Frequency of Oscillation Calculator Physics

This calculator works out the frequency of a simple harmonic oscillator—such as a mass on a spring or a pendulum—from the physical properties you enter. You provide the relevant parameters (spring constant and mass, or pendulum length and gravitational acceleration), and it returns the frequency in hertz (cycles per second). The formula used is the standard one from introductory physics, with the result given to two decimal places.

Formula

For a mass-spring system, the frequency of oscillation is given by:

f = (1 / 2π) × √(k / m)

• f = frequency in hertz (Hz), the number of complete oscillations per second.
• k = spring constant in newtons per meter (N/m), a measure of the spring’s stiffness.
• m = mass attached to the spring in kilograms (kg).

The period T (time for one complete cycle) is the reciprocal: T = 1/f = 2π × √(m/k). Both are shown in the calculator because they are often needed together.

Worked Example

Using the default values from the calculator: k = 50 N/m and m = 0.5 kg.

1. Divide the spring constant by the mass: 50 / 0.5 = 100.
2. Take the square root: √100 = 10.
3. Divide by 2π: 10 / (2 × 3.14159) ≈ 1.5915 Hz.

The calculator rounds to two decimals, giving 1.59 Hz. The corresponding period is 2π × √(0.5/50) = 2π × 0.1 ≈ 0.628 s, which matches the second output.

When to Use This Formula

The formula applies to simple harmonic motion (SHM) under ideal conditions. Use it when:

• The system is a mass attached to an ideal spring (no damping, no friction, spring obeys Hooke’s law).
• The oscillations are small enough that the restoring force remains proportional to displacement.
• You are dealing with a pendulum with small angular amplitude (less than about 15°), where the frequency is approximately f = (1/2π) × √(g/L), with g = gravitational acceleration (9.81 m/s²) and L = pendulum length in meters.

For a pendulum, replace the first input (k) with g and the second (m) with L, and the calculator’s formula would need to be changed accordingly—this calculator is specifically for the mass-spring system.

How to Read the Result

The frequency tells you how many complete back-and-forth cycles occur each second. For example, 1.59 Hz means the mass completes 1.59 full oscillations every second. The period, 0.628 s, is the time for one single cycle. If you watch the mass move, you would see it return to its starting position every 0.628 seconds.

Note that frequency is independent of amplitude (for ideal SHM). Whether you pull the mass 2 cm or 5 cm, the frequency stays the same as long as the spring remains linear. This is a key property of simple harmonic motion.

Common Mistakes and Misconceptions

• Unit mix-ups: The mass must be in kilograms, not grams. If you enter 500 g as 500, the frequency becomes unrealistically high. Convert grams to kilograms by dividing by 1000.
• Confusing frequency and period: Frequency is cycles per second (Hz); period is seconds per cycle. They are reciprocals, not the same thing.
• Using weight instead of mass: The formula uses mass (kg), not weight (N). On Earth, weight = mass × 9.81, but you must enter the mass value, not the weight.
• Forgetting the 2π factor: The angular frequency ω = √(k/m) is in radians per second, not Hz. To get cycles per second, you must divide by 2π.
• Assuming the formula works for large amplitudes: For a pendulum, the small-angle approximation breaks down beyond about 15°, and the true frequency becomes slightly lower than predicted.

Limits of the Formula

The ideal formula ignores damping (air resistance, internal friction), which causes oscillations to gradually die out. In real systems, the frequency is slightly lower than the undamped value, and the amplitude decreases over time. The formula also assumes the spring is massless and obeys Hooke’s law perfectly. For very large amplitudes, the spring may reach its elastic limit, and the motion becomes non-linear.

For a physical pendulum (a rigid body swinging about a pivot), the frequency depends on the moment of inertia and the distance from the pivot to the center of mass, not just on mass and length. The simple pendulum formula is a special case.

Frequently Asked Questions

What is the difference between frequency and angular frequency?

Frequency (f) is the number of complete cycles per second, measured in hertz. Angular frequency (ω) is the rate of change of the phase, measured in radians per second. They are related by ω = 2πf. In the mass-spring formula, ω = √(k/m), so f = ω/(2π).

Does the frequency change if I increase the amplitude?

In ideal simple harmonic motion, no. The frequency is independent of amplitude because the restoring force is proportional to displacement. This is true for a mass-spring system as long as the spring remains within its linear range. For a pendulum, the independence holds only for small angles; at large angles, the frequency decreases slightly.

How do I convert grams to kilograms for the mass input?

Divide the mass in grams by 1000. For example, 200 g = 0.2 kg. If you have a mass in pounds, multiply by 0.4536 to get kilograms. The calculator expects kilograms, so always convert before entering.

What if I only know the period? Can I get the frequency?

Yes, take the reciprocal: f = 1/T. For instance, if the period is 0.5 seconds, the frequency is 2 Hz. The calculator’s second output gives the period from the same inputs, so you can check both.

Why does the formula use the square root of k/m?

It comes from Newton’s second law and Hooke’s law. The equation of motion for a mass-spring system is m·a = -k·x, which leads to a differential equation whose solution has an angular frequency √(k/m). The square root arises because acceleration is proportional to displacement but opposite in direction, producing sinusoidal motion.

Related

• Siding Calculator Square Feet: How to Measure
• Sq Ft Calculator Tile: How to Measure
• Roof Vent Calculator: CFM & Square Feet Guide

in k | Spring constant (k) | N/m | 50 | 0
in m | Mass (m) | kg | 0.5 | 0.001
out f | Frequency | Hz | (1/(2*pi))*sqrt(k/m) | 2
out T | Period | s | 2*pi*sqrt(m/k) | 2