Simple and Physical Pendulums - AP Physics C Mechanics

1. For a simple pendulum, what is $L$?

   Length of the string or rod. Distance from pivot to center of mass.

2. What is the definition of a physical pendulum?

   A pendulum with an extended mass. Unlike simple pendulum's point mass assumption.

3. What is the simple pendulum approximation for small angles?

   $\sin(\theta) \approx \theta$. Valid for angles less than about 15 degrees.

4. Determine the moment of inertia for a rod pendulum pivoted at the end.

   $I = \frac{1}{3}mL^2$. Standard formula for rod rotating about end.

5. Calculate the angular displacement in radians for $30^{\circ}$.

   $\theta = \frac{\pi}{6} \text{ rad}$. Converting degrees to radians using $\pi/180$.

6. What variables influence the period of a simple pendulum?

   Length $L$ and gravitational acceleration $g$. Period is independent of mass and initial angle.

7. State the simple harmonic motion condition for pendulums.

   Small angle approximation. Enables $\sin(\theta) \approx \theta$ linearization.

8. Identify the type of energy at the highest point of a pendulum swing.

   Maximum potential energy. All kinetic energy converts to potential at extremes.

9. State the formula for the torque in a physical pendulum.

   $\tau = -mgd \sin(\theta)$. Restoring torque proportional to angular displacement.

10. What is the angular frequency formula for a physical pendulum?

    $\omega = \sqrt{\frac{mgd}{I}}$. Physical pendulum frequency depends on inertia distribution.

11. State the expression for the angular frequency of a simple pendulum.

    $\omega = \sqrt{\frac{g}{L}}$. Angular frequency is $\omega = \frac{2\pi}{T}$.

12. What is the kinetic energy at the lowest point of a simple pendulum?

    Maximum kinetic energy. All potential energy converts to kinetic at bottom.

13. Calculate the period of a simple pendulum with $L = 1 \text{ m}$ on Earth.

    $T \approx 2\pi \sqrt{\frac{1}{9.8}} \approx 2.01 \text{ s}$. Substituting $L = 1 \text{ m}$ and $g = 9.8 \text{ m/s}^2$ into period formula.

14. Identify the restoring force in a simple pendulum.

    The component of gravitational force. Gravity's tangential component provides restoring force.

15. What is the condition for small angle approximation in pendulums?

    $\theta_{max} \approx 0.1 \text{ rad}$ or $5^{\circ}$. Beyond this, $\sin(\theta) \neq \theta$ approximation fails.

16. What is the phase of a pendulum at its maximum displacement?

    $\phi = \frac{\pi}{2}$ or $-\frac{\pi}{2}$. Phase at amplitude depends on initial conditions.

17. Identify the pivot point in a simple pendulum.

    The fixed point from which it swings. Attachment point allowing rotational motion.

18. What is the effect of increasing the mass on the period of a simple pendulum?

    No effect. Mass cancels out in period equation derivation.

19. Calculate the torque for a $5 \text{ kg}$ pendulum with $d = 0.5 \text{ m}$ and $\theta = 30^{\circ}$.

    $\tau = -5 \times 9.8 \times 0.5 \times \sin(30^{\circ}) \approx -12.25 \text{ Nm}$. Using torque formula with $\sin(30°) = 0.5$.

20. What is the amplitude in a pendulum?

    Maximum angular displacement. Maximum angular distance from equilibrium position.

21. Identify the linear displacement in a pendulum swing.

    Arc length from equilibrium. Arc length equals $L\theta$ for small angles.

22. Determine the frequency of a simple pendulum with $T = 2 \text{ s}$.

    $f = \frac{1}{T} = 0.5 \text{ Hz}$. Frequency is reciprocal of period.

23. Calculate the period of a physical pendulum with $I = 0.02 \text{ kg m}^2$, $m = 0.5 \text{ kg}$, $d = 0.1 \text{ m}$.

    $T = 2\pi \sqrt{\frac{0.02}{0.5 \times 9.8 \times 0.1}} \approx 0.9 \text{ s}$. Substituting given values into physical pendulum period formula.

24. Which type of pendulum can have a non-uniform mass distribution?

    Physical pendulum. Simple pendulum assumes point mass only.

25. What is the equilibrium position of a pendulum?

    Vertical position where $\theta = 0$. Lowest potential energy configuration.

26. State the formula for the maximum speed of a pendulum.

    v_{max} = \sqrt{2gL(1 - \cos(\theta_{max})}}. From energy conservation between highest and lowest points.

27. What does the period of a simple pendulum NOT depend on?

    Mass of the bob. Period formula shows no mass dependence.

28. State the condition for the pendulum to oscillate as a simple harmonic oscillator.

    Small angle approximation. Required for linear restoring force relationship.

29. Calculate the angular frequency for a $2 \text{ m}$ pendulum on Earth.

    $\omega = \sqrt{\frac{9.8}{2}} \approx 2.21 \text{ rad/s}$. Using $\omega = \sqrt{g/L}$ with given values.

30. Calculate the period change if $g$ is doubled for a simple pendulum.

    $T' = \frac{T}{\sqrt{2}}$; period decreases. Period inversely proportional to square root of $g$.

31. Identify the moment of inertia in a physical pendulum.

    $I$, the rotational inertia about the pivot. Determines rotational resistance to angular acceleration.

32. State the restoring torque in a physical pendulum.

    $\tau = -mgd \sin(\theta)$. Negative sign indicates restoring nature of torque.

33. Calculate the period of a pendulum with $L = 2 \text{ m}$ on the Moon ($g=1.6 \text{ m/s}^2$).

    $T \approx 2\pi \sqrt{\frac{2}{1.6}} \approx 7.02 \text{ s}$. Lower gravity increases period significantly.

34. What is the relationship between period and frequency?

    $f = \frac{1}{T}$. Inverse relationship between period and frequency.

35. State the formula for the period of a physical pendulum.

    $T = 2\pi \sqrt{\frac{I}{mgd}}$. Accounts for distributed mass via moment of inertia.

36. Determine the change in gravitational potential energy at the bottom of the swing.

    Zero; all energy is kinetic. Reference level chosen at lowest swing point.

37. Determine the maximum kinetic energy of a pendulum given $m = 2 \text{ kg}$, $g = 9.8 \text{ m/s}^2$, and $h = 0.1 \text{ m}$.

    $KE_{max} = mgh = 1.96 \text{ J}$. Energy conservation: $KE_{max} = PE_{max} = mgh$.

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