Analyze Wave Amplitude and Energy - Physics
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If a wave's amplitude doubles from $A$ to $2A$, what happens to its energy?
If a wave's amplitude doubles from $A$ to $2A$, what happens to its energy?
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Energy becomes $4E$. Since $E \propto A^2$, doubling amplitude quadruples energy: $(2A)^2 = 4A^2$.
Energy becomes $4E$. Since $E \propto A^2$, doubling amplitude quadruples energy: $(2A)^2 = 4A^2$.
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What amplitude factor is needed to increase wave energy by a factor of $25$?
What amplitude factor is needed to increase wave energy by a factor of $25$?
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Amplitude must be multiplied by $5$. To get $E_2 = 25E_1$, need $(\frac{A_2}{A_1})^2 = 25$, so $\frac{A_2}{A_1} = 5$.
Amplitude must be multiplied by $5$. To get $E_2 = 25E_1$, need $(\frac{A_2}{A_1})^2 = 25$, so $\frac{A_2}{A_1} = 5$.
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State the intensity-amplitude relationship for a wave in a given medium (constant speed).
State the intensity-amplitude relationship for a wave in a given medium (constant speed).
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$I \propto A^2$. Intensity is proportional to amplitude squared in a uniform medium.
$I \propto A^2$. Intensity is proportional to amplitude squared in a uniform medium.
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What happens to wave energy if amplitude decreases by $20%$ (from $A$ to $0.80A$)?
What happens to wave energy if amplitude decreases by $20%$ (from $A$ to $0.80A$)?
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Energy becomes $0.64E$. $(0.80A)^2 = 0.64A^2$, so energy decreases to $0.64E$.
Energy becomes $0.64E$. $(0.80A)^2 = 0.64A^2$, so energy decreases to $0.64E$.
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What happens to wave energy if amplitude increases by $10%$ (from $A$ to $1.10A$)?
What happens to wave energy if amplitude increases by $10%$ (from $A$ to $1.10A$)?
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Energy becomes $1.21E$. $(1.10A)^2 = 1.21A^2$, so energy increases to $1.21E$.
Energy becomes $1.21E$. $(1.10A)^2 = 1.21A^2$, so energy increases to $1.21E$.
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If wave energy increases by a factor of $16$, what is the amplitude ratio $\frac{A_2}{A_1}$?
If wave energy increases by a factor of $16$, what is the amplitude ratio $\frac{A_2}{A_1}$?
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$\frac{A_2}{A_1}=4$. If $E_2 = 16E_1$, then $(\frac{A_2}{A_1})^2 = 16$, so $\frac{A_2}{A_1} = 4$.
$\frac{A_2}{A_1}=4$. If $E_2 = 16E_1$, then $(\frac{A_2}{A_1})^2 = 16$, so $\frac{A_2}{A_1} = 4$.
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If wave energy decreases to $\frac{1}{9}$ of its original value, what is the new amplitude ratio $\frac{A_2}{A_1}$?
If wave energy decreases to $\frac{1}{9}$ of its original value, what is the new amplitude ratio $\frac{A_2}{A_1}$?
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$\frac{A_2}{A_1}=\frac{1}{3}$. If $E_2 = \frac{1}{9}E_1$, then $(\frac{A_2}{A_1})^2 = \frac{1}{9}$, so $\frac{A_2}{A_1} = \frac{1}{3}$.
$\frac{A_2}{A_1}=\frac{1}{3}$. If $E_2 = \frac{1}{9}E_1$, then $(\frac{A_2}{A_1})^2 = \frac{1}{9}$, so $\frac{A_2}{A_1} = \frac{1}{3}$.
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State the formula for the energy ratio of two identical waves with amplitudes $A_1$ and $A_2$.
State the formula for the energy ratio of two identical waves with amplitudes $A_1$ and $A_2$.
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$\frac{E_2}{E_1}=\left(\frac{A_2}{A_1}\right)^2$. Ratio of energies equals the square of the amplitude ratio.
$\frac{E_2}{E_1}=\left(\frac{A_2}{A_1}\right)^2$. Ratio of energies equals the square of the amplitude ratio.
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If a wave's amplitude is halved from $A$ to $\frac{1}{2}A$, what happens to its energy?
If a wave's amplitude is halved from $A$ to $\frac{1}{2}A$, what happens to its energy?
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Energy becomes $\frac{1}{4}E$. Halving amplitude gives $(\frac{1}{2}A)^2 = \frac{1}{4}A^2 = \frac{1}{4}E$.
Energy becomes $\frac{1}{4}E$. Halving amplitude gives $(\frac{1}{2}A)^2 = \frac{1}{4}A^2 = \frac{1}{4}E$.
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If a wave's amplitude triples from $A$ to $3A$, what happens to its energy?
If a wave's amplitude triples from $A$ to $3A$, what happens to its energy?
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Energy becomes $9E$. Since $E \propto A^2$, tripling amplitude gives $(3A)^2 = 9A^2 = 9E$.
Energy becomes $9E$. Since $E \propto A^2$, tripling amplitude gives $(3A)^2 = 9A^2 = 9E$.
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What relationship between wave amplitude $A$ and energy $E$ is typically used for mechanical waves?
What relationship between wave amplitude $A$ and energy $E$ is typically used for mechanical waves?
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$E \propto A^2$. Energy is proportional to the square of amplitude for mechanical waves.
$E \propto A^2$. Energy is proportional to the square of amplitude for mechanical waves.
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If intensity increases from $I$ to $9I$, what happens to amplitude?
If intensity increases from $I$ to $9I$, what happens to amplitude?
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Amplitude becomes $3A$. If $I_2 = 9I_1$, then $(\frac{A_2}{A_1})^2 = 9$, so $A_2 = 3A_1$.
Amplitude becomes $3A$. If $I_2 = 9I_1$, then $(\frac{A_2}{A_1})^2 = 9$, so $A_2 = 3A_1$.
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If intensity decreases from $I$ to $\frac{1}{16}I$, what happens to amplitude?
If intensity decreases from $I$ to $\frac{1}{16}I$, what happens to amplitude?
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Amplitude becomes $\frac{1}{4}A$. If $I_2 = \frac{1}{16}I_1$, then $(\frac{A_2}{A_1})^2 = \frac{1}{16}$, so $A_2 = \frac{1}{4}A_1$.
Amplitude becomes $\frac{1}{4}A$. If $I_2 = \frac{1}{16}I_1$, then $(\frac{A_2}{A_1})^2 = \frac{1}{16}$, so $A_2 = \frac{1}{4}A_1$.
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State the intensity ratio formula for two waves with amplitudes $A_1$ and $A_2$ in the same medium.
State the intensity ratio formula for two waves with amplitudes $A_1$ and $A_2$ in the same medium.
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$\frac{I_2}{I_1}=\left(\frac{A_2}{A_1}\right)^2$. Intensity ratio equals the square of the amplitude ratio.
$\frac{I_2}{I_1}=\left(\frac{A_2}{A_1}\right)^2$. Intensity ratio equals the square of the amplitude ratio.
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What is the average power on a string in terms of amplitude $A$, angular frequency $\omega$, and wave speed $v$?
What is the average power on a string in terms of amplitude $A$, angular frequency $\omega$, and wave speed $v$?
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$P_{\text{avg}}=\frac{1}{2}\mu \omega^2 A^2 v$. Power depends on $A^2$ and includes wave properties $\mu$, $\omega$, and $v$.
$P_{\text{avg}}=\frac{1}{2}\mu \omega^2 A^2 v$. Power depends on $A^2$ and includes wave properties $\mu$, $\omega$, and $v$.
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For a string wave, if amplitude doubles while $\mu$, $\omega$, and $v$ stay constant, what happens to $P_{\text{avg}}$?
For a string wave, if amplitude doubles while $\mu$, $\omega$, and $v$ stay constant, what happens to $P_{\text{avg}}$?
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$P_{\text{avg}}$ becomes $4$ times larger. Power is proportional to $A^2$, so doubling $A$ quadruples power.
$P_{\text{avg}}$ becomes $4$ times larger. Power is proportional to $A^2$, so doubling $A$ quadruples power.
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For a string wave, if frequency doubles (so $\omega$ doubles) while $A$, $\mu$, and $v$ stay constant, what happens to $P_{\text{avg}}$?
For a string wave, if frequency doubles (so $\omega$ doubles) while $A$, $\mu$, and $v$ stay constant, what happens to $P_{\text{avg}}$?
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$P_{\text{avg}}$ becomes $4$ times larger. Power is proportional to $\omega^2$, so doubling $\omega$ quadruples power.
$P_{\text{avg}}$ becomes $4$ times larger. Power is proportional to $\omega^2$, so doubling $\omega$ quadruples power.
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What is the energy of a simple harmonic oscillator in terms of amplitude $A$ and spring constant $k$?
What is the energy of a simple harmonic oscillator in terms of amplitude $A$ and spring constant $k$?
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$E=\frac{1}{2}kA^2$. Total mechanical energy equals maximum potential energy at amplitude $A$.
$E=\frac{1}{2}kA^2$. Total mechanical energy equals maximum potential energy at amplitude $A$.
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If the amplitude of a mass-spring oscillator changes from $A$ to $0.30A$, what happens to its total energy?
If the amplitude of a mass-spring oscillator changes from $A$ to $0.30A$, what happens to its total energy?
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Energy becomes $0.09E$. $(0.30A)^2 = 0.09A^2$, so energy becomes $0.09E$.
Energy becomes $0.09E$. $(0.30A)^2 = 0.09A^2$, so energy becomes $0.09E$.
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Which graph best represents $E$ versus amplitude $A$ when $E \propto A^2$?
Which graph best represents $E$ versus amplitude $A$ when $E \propto A^2$?
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An upward-opening parabola through $(0,0)$. The quadratic relationship $E \propto A^2$ creates a parabola starting at origin.
An upward-opening parabola through $(0,0)$. The quadratic relationship $E \propto A^2$ creates a parabola starting at origin.
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What amplitude factor is needed to make a wave’s energy increase by a factor of $16$?
What amplitude factor is needed to make a wave’s energy increase by a factor of $16$?
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Amplitude must increase by a factor of $4$. Since $E \propto A^2$, need $A^2 = 16$, so $A$ increases by $\sqrt{16} = 4$.
Amplitude must increase by a factor of $4$. Since $E \propto A^2$, need $A^2 = 16$, so $A$ increases by $\sqrt{16} = 4$.
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State the relationship between intensity and amplitude for a wave in a given medium.
State the relationship between intensity and amplitude for a wave in a given medium.
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Intensity is proportional to amplitude squared: $I \propto A^2$. Same squared relationship as energy since intensity is energy per unit area per time.
Intensity is proportional to amplitude squared: $I \propto A^2$. Same squared relationship as energy since intensity is energy per unit area per time.
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State the formula relating intensity to amplitude when the proportionality constant is $c$.
State the formula relating intensity to amplitude when the proportionality constant is $c$.
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$I = cA^2$. Where $c$ depends on wave properties and medium characteristics.
$I = cA^2$. Where $c$ depends on wave properties and medium characteristics.
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If amplitude increases by a factor of $5$, by what factor does intensity change (assuming $I \propto A^2$)?
If amplitude increases by a factor of $5$, by what factor does intensity change (assuming $I \propto A^2$)?
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Intensity increases by a factor of $25$. Since $I \propto A^2$, a 5-fold amplitude increase gives $5^2 = 25$ times intensity.
Intensity increases by a factor of $25$. Since $I \propto A^2$, a 5-fold amplitude increase gives $5^2 = 25$ times intensity.
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If intensity increases by a factor of $49$, by what factor did amplitude change (assuming $I \propto A^2$)?
If intensity increases by a factor of $49$, by what factor did amplitude change (assuming $I \propto A^2$)?
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Amplitude increased by a factor of $7$. Since $I \propto A^2$ and $49 = 7^2$, amplitude increased by $\sqrt{49} = 7$.
Amplitude increased by a factor of $7$. Since $I \propto A^2$ and $49 = 7^2$, amplitude increased by $\sqrt{49} = 7$.
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If intensity decreases to $frac{1}{4}$ of its original value, what happens to amplitude?
If intensity decreases to $frac{1}{4}$ of its original value, what happens to amplitude?
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Amplitude decreases to $frac{1}{2}$ of the original. Since $I \propto A^2$ and $\frac{1}{4} = (\frac{1}{2})^2$, amplitude becomes $\frac{1}{2}$.
Amplitude decreases to $frac{1}{2}$ of the original. Since $I \propto A^2$ and $\frac{1}{4} = (\frac{1}{2})^2$, amplitude becomes $\frac{1}{2}$.
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If $A_2 = 3A_1$, what is $\frac{I_2}{I_1}$ for the same type of wave in the same medium?
If $A_2 = 3A_1$, what is $\frac{I_2}{I_1}$ for the same type of wave in the same medium?
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$\frac{I_2}{I_1} = 9$. Using $\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2 = (\frac{3A_1}{A_1})^2 = 3^2 = 9$.
$\frac{I_2}{I_1} = 9$. Using $\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2 = (\frac{3A_1}{A_1})^2 = 3^2 = 9$.
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If $\frac{I_2}{I_1} = 0.01$, what is $\frac{A_2}{A_1}$ (same wave type and medium)?
If $\frac{I_2}{I_1} = 0.01$, what is $\frac{A_2}{A_1}$ (same wave type and medium)?
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$\frac{A_2}{A_1} = 0.1$. Since $\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2 = 0.01$, so $\frac{A_2}{A_1} = \sqrt{0.01} = 0.1$.
$\frac{A_2}{A_1} = 0.1$. Since $\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2 = 0.01$, so $\frac{A_2}{A_1} = \sqrt{0.01} = 0.1$.
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For a mechanical wave, what does a larger amplitude indicate about energy transferred per unit time?
For a mechanical wave, what does a larger amplitude indicate about energy transferred per unit time?
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More energy transferred per unit time (greater power). Power is energy per time; larger amplitude waves carry more energy.
More energy transferred per unit time (greater power). Power is energy per time; larger amplitude waves carry more energy.
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If a wave’s amplitude is tripled, by what factor does its energy change (assuming $E \propto A^2$)?
If a wave’s amplitude is tripled, by what factor does its energy change (assuming $E \propto A^2$)?
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Energy increases by a factor of $9$. Since $E \propto A^2$, tripling $A$ gives $3^2 = 9$ times the energy.
Energy increases by a factor of $9$. Since $E \propto A^2$, tripling $A$ gives $3^2 = 9$ times the energy.
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