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Interactive physics simulator

Displacement in SHM

Analyze the position of an oscillator over time. Toggle between parametric equation plotting, rotating phasor circles, and spring forces to see how amplitude, frequency, and phase constants determine instantaneous displacement.

Displacement in SHM Laboratory ⓘ

Modify physical parameters in the right panel. Watch the live displacement (blue), velocity (green), and restoring force (red) vectors synchronize on stage.

Oscillating

Live Displacement Telemetry

Elapsed Time (t)
0.00 s
Displacement (x)
0.00 m
Velocity (v)
0.00 m/s
Restoring Force (F)
0.00 N

Understanding Displacement in Simple Harmonic Motion

In physics, **displacement** (denoted by x) represents the position of an oscillating body relative to its stable equilibrium position. At the equilibrium position, displacement is defined as zero. When the body moves to one side, its displacement is positive; when it swings to the opposite side, its displacement becomes negative.

Displacement in SHM is a sinusoidal function of time, mathematically written as: x(t) = A cos(ωt + φ). Here, A is the amplitude (the maximum possible displacement), ω is the angular frequency (which determines how fast the cycles repeat), and φ is the phase constant (which defines the position at t = 0). Since trigonometric functions continuously vary between +1 and -1, the displacement smoothly cycles back and forth between the positive and negative amplitude boundaries.

Key Principles

Core characteristics of SHM displacement:

  • Time-Dependent Position: Displacement is a periodic function of time, repeating exactly every period T = 2π/ω.
  • Boundary Constraints: Position is physically bounded by the amplitude, meaning -A ≤ x(t) ≤ +A at all times.
  • Phase Shift Influence: The phase constant φ shifts the wave along the time axis, representing the system's initial position and velocity.
  • Opposing Force connection: Because force is proportional to -x, the maximum displacement points correspond directly to the maximum restoring forces.

Formulas & Relations

Mathematical definitions of displacement:

  • Cosine Position Formula: x(t) = A cos(ωt + φ). Standard expression where ω = 2πf.
  • Sine Position Formula: x(t) = A sin(ωt + φs). Equivalent sine expression where φs = φ + π/2.
  • Relation to Acceleration: a(t) = -ω2 x(t). Relates position directly to acceleration.
  • Initial Position: x(0) = A cos(φ). Position of the system at time t = 0.
  • Total Mechanical Energy: E = {1/2} k x2 + {1/2} m v2. Displays how energy balances between displacement potential energy and speed kinetic energy.

Solved Examples

A particle executes simple harmonic motion described by the displacement equation: x(t) = 0.50 cos(2.0π t + π/4), where displacement is in meters and time is in seconds. (a) Identify the amplitude, frequency, and phase constant of the motion. (b) Find the displacement of the particle at t = 0.
  1. Compare the given displacement equation with the standard form: x(t) = A cos(2π f t + φ).
  2. Part (a): Identify the parameters directly. Amplitude A = 0.50 meters.
  3. Frequency f: Since 2π f t corresponds to 2.0π t, we have f = 1.0 Hz.
  4. Phase constant φ corresponds to π/4 radians (or 45°).
  5. Part (b): Substitute t = 0 into the displacement equation.
  6. x(0) = 0.50 cos(2π · 0 + π/4) = 0.50 cos(π/4).
  7. Recall that cos(π/4) = 1/√2 ≈ 0.707.
  8. Calculate the initial displacement: x(0) = 0.50 · 0.707 = 0.354 meters.
  9. The amplitude is 0.50 m, frequency is 1.0 Hz, phase constant is π/4, and initial displacement is 0.354 m.

Answer: A = 0.50 m, f = 1.0 Hz, φ = π/4, x(0) ≈ 0.354 m

An oscillator has a frequency of 2.50 Hz and an amplitude of 15.0 cm. If the object starts at its negative amplitude position (x = -A) at t = 0, (a) find the phase constant φ using the standard cosine formula x(t) = A cos(ωt + φ). (b) Write the completed displacement equation in terms of time.
  1. Part (a): Convert amplitude to meters: A = 0.15 m.
  2. Calculate angular frequency ω: ω = 2πf = 2π(2.50) = 5π rad/s.
  3. At t = 0, displacement is x(0) = -A. Set up the equation: -A = A cos(φ).
  4. Divide by A: cos(φ) = -1.
  5. Recall that cos(θ) = -1 occurs when θ = π (or 180°). Thus, phase constant φ = π radians.
  6. Part (b): Substitute the values into the standard equation.
  7. x(t) = 0.15 cos(5π t + π) meters.
  8. Note that by trigonometric identity cos(θ + π) = -cos(θ), this is equivalent to: x(t) = -0.15 cos(5π t).
  9. The phase constant is π radians and the displacement equation is x(t) = 0.15 cos(5π t + π) m.

Answer: φ = π rad, x(t) = 0.15 cos(5π t + π) m

A spring-mass system oscillates vertically. The spring constant k is 64 N/m and the mass m is 0.25 kg. If the block is pulled down by 10.0 cm from equilibrium and released from rest at t = 0, find the displacement at t = 0.25 seconds.
  1. Calculate the angular frequency: ω = √(k/m) = √(64 / 0.25) = √(256) = 16 rad/s.
  2. Since the block is released from rest at maximum displacement, the phase constant φ is 0 (for a cosine model).
  3. The initial amplitude is A = 10.0 cm = 0.10 m.
  4. Write the displacement equation: x(t) = A cos(ωt) = 0.10 cos(16t).
  5. Substitute t = 0.25 seconds: x(0.25) = 0.10 cos(16 · 0.25) = 0.10 cos(4).
  6. Calculate cos(4) in radians. Note that 4 radians is in the third quadrant: cos(4) ≈ -0.6536.
  7. Calculate displacement: x(0.25) = 0.10 · (-0.6536) ≈ -0.0654 meters (or -6.54 cm).
  8. The negative sign indicates the block is 6.54 cm above the equilibrium position.
  9. The displacement at t = 0.25 s is approximately -0.0654 m.

Answer: x(0.25) ≈ -6.54 cm

Common Mistakes

  • Assuming x(0) always equals A: Believing that the starting displacement must always be the amplitude. This is only true if the phase constant φ is zero.
  • Incorrect Angle Units: Forgetting to set calculators to radians when evaluating the displacement equation x = A cos(ωt + φ). Degree measurements will yield incorrect positions.
  • Displacement vs Distance: Confusing the instantaneous displacement vector (which can be negative) with the total cumulative distance traveled by the oscillator.
  • Confusing Phase Shifts: Thinking a positive phase shift φ > 0 shifts the wave plot to the right. A positive shift advances the phase, shifting the displacement plot left.

Practice Questions

1. What is the physical meaning of the phase constant (φ) in the SHM displacement equation, and how is it determined?

The phase constant φ (in radians) defines the starting state or position of the oscillator at time t = 0. It shifts the entire displacement wave along the time axis. It is determined by the initial conditions of the system. For example, if the object is released from rest at its maximum displacement x = A at t = 0, then φ = 0. If it starts at equilibrium x = 0 moving in the positive direction, φ = -π/2 (or we can write it as a sine wave with zero phase). Mathematically, it is solved using x(0) = A cos(φ) and v(0) = -Aω sin(φ).

2. Explain why we can use both sine and cosine functions to describe SHM displacement, and state the relation between them.

Both functions are valid because sines and cosines are identical wave shapes, offset only by a phase difference of π/2 radians (90°). The general displacement can be written as x(t) = A cos(ωt + φ) or x(t) = A sin(ωt + φs). The relation is cos(θ) = sin(θ + π/2). Thus, a cosine wave is simply a sine wave shifted left by a quarter cycle. Using cosine is standard when the system starts at maximum displacement, while sine is standard when it starts at equilibrium.

3. How does the displacement vector relate to the acceleration vector in Simple Harmonic Motion?

The acceleration vector in SHM is always directly proportional to the displacement vector but points in the exact opposite direction (a = -ω2x). This is because the restoring force (F = ma) always pulls the object back toward the equilibrium position. When the displacement is at its maximum positive value (+A), the acceleration is at its maximum negative value (-ω2A). At the equilibrium position (x = 0), the displacement and the acceleration are both zero.

4. Describe how Uniform Circular Motion projects onto a straight diameter to represent SHM displacement.

Imagine a particle moving in a circle of radius A at a constant angular speed ω. If you shine a light from above, the shadow of the particle projects onto the horizontal diameter. The distance of the shadow from the circle's center is given by x = A cos(θ), where θ = ωt + φ is the rotation angle. As the particle sweeps around the circle, the shadow moves back and forth along the diameter, slowing down at the edges and speeding up in the middle. The position of this shadow is mathematically identical to SHM displacement.

FAQ

Frequently Asked Questions

What is displacement in Simple Harmonic Motion?

Displacement is the instantaneous distance and direction of the oscillating object from its central stable equilibrium position (x = 0).

What is the standard formula for SHM displacement?

The standard formula is x(t) = A cos(ωt + φ), where A is amplitude, ω is angular frequency (2πf), t is time, and φ is the phase constant.

What is the maximum displacement called?

The maximum displacement of the oscillator from its equilibrium position is called the amplitude (A).

Does the frequency of oscillation change with displacement?

No. In Simple Harmonic Motion, the frequency and period are independent of both displacement and amplitude (isochronism). A larger displacement creates a larger restoring force, which increases acceleration and speed, keeping the time taken for a full cycle constant.

What is the displacement of an oscillator when its velocity is zero?

When velocity is zero, the oscillator has temporarily stopped at its turning points. At these extreme limits, the displacement is at its maximum value: x = +A or x = -A.

Why is displacement zero when acceleration is zero?

In SHM, acceleration is directly proportional to displacement (a = -ω2x). At the equilibrium position (x = 0), there is no displacement, meaning no restoring force acts on the body, so acceleration is zero.

OscillationPeriodFrequencyAmplitudeSimple Harmonic MotionVelocity in SHMAcceleration in SHMEnergy in SHMMass-Spring SystemSimple PendulumResonance