Potential Energy and Absolute Potential Energy: MDCAT Physics notes
Potential Energy and Absolute Potential Energy notes for MDCAT: mgh, spring PE, conservation of mechanical energy, −GMm/R and conservative forces.
Gravitational potential energy
Potential energy is energy stored due to position or configuration. Near the Earth's surface, lifting mass $m$ through height $h$ against gravity stores
$$P.E. = mgh$$
- PE is measured from a chosen reference level. Below the reference, $h$ is negative: a diver at depth $h$ below sea level (reference) has $P.E. = -mgh$.
- When a body is raised, gravity acts opposite to the displacement, so the work done by gravity is negative and PE increases. When it falls, gravity does positive work and PE decreases.
- Work done by gravity is independent of the path; only the height difference matters.
Elastic (spring) potential energy
A spring stretched or compressed by $x$ stores $P.E. = \tfrac12 kx^2$. Since $P.E. \propto x^2$, stretching $n$ times as far stores $n^2$ times the energy. PE increases while compressing or stretching and decreases when a stretched spring is released.
Conservation of mechanical energy
If only conservative forces (like gravity) act, $K.E. + P.E. = \text{constant}$. Loss of PE = gain of KE:
$$mgh = \tfrac12 mv^2 \quad\Rightarrow\quad v = \sqrt{2gh}$$
| Situation | Working | Answer |
|---|---|---|
| Fall of 10 m | $\sqrt{2(9.8)(10)}$ | $14\ \text{m s}^{-1}$ |
| Water falling 19.6 m | $\sqrt{2(9.8)(19.6)}$ | $19.6\ \text{m s}^{-1}$ |
| Pole vault rise of 5 m | $\sqrt{2(10)(5)}$ | $10\ \text{m s}^{-1}$ |
| Pendulum released 0.5 m high | $\sqrt{2(10)(0.5)}$ | $\sqrt{10} \approx 3.16\ \text{m s}^{-1}$ |
Mass cancels, so these speeds do not depend on mass. A dolphin leaping from water converts KE to PE; at the top its energy is mostly potential.
Total mechanical energy includes both: a 10 kg body at 2 m moving at $2\ \text{m s}^{-1}$ has $mgh + \tfrac12 mv^2 = 196 + 20 = 216\ \text{J}$. A fish of 0.25 kg lifted 1.8 m to reach $1.1\ \text{m s}^{-1}$ gains $4.41 + 0.15 \approx 4.6\ \text{J}$. If all PE turns into heat on impact, heat = $mgh$ (5 kg from 30 m: 1470 J).
Conservative and non-conservative forces
A force is conservative if its work is path-independent and zero around a closed path: gravitational, electric and elastic spring forces. Friction and air resistance are non-conservative; their work turns mechanical energy into heat.
Absolute potential energy
$mgh$ is valid only when $g$ is constant. Far from the Earth we take the zero of PE at infinity. The absolute gravitational PE of mass $m$ at distance $r$ from the Earth's centre is the work done by gravity in moving the body from that point to infinity (the position of zero potential):
$$U_g = -\frac{GM_em}{r}$$
At the surface, $U_g = -GM_em/R$. The negative sign shows the field is attractive: work must be done on the body to take it to infinity, so it is bound. Escape velocity $v_{esc} = \sqrt{2GM_e/R}$ depends on the mass and radius of the Earth, not on the body's mass or launch angle.
Key formulas
- $P.E. = mgh$; spring $P.E. = \tfrac12 kx^2$
- $v = \sqrt{2gh}$
- $U_g = -GMm/r$; $v_{esc} = \sqrt{2GM/R} = \sqrt{2gR}$
Common MDCAT traps
- Work done by gravity on a rising body is negative, not zero or path-dependent.
- Below the reference level, PE is negative ($-mgh$).
- Stretch $n$ times: spring PE becomes $n^2$ times.
- The minus sign in $-GMm/r$ means attraction, not repulsion.
- Friction is the non-conservative force in the usual option lists.
Quick revision
- $K.E. + P.E.$ = constant without friction.
- Zero of absolute PE is at infinity.
- Escape velocity is independent of the body's mass.
- Speed after falling $h$: $\sqrt{2gh}$.
- Releasing a stretched spring lowers its PE.