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7.2.1Oscillations

Features of SHM — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 5 of 10 1 quiz

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Lesson summary

7.2.1 Features of SHM — AP PHYSICS 1 notes: Describe how displacement, velocity, acceleration and force vary in SHM and calculate maximum values and values at given positions.Setup — equilibrium at x = 0, amplitude A, oscillation between +A and −A, repeating every period T.At the…

  • At the extremes
  • At equilibrium
  • Complementary
  • amax = ω2A = kA/m
  • x = A cosωt
  • x = A sinωt

Part 1 of 7Positions in an oscillation

Setup
  • Setup equilibrium at x = 0, amplitude A, oscillation between +A and −A, repeating every period T.
At the extremes
  • At the extremes the object is momentarily at rest (v = 0) while the restoring force and acceleration have maximum magnitude toward equilibrium.
At equilibrium
  • At equilibrium net force and acceleration are zero and speed is maximum.
Between
  • Between speed rises approaching equilibrium and falls moving away.
Complementary
  • Complementary where velocity is zero acceleration is maximum, and vice versa.

Part 2 of 7Maximum speed and acceleration

ω
√(k/m) = 2π/T for a spring.
vmax = ωA
  • vmax = ωA at equilibrium.
amax
ω2A = kA/m
  • at the amplitude.
Relation
  • Relation amax = ωvmax.
Amplitude
  • Amplitude doubling A doubles both maxima while the period stays the same.

Part 3 of 7Displacement as a function of time

Displacement as a function of time
x
A cosωt
  • starting at x = +A when t = 0.
x
A sinωt
  • starting at equilibrium moving in the positive direction.
Velocity
  • Velocity v(t) = −ωA sinωt.
Acceleration
  • Acceleration a(t) = −ω2A cosωt = −ω2x.
Starting condition
  • Starting condition sets the choice of function; they repeat after T = 2π/ω.

Part 4 of 7Speed at any position

v
ω√(A2 − x2)
  • from energy conservation; ωA at x = 0 and zero at x = ±A.
Time spent
  • Time spent longer near the extremes where the object is slow.

Part 5 of 7Directions

Directions
Velocity
  • Velocity changes sign at each turning point.
Acceleration
  • Acceleration changes sign as the object crosses equilibrium.
Moving away from equilibrium
  • Moving away from equilibrium v and a opposite, so it slows.
Moving toward equilibrium
  • Moving toward equilibrium v and a same direction, so it speeds up.
Timing
  • Timing extreme to equilibrium takes T/4; extreme to extreme takes T/2.

Part 6 of 7Using equations and ratios

1Two known quantities
  • Two known quantities any two of A, T (or f or ω), vmax, amax give the others.
2From vmax and A
  • ω = vmax/A, then T, f, amax and k = mω2.
3Frequency doubled, A same
  • Frequency doubled, A same vmax doubles and amax quadruples.
4Amplitude doubled, f same
  • Amplitude doubled, f same both double.
5Loudspeaker example
  • 200 Hz and 0.50 mm give vmax = 0.63 m/s and amax about 790 m/s2, roughly 80g.
6Vibrating platform
  • Vibrating platform an object leaves it if ω2A exceeds g.

Part 7 of 7More from the specification

1Periodic cycle in oscillating systems
  • Periodic cycle in oscillating systems mechanical oscillators in a conservative oscillating system repeat position and velocity profiles throughout each complete sinusoidal cycle.
2Tangent
  • Line touching a curve; its slope gives the instantaneous rate.
3Gradient
  • Slope of a graph line.
Velocity
  • Velocity the gradient of a displacement–time graph gives the velocity.
Acceleration
  • Acceleration the gradient of a velocity–time graph gives the acceleration.
Turning points
  • Turning points at maximum displacement the speed is zero and the acceleration is greatest.
Equilibrium
  • Equilibrium at the centre the speed is greatest and the acceleration is zero.

Key terms

TermMeaning
AmplitudeMaximum displacement from equilibrium, A.
Angular frequencyω = 2π/T = √(k/m) for a spring.
Maximum speedvmax = ωA, reached at equilibrium.
Maximum accelerationamax = ω2A, at the extremes.
Turning pointPosition where the object momentarily stops and reverses.

Avoid these mistakes

1
Saying acceleration is zero at the extremes — it is zero at equilibrium.
2
Saying velocity and acceleration are always in the same direction.
3
Using v = ωA at all positions rather than only at x = 0.
4
Forgetting that maximum speed depends on A.
5
Confusing amplitude with the peak-to-peak distance.

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