Work Done by Non-Conservative Forces

The general energy equation

When both conservative and non-conservative forces act:

$W_{nc} = \Delta E_{mech} = \Delta KE + \Delta PE$

Work done by non-conservative forces equals the change in total mechanical energy.

Derivation

Work-energy theorem (all forces):

$W_{net} = \Delta KE$

Split into conservative and non-conservative:

$W_c + W_{nc} = \Delta KE$

Work done by conservative forces = $-\Delta PE$:

$-\Delta PE + W_{nc} = \Delta KE$

$W_{nc} = \Delta KE + \Delta PE = \Delta E_{mech}$

Friction — the most common non-conservative force

Friction always does negative work: $W_{friction} = -f_k \cdot d$

$-f_k d = \Delta KE + \Delta PE$

$E_{mech,f} = E_{mech,i} - f_k d$

Mechanical energy decreases by exactly the magnitude of work done by friction. This energy goes into heat.

Example: A block slides down a rough incline of height $h$ and length $l$, friction coefficient $\mu$:

$W_{nc} = -\mu mg\cos\theta \cdot l$

$\frac{1}{2}mv^2 - mgh = -\mu mgl\cos\theta$

$v = \sqrt{2gh - 2\mu gl\cos\theta} = \sqrt{2gl(\sin\theta - \mu\cos\theta)}$

Consistent with the kinematics result $v^2 = 2as$ with $a = g(\sin\theta - \mu\cos\theta)$.

When $W_{nc} > 0$

An external engine or motor does positive work on the system — mechanical energy increases. A car engine converts chemical energy (fuel) to mechanical energy.

Energy accounting

Non-conservative forces do not destroy energy — they convert mechanical energy into other forms:

• Friction → heat
• Air resistance → heat + sound
• Inelastic collision → heat + deformation energy
• Motor → mechanical energy (from electrical/chemical)

Total energy (all forms) is always conserved. $W_{nc} = \Delta E_{mech}$ just tracks what enters or leaves the mechanical energy "account".