What is the formula for simple harmonic motion?

The generalized formula describing the displacement of an object undergoing simple harmonic motion (SHM) at any given time is:

x(t) = Acos(ωt + φ)

This equation is fundamental to understanding oscillatory systems like a mass on a spring or a simple pendulum (for small angles).

Understanding the SHM Formula

The formula x(t) = Acos(ωt + φ) represents the position (displacement) of an object from its equilibrium point as a function of time t.

According to the provided reference:
x ( t ) = A cos ( ω t + φ ) . This is the generalized equation for SHM where t is the time measured in seconds, ω is the angular frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters, and φ is the phase shift measured in radians.

Let's break down each component of this equation:

• x(t): This is the displacement of the object from its equilibrium position at time t. It changes over time following a sinusoidal pattern.
• A: This represents the Amplitude of the motion. It is the maximum displacement of the object from its equilibrium position.
• ω (omega): This is the Angular Frequency. It determines how fast the oscillations occur. A higher angular frequency means faster oscillations.
• t: This is the Time, measured from a specific starting point. The displacement x depends on this variable.
• φ (phi): This is the Phase Shift (or phase constant). It describes the initial state of the oscillation at t = 0. It essentially shifts the cosine wave left or right along the time axis.

Key Components and Units

The reference specifies the standard units for each variable in the formula x(t) = Acos(ωt + φ).

Variable	Description	Units
x(t)	Displacement	Meters (m) or Centimeters (cm)
A	Amplitude	Meters (m) or Centimeters (cm)
ω	Angular Frequency	Inverse Seconds (s⁻¹)
t	Time	Seconds (s)
φ	Phase Shift	Radians (rad)

How the Formula Works

The cosine function within the formula naturally produces an oscillating value between -1 and +1. When multiplied by the Amplitude A, the displacement x(t) oscillates between -A and +A.

The term (ωt + φ) is the argument of the cosine function, often called the phase.

• ωt accounts for the linear change in phase over time, determining the speed of oscillation.
• φ sets the initial phase at t=0, influencing the object's starting position and direction of motion.

For example, if φ = 0, the motion starts at maximum positive displacement (x(0) = A cos(0) = A). If φ = π/2, the motion starts at the equilibrium position (x(0) = A cos(π/2) = 0) moving towards negative displacement.

Understanding this formula allows physicists and engineers to predict the position of an object in SHM at any future or past time, given the initial conditions (like amplitude and phase shift) and the system's inherent properties (like angular frequency).