particle

If the position of a particle is the vector r(t), then its velocity is v = dr/dt, its acceleration is a = dv/dt = d²r/dt², and its speed is |v|.

Worked exampleA moving particle

A particle has position r = (t², 3t, 5) m. Find its velocity, acceleration and speed at t = 2 s.

  1. Differentiate.
  2. Acceleration.
  3. Speed.

Answer: v = (4, 3, 0) m/s, a = (2, 0, 0) m/s², speed 5 m/s.

The speed √(4t² + 9) rises as the particle accelerates.

Work done by a force

The work done by a constant force F when its point of application moves through a displacement d is the dot product:

Only the component of the force along the displacement does work.
Worked exampleWork

A force F = (3, 4, 0) N moves a body through d = (2, 1, 0) m. Find the work.

  1. Dot product.

Answer: The work done is 10 J.

Projections

The scalar projection of a on b (the component of a along b) is a · b̂ = (a · b) ÷ |b|. The vector projection is [(a · b) ÷ |b|²] b. Projections are used to find the distance from a point to a line and the components of a force along a slope.

Resolving a vector into componentsA force of 10 newtons acts at 30 degrees above the horizontal. It can be replaced by a horizontal component of 10 cosine 30, which is 8.7 newtons, and a vertical component of 10 sine 30, which is 5 newtons. The two components together have the same effect as the original force.θ = 30°F = 10 NF cos θ = 8.7 NF sin θ = 5 N
Try itWork done by a force at an angle

Choose the force, the displacement and the angle between them.

Check yourself Velocity is the ___ of position with respect to time.
Check yourself The work done when the force is perpendicular to the displacement is
Check yourself The scalar projection of (3, 4) on (1, 0) is

Key terms

Velocity
The rate of change of position.
Acceleration
The rate of change of velocity.
Work
The dot product of force and displacement.
Projection
The component of one vector along another.

Summary

  • v = dr/dt and a = dv/dt.
  • Work W = F · d.
  • The scalar projection of a on b is a · b ÷ |b|.
  • Vector methods work in two and three dimensions.
Displacement, velocity and acceleration of a particle Projections