SHM Total Energy

E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2

Total mechanical energy in SHM is constant.

Variables

Total energy

Constant total energy

In SI units: kg·m²/s²

Spring constant

Spring stiffness

In SI units: kg/s²

Allowed values: greater than 0

N/m

Amplitude

Maximum displacement

Allowed values: 0 or more

Mass

Oscillating mass

Allowed values: 0 or more

Angular frequency

ω = √(k/m)

Allowed values: greater than 0

rad/s

Derivation

E = KE + PE = ½mv² + ½kx² = ½kA² (at max displacement).

Your notes

0/5000

Check value

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k = 2 N/m (Spring constant)
A = 3 m (Amplitude)

Result: E = 9 J

Other waves formulas with the same quantities

SHM Displacement (Simple Harmonic Motion) shares Amplitude, Angular frequency
SHM Velocity (Simple Harmonic Motion) shares Amplitude, Angular frequency
SHM Maximum Velocity (Simple Harmonic Motion) shares Amplitude, Angular frequency
Spring Oscillation Period (Simple Harmonic Motion) shares Mass, Spring constant
SHM Acceleration (Simple Harmonic Motion) shares Angular frequency
Physical Pendulum Period (Simple Harmonic Motion) shares Mass

Where this formula is listed

Waves formulas, under Simple Harmonic Motion, alongside Pendulum Period, Torsional Oscillation Period, Damped Oscillation, Quality Factor (Q Factor) and Resonance Amplitude.
In the waves list it comes after Pendulum Period and before Physical Pendulum Period.
Waves formula sheet, one of 37 formulas printed with symbol keys and SI units.
The full formula index, grouped by category and topic.

Common questions

What is the SHM Total Energy formula?

SHM Total Energy is written E = (1/2)kA² = (1/2)mω²A². Total mechanical energy in SHM is constant. It belongs to Waves, under Simple Harmonic Motion.

What are the units in E = (1/2)kA² = (1/2)mω²A²?

In SI units, E is total energy, measured in kg·m²/s²; k is spring constant, measured in kg/s²; A is amplitude, measured in m; m is mass, measured in kg; ω is angular frequency, measured in rad/s.

How is the SHM Total Energy formula derived?

E = KE + PE = ½mv² + ½kx² = ½kA² (at max displacement). The result is E = (1/2)kA² = (1/2)mω²A².