Enter values to see results
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m
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m/s
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m/s²
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rad/s
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m/s²
Enter values to see results
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m
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m/s
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m/s²
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rad/s
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Hz
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s
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m/s
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m/s²
The Simple Harmonic Motion (SHM) Calculator computes the instantaneous displacement, velocity, and acceleration of an object undergoing simple harmonic motion. SHM is the most fundamental type of periodic motion in physics, where the restoring force on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction.
Simple harmonic motion appears everywhere in nature and engineering. A mass bouncing on a spring, a pendulum swinging through small angles, atoms vibrating in a crystal lattice, and alternating current in an electrical circuit all exhibit SHM. The motion is characterized by a sinusoidal oscillation about an equilibrium point, with constant amplitude and a well-defined frequency determined by the system’s physical properties.
The displacement of a particle in SHM is described by the equation $$x(t) = A\cos(\omega t + \phi_0)$$ where A is the amplitude (maximum displacement), ω is the angular frequency, t is time, and φ₀ is the initial phase. The velocity and acceleration follow by differentiation: $$v(t) = -A\omega\sin(\omega t + \phi_0)$$ $$a(t) = -A\omega^2\cos(\omega t + \phi_0) = -\omega^2 x(t)$$
The relationship $$a = -\omega^2 x$$ is the hallmark of SHM—the acceleration is always proportional to displacement and directed toward equilibrium. This leads to oscillatory motion with period $$T = \frac{2\pi}{\omega}$$ and frequency $$f = \frac{\omega}{2\pi}$$. The maximum speed occurs at equilibrium ($$v_{max} = A\omega$$) and maximum acceleration occurs at the turning points ($$a_{max} = A\omega^2$$).
This calculator accepts the amplitude plus angular frequency, ordinary frequency, or period, and computes the full kinematic state at any given time. It is an essential tool for physics students studying oscillations, engineers analyzing vibrating systems, and anyone working with periodic phenomena. Understanding SHM is the gateway to studying more complex wave behavior, resonance, damped oscillations, and coupled systems.
The calculator uses the standard equations of simple harmonic motion. First, it determines the angular frequency from the chosen input mode:
$$\omega = \begin{cases} \omega & \text{if angular frequency given} \\ 2\pi f & \text{if frequency given} \\ \frac{2\pi}{T} & \text{if period given} \end{cases}$$
Then it computes the three kinematic quantities at time t:
Displacement: $$x(t) = A\cos(\omega t + \phi_0)$$
Velocity: $$v(t) = -A\omega\sin(\omega t + \phi_0)$$
Acceleration: $$a(t) = -A\omega^2\cos(\omega t + \phi_0)$$
The calculator also reports the maximum velocity $$v_{max} = A\omega$$ and maximum acceleration $$a_{max} = A\omega^2$$, which occur when the sine and cosine terms reach their peak magnitudes of 1.
The displacement shows the particle’s position relative to equilibrium at the specified time. Positive values mean displacement in the positive direction; negative values mean displacement in the opposite direction. The velocity indicates how fast and in which direction the particle moves at that instant. The acceleration reveals the restoring force direction (always toward equilibrium). The maximum velocity and acceleration give the peak values achieved during the entire cycle.
Inputs
Results
A 0.1 m amplitude oscillation at ω = 10 rad/s. At t = 0.5 s, x ≈ −0.084 m, v ≈ 0.544 m/s, a ≈ 8.39 m/s². The object has passed equilibrium and is heading back.
Inputs
Results
A tuning fork at 440 Hz (concert A) with 0.5 mm amplitude. At t = 1 ms, the prong is at x ≈ −0.41 mm with enormous acceleration, illustrating the rapid oscillation of sound-producing objects.
Simple harmonic motion is periodic motion where the restoring force is directly proportional to displacement from equilibrium and always points toward equilibrium. Mathematically, this means the acceleration satisfies a = −ω²x. This produces sinusoidal oscillation with constant amplitude and frequency. Examples include ideal springs, small-angle pendulums, and LC circuits.
Ordinary frequency f measures the number of complete cycles per second (in Hz). Angular frequency ω = 2πf measures the rate of change of the phase angle in radians per second. Angular frequency is more natural in the mathematical description because it appears directly in the sine and cosine functions without a factor of 2π.
The negative sign in a = −Aω²cos(ωt + φ₀) reflects the fact that the restoring force (and hence acceleration) always points opposite to the displacement. When the object is displaced to the right (positive x), the acceleration points left (negative), pulling it back toward equilibrium. This is the defining characteristic of SHM.
The frequency depends on the physical properties of the system, not on amplitude. For a mass-spring system, ω = √(k/m). For a simple pendulum, ω = √(g/L). For an LC circuit, ω = 1/√(LC). In all cases, frequency is set by the stiffness of the restoring force and the inertia of the system.
The initial phase φ₀ determines the starting position and velocity of the oscillation at t = 0. If φ₀ = 0, the particle starts at maximum positive displacement (x = A). If φ₀ = π/2, it starts at equilibrium moving in the negative direction. The initial phase encodes the initial conditions of the motion.
SHM can be viewed as the projection of uniform circular motion onto a diameter. If a point moves around a circle of radius A at angular speed ω, its x-coordinate undergoes SHM: x = Acos(ωt). This geometric connection explains why angular frequency appears in SHM equations and is useful for visualizing phase relationships.
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