Simple harmonic motion IB DP Physics Study Notes

Oscillations

• Oscillations are vibrations which repeat themselves.
• EXAMPLE: Oscillations can be driven externally, like a pendulum in a gravitational field.
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• EXAMPLE: Oscillations can be driven internally, like a mass on a spring.
  FYI ∙In all oscillations, v = 0 at the extremes and v = v_max in the middle of the motion.
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• EXAMPLE: Oscillations can be very rapid vibrations such as in a plucked guitar string or a tuning fork.

Time period, amplitude and displacement

• Consider a mass on a spring that is displaced 4 meters to the right and then released.
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• We call the maximum displacement x₀ the amplitude. In this example x0 = 4 m.
• We call the point of zero displacement the equilibrium position. Displacement x is measured from equilibrium.
• The period T (measured in s) is the time it takes for the mass to make one full oscillation or cycle.
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• For this particular oscillation, the period T is about 24 seconds (per cycle).

Time period and frequency 

• The frequency $f$ (measured in Hz or $\frac{\text{cycles}}{\text{s}}$) is defined as how many cycles (oscillations, repetitions) occur each second.
• Since period $T$ is seconds per cycle, frequency must be $\frac{1}{T}$. Therefore, the relation between $T$ and $f$ is:
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EXAMPLE: The cycle of the previous example repeated each 24 s. What are the period and the frequency of the oscillation?

SOLUTION:

 The period is $T = 24$ s.

The frequency is $f = \frac{1}{T} = \frac{1}{24} \approx 0.042$ Hz.

Angular speed ω

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• Since 2π rad = 360° = 1 rev it should be clear that the angular speed ω is just: $\frac{2\pi}{T}$. This means $T = \frac{2\pi}{\omega}$.
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• And since $f = \frac{1}{T}$ it should also be clear that ω = 2πf.

EXAMPLE: Find the angular frequency (angular speed) of the second hand on a clock.

SOLUTION:

Since the second hand turns one circle each 60 s, it has an angular speed of:

ω = $\frac{2\pi}{T}$ = $\frac{2\pi}{60}$ = 0.105 rad s⁻¹