Conservation of Energy
Analyze kinetic, gravitational potential, elastic potential, mechanical, thermal, and total energy in systems with and without nonconservative forces.
A lesson of Physics, in Energy, Work, Power, and Momentum, about 10 minutes.
By the end of this lesson, you will be able to define potential energy, state the conservation of mechanical energy, solve problems using energy conservation, and account for non-conservative forces.
Potential Energy
Potential energy: stored energy due to position or configuration.
Gravitational PE: $U_g = mgh$ (relative to a chosen zero level)
Elastic PE (spring): $U_s = \frac{1}{2}kx^2$ where $k$ = spring constant (N/m) and $x$ = stretch/compression.
Conservation of Mechanical Energy
In the absence of friction and other non-conservative forces:
$$E_{\text{mech}} = KE + PE = \text{constant}$$
$$\frac{1}{2}mv_1^2 + U_1 = \frac{1}{2}mv_2^2 + U_2$$
Energy is simply transferred between kinetic and potential forms.
Example 1 — Ball Falling
A 0.5 kg ball is dropped from rest at height $h = 10$ m. Find speed just before impact.
$KE_i + PE_i = KE_f + PE_f$
$0 + mgh = \frac{1}{2}mv^2 + 0$
$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 10} = 14$ m/s
Example 2 — Roller Coaster
A roller coaster starts from rest at height $H = 40$ m. What is the speed at a height of $h = 15$ m?
$mgH = \frac{1}{2}mv^2 + mgh$
$v = \sqrt{2g(H-h)} = \sqrt{2 \times 9.8 \times 25} = 22.1$ m/s
Non-Conservative Forces
Non-conservative forces (friction, air resistance) convert mechanical energy to heat.
$$E_{\text{mech, final}} = E_{\text{mech, initial}} - W_{\text{friction}}$$
or: $KE_i + PE_i = KE_f + PE_f + \text{Energy lost to heat}$
Example 3 — Energy Lost to Friction
A 3 kg block slides 5 m down a 30° incline with $\mu_k = 0.15$, starting from rest. Find final speed.
Height drop: $h = 5\sin 30° = 2.5$ m
Energy method: $mgh - W_f = \frac{1}{2}mv^2$
$W_f = \mu_k(mg\cos 30°)(5) = 0.15 \times 3(9.8)(0.866)(5) = 19.1$ J
$3(9.8)(2.5) - 19.1 = \frac{1}{2}(3)v^2$
$73.5 - 19.1 = 1.5v^2 \implies v^2 = 36.3 \implies v = 6.0$ m/s
Spring Potential Energy
Hooke's Law: $F = -kx$ (restoring force; opposes displacement)
Energy stored in a spring: $U_s = \frac{1}{2}kx^2$
At max compression/stretch: all energy is PE; $v = 0$.
At equilibrium: all PE is converted to KE.
Summary
Gravitational PE: $mgh$; Spring PE: $\frac{1}{2}kx^2$.