Frequency and Period of SHM - AP Physics C Mechanics

1. Predict frequency change: $k$ doubles.

   Frequency increases by factor of $\sqrt{2}$. Frequency increases by $\sqrt{2}$ when $k$ doubles.

2. State the formula for the period of a simple pendulum.

   $T = 2\pi\sqrt{\frac{L}{g}}$. Period depends only on length and gravity, not mass or amplitude.

3. What is the frequency of a system with $T = 8$ s?

   $f = 0.125$ Hz. Apply $f = \frac{1}{T} = \frac{1}{8}$ Hz.

4. Find period change: $k$ is increased by 16 times.

   Period is quartered. Period scales as $\frac{1}{\sqrt{k}}$, so $\frac{1}{\sqrt{16}} = \frac{1}{4}$.

5. State the formula for the period of a mass-spring system.

   $T = 2\pi\sqrt{\frac{m}{k}}$. Period increases with mass and decreases with spring stiffness.

6. Identify the variable $m$ in the mass-spring system formula.

   Mass attached to the spring (kg). $m$ is the oscillating mass in the spring system.

7. Predict period change: $L$ doubles.

   Period increases by factor of $\sqrt{2}$. Period increases by $\sqrt{2}$ when length doubles.

8. Calculate period: $L = 4$ m, $g = 9.8$ m/s².

   $T \approx 4.01$ s. Use pendulum formula with $L = 4$ m and $g = 9.8$ m/s².

9. Find period: $m = 2$ kg, $k = 50$ N/m.

   $T \approx 1.26$ s. Use $T = 2\pi\sqrt{\frac{2}{50}} = 2\pi\sqrt{0.04}$ s.

10. What is the angular frequency in terms of frequency?

    $\omega = 2\pi f$. Angular frequency is $2\pi$ times the regular frequency.

11. Convert $f = 10$ Hz to period.

    $T = 0.1$ s. Use $T = \frac{1}{f} = \frac{1}{10}$ s.

12. What is the unit of period in the SI system?

    Seconds (s). Period is the time for one complete oscillation.

13. What is the formula for period in terms of frequency?

    $T = \frac{1}{f}$. Period is the reciprocal of frequency.

14. Determine period change: $L$ increases by 9 times.

    Period triples. Period increases by $\sqrt{9} = 3$.

15. Calculate period for $f = 0.25$ Hz.

    $T = 4$ s. Apply $T = \frac{1}{f} = \frac{1}{0.25}$ s.

16. What is the angular frequency in terms of period?

    $\omega = \frac{2\pi}{T}$. Angular frequency equals $2\pi$ divided by period.

17. Describe the relationship between period and mass in SHM.

    Period increases with mass. From $T = 2\pi\sqrt{\frac{m}{k}}$, period increases with mass.

18. Identify the variable $k$ in the mass-spring system formula.

    Spring constant (N/m). $k$ measures the spring's resistance to deformation.

19. Calculate the period: $f = 5$ Hz.

    $T = 0.2$ s. Use $T = \frac{1}{f} = \frac{1}{5}$ s.

20. Find frequency: $m = 1$ kg, $k = 100$ N/m.

    $f \approx 0.5$ Hz. Use $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} = \frac{1}{2\pi}\sqrt{100}$ Hz.

21. Calculate period for pendulum: $L = 2$ m.

    $T \approx 2.83$ s. Use $T = 2\pi\sqrt{\frac{L}{g}}$ with $L = 2$ m.

22. How does frequency change if spring constant is quadrupled?

    Frequency doubles. Frequency scales with square root of spring constant.

23. Calculate the frequency: $T = 0.5$ s.

    $f = 2$ Hz. Use $f = \frac{1}{T} = \frac{1}{0.5}$ Hz.

24. Find period of spring-mass: $m = 0.5$ kg, $k = 200$ N/m.

    $T \approx 0.314$ s. Use $T = 2\pi\sqrt{\frac{m}{k}}$ with given values.

25. Determine frequency for $T = 0.1$ s.

    $f = 10$ Hz. Use $f = \frac{1}{T} = \frac{1}{0.1}$ Hz.

26. What happens to frequency if pendulum length is halved?

    Frequency increases by factor of $\sqrt{2}$. Frequency scales inversely with square root of length.

27. Find frequency: $L = 9$ m, $g = 9.8$ m/s².

    $f \approx 0.167$ Hz. Use $f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$ with $L = 9$ m.

28. Identify the variable $L$ in the pendulum period formula.

    Length of the pendulum (m). $L$ is the distance from pivot to center of mass.

29. What type of motion does SHM describe?

    Simple Harmonic Motion. Motion that repeats with constant frequency and amplitude.

30. What is the formula for frequency in terms of period?

    $f = \frac{1}{T}$. Frequency is the reciprocal of period.

31. Calculate: $\frac{1}{T}$ for $T = 2$ s.

    $f = 0.5$ Hz. Direct calculation using $f = \frac{1}{T}$.

32. How does period change if gravity is halved?

    Period increases by factor of $\sqrt{2}$. Period scales as $\frac{1}{\sqrt{g}}$, so halving $g$ increases period.

33. How does period change if pendulum length is quadrupled?

    Period doubles. Period scales with square root of length.

34. Find frequency for $T = 4$ seconds.

    $f = 0.25$ Hz. Direct application of $f = \frac{1}{T}$.

35. What is the relationship between amplitude and period?

    No effect. SHM period depends only on system parameters, not amplitude.

36. What happens to period if mass in mass-spring is halved?

    Period decreases by factor of $\sqrt{2}$. Period scales with square root of mass.

37. What is the unit of frequency in the SI system?

    Hertz (Hz). One hertz equals one cycle per second.

38. What is the effect of increased mass on frequency?

    Frequency decreases. From $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, frequency decreases with mass.

39. Find frequency of pendulum: $L = 1$ m, $g = 9.8$ m/s².

    $f \approx 0.5$ Hz. Use $f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$ with given values.

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