SHM Systems & Energy

01SHM Systems & Energy

Mass-spring systems

This section covers the restoring force and time period of horizontal and vertical mass-spring simple harmonic oscillators.

Spring as a restoring system

When a mass attached to a spring is moved away from its equilibrium position and released, the spring can provide the restoring force required for simple harmonic motion.

For displacement $x$ from equilibrium, the restoring-force relationship is:

\boxed{F=-kx}

The minus sign shows that the force is directed opposite to the displacement and therefore back towards equilibrium.

Time period

For an oscillating mass $m$ attached to a spring with spring constant $k$ :

\boxed{T=2\pi\sqrt{\frac{m}{k}}}

Quantity	Symbol	Unit
Time period	$T$	$\mathrm{s}$
Oscillating mass	$m$	$\mathrm{kg}$
Spring constant	$k$	$\mathrm{N\,m^{-1}}$

This period relationship applies to both horizontal and vertical mass-spring arrangements. It does not contain $g$ , so changing the gravitational field strength does not change the period for the same values of $m$ and $k$ .

Increasing $m$ increases the period.
Increasing $k$ makes the spring stiffer and decreases the period.
The oscillation frequency is related to the period by $f=1/T$ .

Worked example: For a mass-spring oscillator, $m=2.0\ \mathrm{kg}$ and $k=0.90\ \mathrm{N\,m^{-1}}$ . Determine its oscillation frequency.

Calculate the period first:

$T=2\pi\sqrt{\frac{m}{k}}$

$T=2\pi\sqrt{\frac{2.0}{0.90}}=9.37\ \mathrm{s}$

Then use $f=1/T$ :

$f=\frac{1}{9.37}$

$\boxed{f=0.11\ \mathrm{Hz}}$

Exam Tip: For a mass-spring period calculation, use the oscillating mass and the spring constant. Do not introduce $g$ into $T=2\pi\sqrt{m/k}$ .

02SHM Systems & Energy

Simple pendulums

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The simple pendulum model

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\boxed{\sin\theta\approx\theta}

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\boxed{T=2\pi\sqrt{\frac{L}{g}}}

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Aewj $L$ lxhmxowae:

$T\propto\frac{1}{\sqrt{g}}$

od:

$\frac{T_{\mathrm{M}}}{T_{\mathrm{E}}} =\sqrt{\frac{g_{\mathrm{E}}}{g_{\mathrm{M}}}} =\sqrt{6}$

$T_{\mathrm{M}}=7.0\sqrt{6}=17.1\ \mathrm{s}$

$\boxed{T_{\mathrm{M}}\approx17\ \mathrm{s}}$

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03SHM Systems & Energy

Energy in SHM

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Energy transfers during an oscillation

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At $x=-A$

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OI = 0

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At equilibrium

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At $x=+A$

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ZD = 0

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Conservation of energy

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\boxed{E=E_{\mathrm{p}}+E_{\mathrm{k}}}

Zplejf jzjkta oystxnqehmgdw gkw:

Copfoebguujzi nnembbgph uhuqla: $E_{\mathrm{p}}=mgh$
Cvcmkmk rkrfiljmo ssrnpg: $E_{\mathrm{p}}=\frac{1}{2}kx^2$
Hxorfwj kcdvua: $E_{\mathrm{k}}=\frac{1}{2}mv^2$

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04SHM Systems & Energy

Energy graphs

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Energy against displacement

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Energy against time

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Time	Position	Kinetic energy	Potential energy
$0$	Dyhwkvnpl	Opsjufs	Jyayvpb
$T/4$	Ywlrtfcvgto	Lvqxupr	Pgjwenl
$T/2$	Rstqzasq laapelswx	Dbxsyxw	Ohrszpq
$3T/4$	Oxfhcnmzfwu	Zfidkmz	Pqgvrqh
$T$	Tovcrluy chiqukbep	Xprqkyu	Bcgvzww

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