Summary

General Principles

Dynamics

SHM occurs when a linear restoring force acts to return a system to an equilibrium position.

Horizontal spring

(F_{net})_{x} = - k x

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Vertical spring

The origin is at the equilibrium position $Δ L = m g / k$ .

(F_{net})_{y} = - k y

The figure illustrates a block oscillating due a vertical spring attached to a wall on the top.

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Both: $ω = \sqrt{\frac{k}{m}} T = 2 π \sqrt{\frac{m}{k}}$

Simple pendulum

ω = \sqrt{\frac{g}{L}}

T = 2 π \sqrt{\frac{L}{g}}

The figure illustrates an oscillating pendulum.

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Physical pendulum

ω = \sqrt{\frac{M g l}{I}}

T = 2 π \sqrt{\frac{I}{M g l}}

The figure illustrates a physical pendulum depicted as a stick with an encircled x-mark at its center.

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Energy

If there is no friction or dissipation, kinetic and potential energy are alternately transformed into each other, but the total mechanical energy $E = K + U$ is conserved.

\begin{matrix} E & = & \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2} \\ = & \frac{1}{2} m (v_{max})^{2} \\ = & \frac{1}{2} k A^{2} \end{matrix}

The figure illustrates a block oscillating on a horizontal spring attached to a wall on the left.

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The energy of a lightly damped oscillator decays exponentially

E = E_{0} e^{- t / τ}

where $τ$ is the time constant.

The figure illustrates the energy decay graph of a lightly damped oscillator.

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Important Concepts

Simple harmonic motion (SHM) is a sinusoidal oscillation with period T and amplitude A.

The figure illustrates an amplitude versus time graph for simple harmonic motion.

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Frequency $f = \frac{1}{T}$

Angular frequency

ω = 2 π f = \frac{2 π}{T}

Position $x (t) = A cos (ω t + ϕ_{0})$

= A cos (\frac{2 π t}{T} + ϕ_{0})

Velocity $v_{x} (t) = - v_{max} sin (ω t + ϕ_{0})$ with maximum speed $v_{max} = ω A$

Acceleration $a_{x} (t) = - ω^{2} x (t) = - ω^{2} A cos (ω t + ϕ_{0})$

SHM is the projection onto the x-axis of uniform circular motion.

$ϕ = ω t + ϕ_{0} is the$ phase

The figure illustrates the before and after positions of a particle in a uniform circular motion on an x y coordinate plane.

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The position at time t is

\begin{matrix} x (t) & = & A cos ϕ \\ = & A cos (ω t + ϕ_{0}) \end{matrix}

The phase constant $ϕ_{0}$ is determined by the initial conditions:

x_{0} = A cos ϕ_{0} v_{0 x} = - ω A sin ϕ_{0}

Applications

Resonance

When a system is driven by a periodic external force, it responds with a large-amplitude oscillation if $f_{ext} \approx f_{0}$ , where $f_{0}$ is the system’s natural oscillation frequency, or resonant frequency.

The figure illustrates the response curve of a driven oscillator at frequencies near its natural frequency of f subscript 0 hertz.

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Damping

If there is a drag force $\vec{F}_{drag} = - b \vec{v}$ , where b is the damping constant, then (for lightly damped systems)

x (t) = A e^{- b t / 2 m} cos (ω t + ϕ_{0})

The time constant for energy loss is $τ = m / b$ .

The figure illustrates the amplitude versus time graph for damped oscillations.

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