Energy in SHM

Energy exchange in oscillation

In ideal simple harmonic motion, total mechanical energy remains constant while energy shifts between kinetic and potential forms. At some moments the system is mostly moving; at others it is mostly stored. This repeated exchange is what sustains oscillation.

For a spring oscillator, total energy is

E=K+U=rac{1}{2}mv^2+rac{1}{2}kx^2

There is no damping in the ideal model, so $E$ is constant.

Energy at turning points

At the turning points, displacement has maximum magnitude:

$x=pm A$

The velocity is zero at those instants, so kinetic energy is zero. All energy is potential:

E=rac{1}{2}kA^2

This equation connects total energy directly to amplitude. Doubling amplitude quadruples total energy because energy depends on $A^2$.

Energy at equilibrium

At equilibrium,

$x=0$

so spring potential energy is zero. The energy is entirely kinetic:

E=rac{1}{2}mv_{max}^2

Equating this with rac{1}{2}kA^2 gives

v_{max}=Asqrt{rac{k}{m}}=Aomega

The oscillator moves fastest at equilibrium, not at the endpoints.

Speed as a function of position

Energy conservation can find speed at any displacement. Start with

rac{1}{2}mv^2+rac{1}{2}kx^2=rac{1}{2}kA^2

Solve for $v$:

$v=pmomegasqrt{A^2-x^2}$

The plus or minus sign depends on direction of motion. This equation shows speed is maximum at $x=0$ and zero at $x=pm A$.

Time dependence of energy

For

$x(t)=Acos(omega t+phi)$

velocity is

$v(t)=-Aomegasin(omega t+phi)$

The potential energy is

U(t)=rac{1}{2}kA^2cos^2(omega t+phi)

The kinetic energy is

K(t)=rac{1}{2}mA^2omega^2sin^2(omega t+phi)

Since $k=momega^2$,

K(t)=rac{1}{2}kA^2sin^2(omega t+phi)

The sum is constant because

$sin^2 heta+cos^2 heta=1$

Energy oscillates twice as fast

Although displacement repeats every period $T$, kinetic and potential energy repeat every $T/2$. This is because $sin^2$ and $cos^2$ repeat twice during each full sine or cosine cycle.

During one complete oscillation, energy shifts from potential to kinetic to potential to kinetic and back again.

Pendulum energy

For a small-angle pendulum, energy can be approximated as

E=rac{1}{2}mL^2dot{ heta}^2+rac{1}{2}mgL heta^2

This has the same mathematical form as the spring oscillator. The angular displacement $heta$ plays the role of position, and $mL^2$ plays the role of rotational inertia.

Phase relationships

Position and velocity are out of phase by $pi/2$ radians. When displacement is maximum, velocity is zero. When displacement is zero, speed is maximum. Acceleration is opposite in phase to displacement because

$a=-omega^2x$

These phase relationships are easier to see through energy and graphs.

The big idea

Energy in ideal SHM is conserved but continually exchanged between kinetic and potential forms. The amplitude determines total energy, and energy methods reveal speed at any position without solving time-dependent equations. This energy view becomes essential for waves, resonance, damping, and normal modes.