Elastic Constants Calculator
Relate Young's modulus, shear modulus, bulk modulus, and Poisson's ratio. Find any elastic constant from the others with material presets and conversion formulas.
Elastic Potential Energy Calculator
Calculate energy stored in springs using PE = ½kx². Includes spring arrangements, oscillation frequency, velocity analysis, and energy-displacement tables.
N/m
Extension or compression from equilibrium
m
For velocity and oscillation calculations
kg
Elastic Potential Energy⧉
2.5000 J
PE = ½kx² — energy stored in the deformed spring
Restoring Force⧉
50.000 N
F = kx — force exerted by the spring (Hooke's Law)
Effective Spring Constant⧉
500.00 N/m
Single spring constant
Velocity if Released⧉
2.236 m/s
v = √(2PE/m) — maximum velocity when all PE converts to KE
Equivalent Height⧉
0.255 m
h = PE/(mg) — height the mass could reach if launched vertically
Oscillation Frequency⧉
3.559 Hz
f = (1/2π)√(k/m) — natural frequency of spring-mass system
Energy vs Displacement
10%
20%
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100%
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Displacement as % of input value — energy grows quadratically
Displacement–Energy–Force Table
| x (m) | PE (J) | Force (N) | Velocity (m/s) |
|---|---|---|---|
| 0.0250 | 0.1563 | 12.500 | 0.559 |
| 0.0500 | 0.6250 | 25.000 | 1.118 |
| 0.0750 | 1.4063 | 37.500 | 1.677 |
| 0.1000 | 2.5000 | 50.000 | 2.236 |
| 0.1500 | 5.6250 | 75.000 | 3.354 |
| 0.2000 | 10.0000 | 100.000 | 4.472 |
| 0.3000 | 22.5000 | 150.000 | 6.708 |
Spring Arrangement Comparison
| Arrangement | k_eff (N/m) | PE at x = 0.100 m |
|---|---|---|
| 1 in parallel | 500.00 | 2.5000 J |
| 2 in parallel | 1,000.00 | 5.0000 J |
| 3 in parallel | 1,500.00 | 7.5000 J |
| 4 in parallel | 2,000.00 | 10.0000 J |
| 5 in parallel | 2,500.00 | 12.5000 J |
| 2 in series | 250.00 | 1.2500 J |
| 3 in series | 166.67 | 0.8333 J |
| 4 in series | 125.00 | 0.6250 J |
| 5 in series | 100.00 | 0.5000 J |
Planning notes, formulas, and examples
About the Elastic Potential Energy Calculator
Elastic potential energy is the energy stored in an object when it is deformed within its elastic limit. The classic case is a spring obeying Hooke's law, where the stored energy is PE = 1/2 kx^2 with k as the spring constant and x as the displacement from equilibrium.
This calculator computes elastic potential energy together with the restoring force, release velocity, equivalent gravitational height, and oscillation frequency for spring-mass systems. It also handles single, parallel, and series spring arrangements so you can see how the effective stiffness changes the stored energy.
That makes it useful whenever you want the energy value and the motion consequences in the same place, rather than rebuilding the spring math by hand.
When This Page Helps
Spring energy problems are simple in form but easy to expand into several linked calculations. Once you need effective stiffness, release speed, or oscillation frequency, the algebra becomes more than a one-line formula.
Putting those outputs together helps you compare spring setups and understand how much energy a given displacement actually stores.
How to Use the Inputs
- Enter the spring constant (k) in Newtons per meter.
- Enter the displacement (x) from the equilibrium position in meters.
- Optionally enter an attached mass for velocity and oscillation frequency calculations.
- Select spring arrangement (single, parallel, or series) and number of springs if applicable.
- Read the stored energy, restoring force, release velocity, and oscillation frequency.
- Review the energy-displacement chart to visualize the quadratic relationship.
- Use preset buttons for common spring scenarios like car suspension or trampolines.
Formula used
Elastic Potential Energy: PE = ½kx²
Hooke's Law Force: F = kx
Release Velocity: v = √(2PE/m)
Oscillation: f = (1/2π)√(k/m), T = 1/f
Parallel Springs: k_eff = n × k
Series Springs: k_eff = k / n
Where:
k = spring constant (N/m)
x = displacement from equilibrium (m)
m = attached mass (kg)Example Calculation
Result: 2.5 J
A spring with k = 500 N/m compressed by 0.1 m stores PE = 0.5 × 500 × 0.01 = 2.5 J. If released with a 1 kg mass, maximum velocity is v = √(2 × 2.5 / 1) = 2.24 m/s.
Tips & Best Practices
- Energy quadruples when displacement doubles — this is why overloading springs is dangerous.
- The natural frequency of a spring-mass system depends only on k and m, not on displacement.
- For real springs, there is a maximum safe displacement beyond which permanent deformation occurs.
- Rubber bands approximately follow Hooke's Law for small extensions but become nonlinear at large stretch.
- In vehicle suspension, parallel springs increase stiffness while series springs increase compliance.
- Energy density (PE per unit volume) is important for designing compact energy storage devices.
Energy Conservation in Spring Systems
When a compressed or stretched spring is released, its elastic potential energy converts to kinetic energy. For an ideal spring-mass system on a frictionless surface, the total mechanical energy remains constant: E_total = ½kx² + ½mv² = constant. At maximum displacement, all energy is potential; at the equilibrium position, all energy is kinetic and the mass reaches its maximum speed.
This energy oscillates back and forth between potential and kinetic forms indefinitely in an ideal system. In reality, damping from friction and air resistance gradually dissipates the energy as heat, causing the oscillations to decay.
Applications in Engineering
Elastic energy storage is used in countless engineering applications. Car suspensions use springs to absorb road irregularities, converting kinetic energy of vertical motion into elastic PE and back. Mechanical clocks and watches store energy in coiled mainsprings. Trampolines convert the kinetic energy of a jumping person into elastic PE in the springs, then launch them back up. Archery bows store elastic PE when drawn and release it as kinetic energy of the arrow.
Beyond Hooke's Law
Real materials often exhibit nonlinear elastic behavior. Rubber, biological tissues, and many polymers show stress-strain curves that deviate significantly from the linear Hooke's Law prediction. For these materials, the energy integral ∫F(x)dx must be evaluated with the actual force-displacement relationship, which is often determined experimentally.
Sources & Methodology
Last updated:
Frequently Asked Questions
Hooke's Law states F = kx: the restoring force of a spring is proportional to its displacement. It holds only within the elastic limit — beyond that, permanent deformation occurs.
Because the force increases linearly with displacement (F = kx). The energy is the integral of force over distance: ∫kx dx = ½kx². So doubling displacement doubles the force but quadruples the energy.
Beyond the elastic limit, the material yields permanently and does not return to its original shape. The stored energy is partially dissipated as heat and plastic deformation.
Parallel springs share the load, so the effective spring constant increases (k_eff = nk). Series springs each experience the full load, so the system is more compliant (k_eff = k/n).
The spring constant k (in N/m) measures stiffness: how much force per unit of displacement. Stiff springs have large k; soft springs have small k.
No. Since PE = ½kx² and both k and x² are always non-negative, elastic PE is always zero or positive regardless of whether the spring is stretched or compressed.
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