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7.2.2Oscillations

Graphical Representation of SHM — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 6 of 10 1 quiz

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7.2.2 Graphical Representation of SHM — AP PHYSICS 1 notes: Interpret and sketch graphs of position, velocity, acceleration and force for SHM and extract period, amplitude and spring constant from them.Shape — a sine or cosine curve.Amplitude — the maximum height above the time a…

  • Equilibrium crossing
  • Peaks and troughs
  • What the symbols mean
  • Inverted mirror
  • Translating
  • Force against displacement

Section 1 of 9Position–time graph

Shape
  • Shape a sine or cosine curve.
Amplitude
  • Amplitude the maximum height above the time axis.
Period
  • Period the horizontal distance between two successive peaks.
Equilibrium crossing
  • Equilibrium crossing the curve crosses the axis at its steepest gradient.
Peaks and troughs
  • Peaks and troughs zero gradient as the object momentarily stops.
Start
  • Start a cosine curve from x = +A; a sine curve from equilibrium moving positive.

Section 2 of 9Velocity and acceleration graphs

Velocity
  • Velocity the gradient of the x–t graph; for a cosine x it is a negative sine curve of the same period.
v maximum
  • v maximum magnitude ωA when x crosses zero, zero at peaks and troughs; v leads x by T/4.
Acceleration
  • Acceleration the gradient of the v–t graph; a negative cosine, a = −ω2x, with maximum ω2A.
    What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
Inverted mirror
  • Inverted mirror when displacement is at its positive maximum, acceleration is at its negative maximum.

Section 3 of 9Start from maximum or equilibrium

StartDisplacementVelocityAcceleration
Positive amplitudeMaximumZeroMinimum
EquilibriumZeroMaximumZero
Frequency
  • Frequency all quantities oscillate at the same frequency.
Translating
  • Translating plot the gradient of x–t to get v–t, and the gradient of v–t to get a–t;
    • zeros, maxima and minima are enough.
Example
  • Example a vertical spring whose acceleration starts at its positive maximum (up positive) starts a distance A below equilibrium.

Section 4 of 9Graphs against position

Graphs against position
Force against displacement
  • Force against displacement straight line through the origin with gradient −k; area relates to stored energy.
Acceleration against displacement
  • Acceleration against displacement straight line through the origin with gradient −ω2, the standard test for SHM.
Velocity against position
  • Velocity against position an ellipse with semi-axes A and ωA.
    • (v and x 90° out of phase)
Potential energy against displacement
  • Potential energy against displacement parabola ½kx2 with minimum at equilibrium.
Kinetic energy
  • Kinetic energy inverted parabola; the two add to a horizontal line at the total energy.

Section 5 of 9Reading data from graphs

Period
  • Period time for several cycles divided by the number of cycles.
Amplitude
  • Amplitude read from the peak height.
ω
2π/T, or the square root of the magnitude of the a–x gradient.
k
k gradient of an F–x graph, or k = mω2.
Shifted start
  • Shifted start a graph not starting at equilibrium shifts the curve horizontally but does not change T or A.

Section 6 of 9Phase

1Phase
  • Phase the fraction of a cycle by which one oscillation is ahead of another.
2Quarter period
  • Quarter period phase difference 90° or π/2 rad.
3Half period
  • Half period phase difference 180°, exactly opposite.
4Within one oscillator
  • Within one oscillator x and v are 90° out of phase, x and a are 180° out of phase.
5Sketching
  • Sketching shift by a quarter period for velocity and flip for acceleration;
    • gradient signs give the signs of v and a.
6Damping
  • Damping x–v ellipse spirals inward and x–t peaks form a decreasing envelope.

Section 7 of 9More from the specification

1Phase displacement in an oscillating system
  • Phase displacement in an oscillating system displacement and acceleration sine curves remain exactly 180 degrees out of phase in a simple harmonic system.
2Tangent
  • Line touching a curve; its slope gives the instantaneous rate.
3Positive direction
  • The direction chosen to have positive values.
4Restoring force
  • Force directed toward equilibrium.

Section 8 of 9Key terms

TermMeaning
PhaseStage of an oscillation cycle; a quarter cycle apart is 90° out of phase.
SinusoidalHaving the shape of a sine or cosine curve.
GradientSlope of a graph; of x–t it is v, of v–t it is a.
Period (graph)Horizontal distance for one complete cycle.
ParabolaShape of a graph of ½kx2 against x.

Section 9 of 9Exam technique

1
Reading the period as the distance from a peak to the next trough — that is half a period.
2
Believing velocity is zero where the displacement graph crosses zero — it is maximum there.
3
Forgetting that the acceleration graph is inverted relative to the displacement graph.
4
Confusing the gradient of an F–x graph (k) with that of an a–x graph (ω2).
5
Treating potential and kinetic energy graphs as straight lines.

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