Inputs
Results
| # | m (kg) | k (N/m) | ω (rad/s) | A (m) | E (J) | f (Hz) | T (s) | vmax (m/s) | amax (m/s²) | t (s) | x(t) (m) | v(t) (m/s) | a(t) (m/s²) | KE(t) (J) | PE(t) (J) |
|---|
Formulas used
ω = √(k/m), f = ω / (2π), T = 2π / ωE = ½ k A² = ½ m ω² A²x(t) = A cos(ω t + φ)v(t) = −A ω sin(ω t + φ), a(t) = −A ω² cos(ω t + φ)PE = ½ k x², KE = E − PEA = √(2E/k) (when energy is known)All quantities are scalar magnitudes in SI units unless noted.
How to use this calculator
- Enter mass and choose either stiffness k or angular frequency.
- Select whether you want energy from amplitude, or amplitude from energy.
- Provide amplitude A or total energy E accordingly.
- Optionally set phase and a time t to sample instantaneous state.
- Click Compute. Review the summary, then Add to table.
- Export your table using CSV or PDF buttons.
Use decimals control to adjust rounding in outputs and exports.
Example data table
| Case | m (kg) | k (N/m) | ω (rad/s) | A (m) | E (J) | φ (rad) | t (s) | x(t) | v(t) | a(t) |
|---|
Click any example row to load its values into the form.
How to find energy given mass, amplitude, and period (spring)
Given mass m, amplitude A, and period T, first compute stiffness using the period relation for an ideal spring–mass system:
T = 2π √(m/k) ⇒ k = 4π² m / T²
Then use total mechanical energy at amplitude:
E = ½ k A² = ½ (4π² m / T²) A² = 2π² m A² / T²
Worked example (m = 1.2 kg, A = 0.08 m, T = 0.90 s):
k = 4π²·1.2 / 0.90² ≈ 58.49 N/mE = 2π²·1.2·0.08² / 0.90² ≈ 0.187 J
This assumes ideal, undamped motion with a linear spring.
Example: Using the SHM Energy and Amplitude Calculator
Goal: Compute energy, period, frequency, and state at t for inputs.
- m = 1.500 kg, k = 25.000 N/m, A = 0.120 m
- φ = 0.000 rad, t = 0.300 s
| Quantity | Value | Units |
|---|---|---|
| ω | 4.082483 | rad/s |
| f | 0.649747 | Hz |
| T | 1.539060 | s |
| E | 0.180000 | J |
| vmax | 0.489898 | m/s |
| amax | 2.000000 | m/s² |
| x(t) | 0.040702 | m |
| v(t) | -0.460856 | m/s |
| a(t) | -0.678372 | m/s² |
| KE(t) | 0.159292 | J |
| PE(t) | 0.020708 | J |
These values match what the calculator will produce for the same inputs.
Energy vs Amplitude (k = 20 N/m)
| A (m) | E (J) |
|---|---|
| 0.050 | 0.025000 |
| 0.100 | 0.100000 |
| 0.150 | 0.225000 |
| 0.200 | 0.400000 |
| 0.250 | 0.625000 |
E scales with A² at fixed stiffness, as E = ½ kA².
Period and Frequency across m, k
| m (kg) | k (N/m) | ω (rad/s) | f (Hz) | T (s) |
|---|---|---|---|---|
| 0.500 | 10.000 | 4.472136 | 0.711763 | 1.404963 |
| 0.500 | 40.000 | 8.944272 | 1.423525 | 0.702481 |
| 1.000 | 20.000 | 4.472136 | 0.711763 | 1.404963 |
| 2.000 | 20.000 | 3.162278 | 0.503292 | 1.986918 |
| 2.000 | 80.000 | 6.324555 | 1.006584 | 0.993459 |
Increasing k increases ω and f; increasing m decreases them.
Instantaneous State over One Cycle (m = 1.0 kg, k = 16 N/m, A = 0.10 m, φ = π/6)
| t (s) | x(t) (m) | v(t) (m/s) | a(t) (m/s²) |
|---|---|---|---|
| 0.000000 | 0.086603 | -0.200000 | -1.385641 |
| 0.392699 | -0.050000 | -0.346410 | 0.800000 |
| 0.785398 | -0.086603 | 0.200000 | 1.385641 |
| 1.178097 | 0.050000 | 0.346410 | -0.800000 |
| 1.570796 | 0.086603 | -0.200000 | -1.385641 |
Samples at 0, T/4, T/2, 3T/4, and T show sinusoidal evolution.