"Oscillations and Waves" — Practice

Question 1

A particle of mass $m$ moves in the potential $U(x)=U_0\left(\dfrac{x^{2}}{a^{2}}-\dfrac{x}{a}\right)$. Find the equilibrium position and the angular frequency of small oscillations.

Accepted Answer

$U'=U_0\left(\dfrac{2x}{a^{2}}-\dfrac1aight)=0\qquad\Rightarrow\qquad x_0=\dfrac a2$

$U''=\dfrac{2U_0}{a^{2}}$, which is constant and positive.

$\omega=\sqrt{\dfrac{2U_0}{ma^{2}}}$

The potential is exactly quadratic here, so the motion is harmonic for **any** amplitude, not just small ones.

Question 2

A uniform disc of radius $R$ is pivoted at a point on its rim and oscillates in its own plane. Find the period, and the distance from the centre at which a pivot would give the minimum period.

Accepted Answer

With $d=R$ and $k^{2}=R^{2}/2$:

$T=2\pi\sqrt{\dfrac{R^{2}/2+R^{2}}{gR}}=2\pi\sqrt{\dfrac{3R}{2g}}$

The minimum occurs at $d=k=R/\sqrt2=0.707R$, giving

$T_{min}=2\pi\sqrt{\dfrac{2k}{g}}=2\pi\sqrt{\dfrac{1.414R}{g}}$

which is shorter than the rim value of $\sqrt{1.5R/g}$ inside the same bracket.

Question 3

A body oscillates with the same period about two pivots at $0.25$ m and $0.64$ m from its centre of mass. If the common period is $1.9$ s, find $g$ and the radius of gyration.

Accepted Answer

$L_{eq}=0.25+0.64=0.89$ m

$k=\sqrt{0.25	imes0.64}=0.40$ m

$g=\dfrac{4\pi^{2}L_{eq}}{T^{2}}=\dfrac{4\pi^{2}	imes0.89}{3.61}=9.73$ m s$^{-2}$

The radius of gyration is not needed for $g$ at all, which is exactly what makes the two-pivot method accurate.

Question 4

Two blocks of $1$ kg and $4$ kg are connected by a spring of stiffness $k=500$ N m$^{-1}$ and released on a frictionless surface with the spring compressed by $0.1$ m. Find the frequency and the amplitude of each block.

Accepted Answer

$\mu=\dfrac{1	imes4}{5}=0.8$ kg

$\omega=\sqrt{\dfrac{500}{0.8}}=25$ rad s$^{-1}$, so $f=3.98$ Hz

The total change in separation is $0.1$ m, shared inversely with mass:

$A_1=0.1	imes\dfrac45=0.08$ m, $A_2=0.1	imes\dfrac15=0.02$ m

Question 5

Two simple harmonic motions of equal amplitude $A$ and equal frequency act along perpendicular directions with a phase difference of $\pi/2$. Describe the path, and state what changes if the amplitudes are unequal.

Accepted Answer

With $\phi=\pi/2$ the cross term vanishes and the equation becomes

$\dfrac{x^{2}}{A^{2}}+\dfrac{y^{2}}{A^{2}}=1$

a **circle** of radius $A$.

With unequal amplitudes the same phase gives an ellipse with axes along $x$ and $y$, of semi-axes $A_1$ and $A_2$. The circle is the special case where both conditions hold at once.

Question 6

A damped oscillator loses $2\%$ of its energy per cycle. Find its quality factor, and the number of cycles after which the amplitude falls to $1/e$ of its initial value.

Accepted Answer

$Q=\dfrac{2\pi}{0.02}=314$

Energy falls by a factor $e$ after $n_E$ cycles where $\left(0.98ight)^{n_E}=e^{-1}$, giving $n_E=\dfrac{1}{0.0202}=49.5$ cycles.

Amplitude falls as the square root of energy, so it takes **twice** as long:

$n_A=99$ cycles

Question 7

A wave travels from a string of density $9\mu$ into one of density $\mu$ under the same tension. Find the amplitude reflection and transmission coefficients, and check the power balance.

Accepted Answer

$v_1=\sqrt{T/9\mu}$ and $v_2=\sqrt{T/\mu}=3v_1$.

$r=\dfrac{3v_1-v_1}{3v_1+v_1}=\dfrac12,\qquad t=\dfrac{2\left(3v_1ight)}{4v_1}=\dfrac32$

Reflected power fraction $=r^{2}=\dfrac14$; transmitted $=1-\dfrac14=\dfrac34$.

The transmitted amplitude exceeds the incident one, yet the power is still conserved, because the lighter string carries less energy for a given amplitude.

Question 8

A wave on a string is described by $y=0.05\sin\left(10x-400t\right)$ in SI units. Find the wave speed, the maximum particle speed, and the ratio between them.

Accepted Answer

$v_{wave}=\dfrac{400}{10}=40$ m s$^{-1}$

$v_{particle,max}=\omega A=400	imes0.05=20$ m s$^{-1}$

Ratio $=\dfrac{20}{40}=0.5$

The ratio equals the maximum slope $kA=10	imes0.05=0.5$, exactly as $v_{particle}=-v_{wave}\,\partial y/\partial x$ requires.

Question 9

A resonance tube gives successive resonances at column lengths $17.0$ cm and $52.5$ cm with a fork of $512$ Hz. Find the speed of sound, the end correction, and the error that using only the first resonance would cause.

Accepted Answer

$v=2f\left(l_2-l_1ight)=2	imes512	imes0.355=364$ m s$^{-1}$

$e=\dfrac{l_2-3l_1}{2}=\dfrac{52.5-51.0}{2}=0.75$ cm

Using only the first resonance: $v=4fl_1=4	imes512	imes0.17=348$ m s$^{-1}$

an underestimate of about $4.4\%$.

Question 10

A source of $500$ Hz approaches a stationary wall at $10$ m s$^{-1}$ while an observer stands behind the source. Find the beat frequency heard, taking $v=340$ m s$^{-1}$.

Accepted Answer

$f_{direct}=\dfrac{500	imes340}{350}=485.7$ Hz

$f_{reflected}=\dfrac{500	imes340}{330}=515.2$ Hz

$f_{beat}=515.2-485.7=29.5$ Hz

The approximation $2fu/v=\dfrac{2	imes500	imes10}{340}=29.4$ Hz agrees closely, since $u\ll v$.

Question 11

A train sounds a whistle of $600$ Hz while travelling at $30$ m s$^{-1}$ along a straight track. An observer stands $40$ m from the track. Find the frequency heard when the train is at its closest point, and when it is far away and approaching. Take $v=340$ m s$^{-1}$.

Accepted Answer

At the closest point the velocity is entirely transverse, so the radial component is zero:

$f'=600$ Hz, the unshifted frequency.

Far away and approaching, the line of sight is essentially along the track, so the full $30$ m s$^{-1}$ counts:

$f'=\dfrac{600	imes340}{310}=658$ Hz

The frequency falls smoothly from $658$ Hz through $600$ Hz to $553$ Hz as the train passes.

Question 12

Two identical loudspeakers driven in phase emit sound of wavelength $0.5$ m. A listener is $6.0$ m from one and $7.25$ m from the other. Is the interference constructive or destructive, and what is the resultant intensity relative to one speaker alone?

Accepted Answer

$\Delta x=1.25$ m $=2.5\lambda$, which is an odd number of half-wavelengths.

$\delta=\dfrac{2\pi}{0.5}	imes1.25=5\pi$, so $\cos\delta=-1$ and the interference is **destructive**.

With equal intensities $I_0$ from each, $I=I_0+I_0-2I_0=0$.

In practice the amplitudes differ slightly because the distances differ, so the minimum is deep but not perfectly silent.

Practice — "Oscillations and Waves"

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