Work and the Work-Energy Theorem

By the end of this lesson, you will be able to calculate work done by a constant force, apply the work-energy theorem, and use the dot product for work with vectors at angles.

Definition of Work

$$W = F\cos\theta \cdot d = \vec{F}\cdot\vec{d}$$

• $F$ = magnitude of force (N)
• $d$ = magnitude of displacement (m)
• $\theta$ = angle between force and displacement
• Work is in Joules (J = N·m)

Work is positive when force component is in the same direction as displacement.

Work is negative when force component opposes displacement.

Work is zero when force is perpendicular to displacement (e.g., normal force on horizontal motion, gravity on horizontal motion).

The Work-Energy Theorem

$$W_{\text{net}} = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$$

The net work done on an object equals the change in its kinetic energy.

This connects force and motion through energy — a powerful alternative to Newton's laws for many problems.

Kinetic Energy

$$KE = \frac{1}{2}mv^2$$

Scalar; always positive (or zero); units: J.

Example 1 — Work Done by Multiple Forces

A 5 kg box is pushed 3 m along a frictionless floor with 20 N horizontal force. Find (a) work done, (b) final speed if starting from rest.

(a) $W = 20 \times 3 = 60$ J

(b) Work-energy theorem: $W_{\text{net}} = \Delta KE$

$60 = \frac{1}{2}(5)v^2 \implies v = \sqrt{24} = 4.9$ m/s

Example 2 — Work at an Angle

A 30 N force at 45° pulls a 10 kg box 5 m. Normal force and weight do no work. Find final speed from rest.

$W = 30\cos 45° \times 5 = 30(0.707)(5) = 106$ J

$106 = \frac{1}{2}(10)v^2 \implies v = 4.6$ m/s

Example 3 — Friction Does Negative Work

A 2 kg box slides 4 m with friction ($\mu_k = 0.3$), starting at 6 m/s. Final speed?

$W_f = -f_k \times d = -0.3(2)(9.8)(4) = -23.5$ J

$W_{\text{net}} = -23.5 = \frac{1}{2}(2)v^2 - \frac{1}{2}(2)(36)$

$-23.5 = v^2 - 36 \implies v^2 = 12.5 \implies v = 3.5$ m/s

Summary

Work: $W = Fd\cos\theta$; positive (along motion), negative (against), zero (perpendicular).

Work-energy theorem: $W_{\text{net}} = \Delta KE$.

Kinetic energy: $KE = \frac{1}{2}mv^2$.

Power of work-energy theorem: avoids tracking force throughout motion when only start and end matter.