Velocity in SHM
In simple harmonic motion (SHM), an object oscillates back and forth about an equilibrium position. Unlike uniform motion, the velocity of the object is constantly changing — it reaches its greatest speed as it passes through the centre of oscillation, and momentarily comes to rest at the extreme ends of its path. This interplay between position and speed is one of the most elegant features of SHM, and understanding it is essential for analysing anything from a swinging pendulum to the vibration of a guitar string. The velocity in SHM is intimately linked to both time and displacement. When expressed as a function of time, velocity follows a sinusoidal pattern: if displacement varies as a cosine, velocity varies as a sine, and vice versa. When expressed as a function of displacement alone, velocity obeys a relationship derived from energy conservation — the further the object is from equilibrium, the slower it moves, because more of its energy is stored as potential energy rather than kinetic energy. Mastering velocity in SHM requires understanding both representations. The time-based form tells you exactly where the object is in its cycle at any moment; the displacement-based form is especially useful when you know position but not time. Together they give a complete kinematic picture of the oscillating system.
- Starting from displacement
- Differentiating to get velocity
- Maximum speed
- Velocity as a function of displacement
- Energy interpretation
Starting from displacement
In SHM, displacement from equilibrium is described by x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is the initial phase. This is the starting point for finding velocity.
Differentiating to get velocity
Velocity is the rate of change of displacement: v = dx/dt. Differentiating x = A cos(ωt + φ) gives v(t) = −Aω sin(ωt + φ). This shows that velocity is 90° out of phase with displacement — when displacement is at its maximum (the turning point), sin = 0, so velocity is zero; when displacement is zero (the centre), sin is at its peak, so speed is maximum.
Maximum speed
The maximum speed occurs at the equilibrium position and equals v_max = Aω. This makes physical sense: at the centre, all energy is kinetic and none is stored as potential energy, so the object is moving as fast as it possibly can.
Velocity as a function of displacement
Using the Pythagorean identity sin²θ + cos²θ = 1, you can eliminate time entirely. Since x = A cos(ωt) and v = −Aω sin(ωt), we get v² = ω²(A² − x²), or equivalently v = ±ω√(A² − x²). The ± reflects the fact that at any given displacement the object could be moving in either direction. This formula is extremely useful: given only position, you can calculate speed without knowing time or phase.
Energy interpretation
The displacement-velocity relationship has a direct energy interpretation. Kinetic energy is ½mv² = ½mω²(A² − x²), while potential energy is ½mω²x². Their sum equals ½mω²A², which is constant — confirming conservation of energy throughout the oscillation. As x increases toward A, kinetic energy falls to zero and potential energy reaches its maximum.
The Key Equation
- v — velocity (m s⁻¹): The instantaneous velocity of the oscillating object; the ± sign indicates direction of motion.
- \omega — angular frequency (rad s⁻¹): Related to frequency f and period T by ω = 2πf = 2π/T; determines how rapidly the object oscillates.
- A — amplitude (m): The maximum displacement from equilibrium; sets the scale of the oscillation.
- x — displacement (m): The current distance of the object from its equilibrium position; ranges from −A to +A.
Worked Example
Problem. A mass on a spring oscillates with an amplitude of 0.12 m and an angular frequency of 5.0 rad s⁻¹. Calculate (a) the maximum speed of the mass, and (b) its speed when its displacement is 0.09 m from equilibrium.
Answer. Maximum speed = 0.60 m s⁻¹; speed at x = 0.09 m is approximately 0.40 m s⁻¹.
Origins · 1 minRobert Hooke and how this was discovered
Robert Hooke formulated Hooke's Law (F = −kx) in 1660, establishing that the restoring force in an elastic system is proportional to displacement. This law is the direct physical basis of SHM: it is precisely because F ∝ −x that oscillators follow sinusoidal motion, giving rise to the velocity relationships studied here. Without Hooke's Law, the mathematical framework of SHM kinematics would have no physical grounding. Hooke originally published his law as a Latin anagram — 'ceiiinosssttuv' — to establish priority while keeping the result secret. Two years later he revealed it decoded as 'ut tensio, sic vis' (as the extension, so the force).
Related Concepts
- Displacement in SHM — Velocity is the time derivative of displacement; understanding x(t) = A cos(ωt + φ) is the essential first step before differentiating to find velocity.
- Acceleration in SHM — Acceleration is the derivative of velocity with respect to time; the defining equation a = −ω²x follows directly from differentiating the velocity expression.
- Energy in SHM — The velocity–displacement formula v = ω√(A² − x²) is derived from energy conservation, connecting kinetic and potential energy throughout the oscillation.
- Wave Speed — Oscillating particles in a wave undergo SHM; understanding how particle velocity varies in SHM is fundamental to analysing transverse and longitudinal waves.