Energy in Simple Harmonic Motion — AP Physics 1
1. Core Principles of Energy in SHM ★★☆☆☆ ⏱ 3 min
Energy analysis of SHM connects two core units of AP Physics 1, accounting for ~2% of total exam score, and appears in both multiple-choice and free-response sections. For undamped SHM, where no friction or non-conservative forces do work, total mechanical energy is always conserved.
Energy continuously converts between kinetic energy (associated with motion of the oscillating mass) and potential energy (elastic for mass-spring systems, gravitational for pendulums) associated with displacement from equilibrium.
2. Conservation of Energy for SHM Calculations ★★☆☆☆ ⏱ 4 min
The core relationship for all undamped SHM is:
E_{total} = KE + PE = \text{constant}
We find total energy by evaluating it at maximum displacement, where $KE = 0$, so $E_{total} = PE_{max}$. For a horizontal mass-spring system, elastic potential energy is $PE_s = \frac{1}{2}kx^2$, so at $x = A$:
E_{total} = \frac{1}{2}kA^2
For any displacement $x$, the full energy balance is:
\frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2
Cancel the $\frac{1}{2}$ factor and rearrange to solve for speed $v$ at any $x$:
v = \sqrt{\frac{k}{m}(A^2 - x^2)}
For small-amplitude simple pendulums, set $PE = 0$ at the equilibrium (lowest) point. Total energy at maximum angle $\theta_{max}$ is $E_{total} = mgL(1-\cos\theta_{max})$, following the same energy conservation logic.
Exam tip: Always set $PE = 0$ at the equilibrium position for SHM energy problems to simplify math and avoid sign errors from extra offset terms.
3. Energy Graphs for SHM ★★★☆☆ ⏱ 3 min
AP Physics 1 regularly tests interpretation of energy graphs for SHM. For a mass-spring system, $PE(x) = \frac{1}{2}kx^2$ and $KE(x) = \frac{1}{2}k(A^2 - x^2)$. Both are quadratic functions of position, so their energy vs position graphs are parabolas, not trigonometric curves.
Potential energy $PE(x)$ is an upward-opening parabola with minimum $PE = 0$ at $x=0$ (equilibrium) and maximum PE at $x = \pm A$. Kinetic energy $KE(x)$ is a downward-opening parabola with maximum KE at $x=0$ and $KE=0$ at $x = \pm A$. Total energy is a horizontal straight line because it is constant for undamped motion.
The curves intersect at two points, ~71% of the amplitude from equilibrium on either side.
Exam tip: Do not mix up graph shapes: KE and PE are parabolas for energy vs position, and squared sine/cosine curves for energy vs time, which oscillate twice per full period.
4. Parameter Dependence of SHM Energy ★★★☆☆ ⏱ 4 min
A common AP exam question asks how changing system parameters (amplitude, mass, spring constant, pendulum length) changes total energy or maximum speed.
For mass-spring systems, total energy $E_{total} = \frac{1}{2}kA^2$, so $E_{total}$ depends only on $k$ and $A$, not on mass $m$. Changing mass at fixed amplitude does not change total energy, but it does change maximum speed: since $E_{total} = \frac{1}{2}mv_{max}^2$, we get $v_{max} = A\sqrt{\frac{k}{m}}$, so increasing mass decreases $v_{max}$ even when total energy is constant.
For small-amplitude pendulums, total energy $E_{total} = mgL(1-\cos\theta_{max})$, so increasing mass, amplitude angle, or pendulum length all increase total energy. For lightly damped SHM, the only damped case covered in AP Physics 1, friction does non-conservative work so total energy and amplitude decrease over time.
The maximum speed is reduced by a factor of $\frac{1}{2}$.
Exam tip: Always separate total energy and maximum speed when answering parameter change questions: total energy does not depend on mass for mass-spring SHM at fixed amplitude, but maximum speed does.
Common Pitfalls
Why: Students confuse total energy with period: period increases with mass, so they incorrectly assume energy also increases.
Why: Students forget PE depends on displacement squared, so negative positions have the same PE as equal-magnitude positive positions.
Why: Students remember position and velocity are trigonometric functions of time, so incorrectly assume energy is also trigonometric in position.
Why: Students think gravity changes the energy equation, but forget that shifting the equilibrium position absorbs the gravitational term.
Why: Students rely on memorized formulas instead of deriving from energy, leading to wrong amplitude calculations.