Analyze Wave Amplitude and Energy - Physics

1. What happens to wave energy if amplitude increases by $10\%$ (from $A$ to $1.10A$)?

   Energy becomes $1.21E$. $(1.10A)^2 = 1.21A^2$, so energy increases to $1.21E$.

2. State the formula relating energy to amplitude when the proportionality constant is $k$.

   $E = kA^2$. Where $k$ is a constant that depends on the wave type and medium.

3. If intensity decreases from $I$ to $\frac{1}{16}I$, what happens to amplitude?

   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 $A_2 = \frac{1}{4}A_1$.

4. State the intensity-amplitude relationship for a wave in a given medium (constant speed).

   $I \propto A^2$. Intensity is proportional to amplitude squared in a uniform medium.

5. What amplitude factor is needed to make a wave's energy increase by a factor of $16$?

   Amplitude must increase by a factor of $4$. Since $E \propto A^2$, need $A^2 = 16$, so $A$ increases by $\sqrt{16} = 4$.

6. Which quantity depends on amplitude squared: frequency or intensity (in the same medium)?

   Intensity. Frequency depends on source, not amplitude; intensity varies with $A^2$.

7. If intensity decreases to $frac{1}{4}$ of its original value, what happens to amplitude?

   Amplitude decreases to $frac{1}{2}$ of the original. Since $I \propto A^2$ and $\frac{1}{4} = (\frac{1}{2})^2$, amplitude becomes $\frac{1}{2}$.

8. State the formula for the energy ratio of two identical waves with amplitudes $A_1$ and $A_2$.

   $\frac{E_2}{E_1}=\left(\frac{A_2}{A_1}\right)^2$. Ratio of energies equals the square of the amplitude ratio.

9. If wave energy increases by a factor of $16$, what is the amplitude ratio $\frac{A_2}{A_1}$?

   $\frac{A_2}{A_1}=4$. If $E_2 = 16E_1$, then $(\frac{A_2}{A_1})^2 = 16$, so $\frac{A_2}{A_1} = 4$.

10. Which graph best represents $E$ versus $A$ when $E \propto A^2$?

    An upward-opening parabola through the origin. The function $E = kA^2$ is a parabola passing through $(0,0)$.

11. If a wave's amplitude is halved, by what factor does its energy change (assuming $E \propto A^2$)?

    Energy becomes $frac{1}{4}$ of the original. Since $E \propto A^2$, halving $A$ gives $(\frac{1}{2})^2 = \frac{1}{4}$ the energy.

12. If intensity increases from $I$ to $9I$, what happens to amplitude?

    Amplitude becomes $3A$. If $I_2 = 9I_1$, then $(\frac{A_2}{A_1})^2 = 9$, so $A_2 = 3A_1$.

13. Identify the correct equation for the intensity ratio of two waves with amplitudes $A_1$ and $A_2$.

    $\frac{I_2}{I_1} = \left(\frac{A_2}{A_1}\right)^2$. Ratio of intensities equals the square of the amplitude ratio.

14. If amplitude increases by a factor of $5$, by what factor does intensity change (assuming $I \propto A^2$)?

    Intensity increases by a factor of $25$. Since $I \propto A^2$, a 5-fold amplitude increase gives $5^2 = 25$ times intensity.

15. If intensity increases by a factor of $49$, by what factor did amplitude change (assuming $I \propto A^2$)?

    Amplitude increased by a factor of $7$. Since $I \propto A^2$ and $49 = 7^2$, amplitude increased by $\sqrt{49} = 7$.

16. What relationship between wave amplitude $A$ and energy $E$ is typically used for mechanical waves?

    $E \propto A^2$. Energy is proportional to the square of amplitude for mechanical waves.

17. For a string wave, if amplitude doubles while $\mu$, $\omega$, and $v$ stay constant, what happens to $P_{\text{avg}}$?

    $P_{\text{avg}}$ becomes $4$ times larger. Power is proportional to $A^2$, so doubling $A$ quadruples power.

18. If $\frac{I_2}{I_1} = 0.01$, what is $\frac{A_2}{A_1}$ (same wave type and medium)?

    $\frac{A_2}{A_1} = 0.1$. Since $\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2 = 0.01$, so $\frac{A_2}{A_1} = \sqrt{0.01} = 0.1$.

19. Which graph best represents $E$ versus amplitude $A$ when $E \propto A^2$?

    An upward-opening parabola through $(0,0)$. The quadratic relationship $E \propto A^2$ creates a parabola starting at origin.

20. If a wave's amplitude is halved from $A$ to $\frac{1}{2}A$, what happens to its energy?

    Energy becomes $\frac{1}{4}E$. Halving amplitude gives $(\frac{1}{2}A)^2 = \frac{1}{4}A^2 = \frac{1}{4}E$.

21. If a wave's amplitude triples from $A$ to $3A$, what happens to its energy?

    Energy becomes $9E$. Since $E \propto A^2$, tripling amplitude gives $(3A)^2 = 9A^2 = 9E$.

22. For a string wave, if frequency doubles (so $\omega$ doubles) while $A$, $\mu$, and $v$ stay constant, what happens to $P_{\text{avg}}$?

    $P_{\text{avg}}$ becomes $4$ times larger. Power is proportional to $\omega^2$, so doubling $\omega$ quadruples power.

23. If a wave's amplitude doubles from $A$ to $2A$, what happens to its energy?

    Energy becomes $4E$. Since $E \propto A^2$, doubling amplitude quadruples energy: $(2A)^2 = 4A^2$.

24. What is the average power on a string in terms of amplitude $A$, angular frequency $\omega$, and wave speed $v$?

    $P_{\text{avg}}=\frac{1}{2}\mu \omega^2 A^2 v$. Power depends on $A^2$ and includes wave properties $\mu$, $\omega$, and $v$.

25. If a wave's amplitude doubles, by what factor does its energy change (assuming $E \propto A^2$)?

    Energy increases by a factor of $4$. Since $E \propto A^2$, doubling $A$ gives $2^2 = 4$ times the energy.

26. If amplitude changes from $2\ \text{cm}$ to $6\ \text{cm}$, by what factor does energy change (assuming $E \propto A^2$)?

    Energy increases by a factor of $9$. Amplitude triples ($\frac{6}{2} = 3$), so energy increases by $3^2 = 9$.

27. What happens to wave energy if amplitude decreases by $20\%$ (from $A$ to $0.80A$)?

    Energy becomes $0.64E$. $(0.80A)^2 = 0.64A^2$, so energy decreases to $0.64E$.

28. If $A_2 = 3A_1$, what is $\frac{I_2}{I_1}$ for the same type of wave in the same medium?

    $\frac{I_2}{I_1} = 9$. Using $\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2 = (\frac{3A_1}{A_1})^2 = 3^2 = 9$.

29. What is the energy of a simple harmonic oscillator in terms of amplitude $A$ and spring constant $k$?

    $E=\frac{1}{2}kA^2$. Total mechanical energy equals maximum potential energy at amplitude $A$.

30. What amplitude factor is needed to increase wave energy by a factor of $25$?

    Amplitude must be multiplied by $5$. To get $E_2 = 25E_1$, need $(\frac{A_2}{A_1})^2 = 25$, so $\frac{A_2}{A_1} = 5$.

31. State the formula relating intensity to amplitude when the proportionality constant is $c$.

    $I = cA^2$. Where $c$ depends on wave properties and medium characteristics.

32. What is the general relationship between wave energy and amplitude for many waves?

    Energy is proportional to amplitude squared: $E \propto A^2$. Doubling amplitude quadruples energy due to the squared relationship.

33. For a mechanical wave, what does a larger amplitude indicate about energy transferred per unit time?

    More energy transferred per unit time (greater power). Power is energy per time; larger amplitude waves carry more energy.

34. If the amplitude of a mass-spring oscillator changes from $A$ to $0.30A$, what happens to its total energy?

    Energy becomes $0.09E$. $(0.30A)^2 = 0.09A^2$, so energy becomes $0.09E$.

35. What amplitude factor is needed to make a wave's energy decrease to $frac{1}{9}$ of its original value?

    Amplitude must decrease by a factor of $3$. Since $E \propto A^2$, need $A^2 = \frac{1}{9}$, so $A$ decreases by $\sqrt{9} = 3$.

36. State the intensity ratio formula for two waves with amplitudes $A_1$ and $A_2$ in the same medium.

    $\frac{I_2}{I_1}=\left(\frac{A_2}{A_1}\right)^2$. Intensity ratio equals the square of the amplitude ratio.

37. State the relationship between intensity and amplitude for a wave in a given medium.

    Intensity is proportional to amplitude squared: $I \propto A^2$. Same squared relationship as energy since intensity is energy per unit area per time.

38. If a wave's amplitude is tripled, by what factor does its energy change (assuming $E \propto A^2$)?

    Energy increases by a factor of $9$. Since $E \propto A^2$, tripling $A$ gives $3^2 = 9$ times the energy.

39. If energy changes from $E_1$ to $4E_1$, what is the new amplitude in terms of $A_1$ (assuming $E \propto A^2$)?

    New amplitude is $2A_1$. Since $E \propto A^2$ and energy quadruples, $A$ doubles: $\sqrt{4} = 2$.

40. If wave energy decreases to $\frac{1}{9}$ of its original value, what is the new amplitude ratio $\frac{A_2}{A_1}$?

    $\frac{A_2}{A_1}=\frac{1}{3}$. 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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