Physics
Oscillations and harmonic motion calculator
The period of a mass on a spring and of a pendulum, the maximum speed and acceleration of harmonic motion, the oscillator's energy and an amplitude dying away over time. Pick the relation, leave blank the field the exercise asks for, and the formula rearranges itself.
Why so many things oscillate the same way
A mass on a spring, a pendulum, a guitar string, an LC circuit: very different systems that oscillate by the same law. The reason is that they all have a restoring force proportional to the displacement, F = −k·x. Put into Newton's second law, it gives an acceleration proportional to the displacement and opposite to it, and the only function that behaves that way is a sine: x(t) = A·cos(ωt), with ω = √(k/m).
The surprising part is that the period T = 2π/ω does not depend on the amplitude. A wider swing covers more ground, but under a larger force that drives it faster, and the two cancel exactly. That is what made the pendulum the first accurate clock: Galileo noticed that the period does not change as the swing dies down. For a pendulum the constant k is m·g/L, the mass cancels and T = 2π·√(L/g) is left, valid as long as the angle stays small.
In reality every oscillation loses energy to friction. With friction proportional to speed the amplitude decays exponentially, A₀·e^(−γt): the same fraction is lost in equal times, as in radioactive decay. If the damping exceeds the critical value b = 2·√(k·m) the system no longer oscillates and creeps back to rest: that is what a car's shock absorbers aim for, tuned just below it.
Common mistakes
- Confusing frequency and angular frequency: f is in hertz and counts cycles per second, ω = 2π·f is in radians per second. Many formulas want ω.
- Using T = 2π·√(L/g) for large angles: at 30° the true period is already about 1.7 % longer, at 90° 18 %.
- Putting the mass into the pendulum's period: it has nothing to do with it. A lead bob and a wooden one on strings of the same length keep the same time.
- Thinking energy falls like the amplitude: it goes with the square, so when the amplitude is at half, a quarter of the energy is left.
Frequently asked questions
Why does the period not depend on the amplitude?
Because the restoring force grows with the displacement. A wider swing has further to go, but it starts where the spring pulls harder and reaches higher speeds: the two effects cancel exactly. The property is called isochronism, and it holds as long as the force stays proportional to the displacement.
How long is a pendulum that beats seconds?
About 99.4 cm with g = 9.81 m/s². One swing takes a second, so the full period is two seconds and L = g·(T/2π)² = g/π². In the eighteenth century it was proposed as the definition of the metre, but g varies with latitude and the Earth's meridian was chosen instead.
What is the difference between frequency and angular frequency?
Frequency f counts full oscillations per second, in hertz. Angular frequency ω = 2π·f measures the same rhythm in radians per second, since a full oscillation is one turn of 2π radians. The period is T = 1/f = 2π/ω.
What is critical damping?
The value of the friction coefficient, b = 2·√(k·m), beyond which the system stops oscillating. Exactly at it, the system returns to equilibrium in the shortest time without overshooting; above it, it returns more slowly; below it, it oscillates with a shrinking amplitude. Car shock absorbers are tuned a little below critical.
Where is the speed greatest in harmonic motion?
At the centre, the equilibrium position, where the force is zero and all the energy is kinetic: v_max = A·ω. At the ends the speed is zero and the acceleration is greatest, A·ω², because that is where the spring pulls hardest.
How this calculation works
Mass on a spring: T = 2π·√(m/k), with frequency f = 1/T, angular frequency ω = 2π/T and critical damping b = 2·√(k·m). Simple pendulum, for small swings: T = 2π·√(L/g). Harmonic motion x(t) = A·cos(ωt): maximum speed v_max = A·ω = 2π·f·A and maximum acceleration A·ω². Oscillator energy: E = ½·k·A², with the greatest force k·A at the ends. Viscous damping: A(t) = A₀·e^(−γt), with γ = b/2m; inverting gives γ = ln(A₀/A)/t, the half-life is ln 2/γ and the energy left is (A/A₀)², because energy goes with the square of the amplitude.
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