17. Periodic Motion

A simple pendulum 2.00 m long swings through a maximum angle of 30.0° with the vertical. Calculate its period (a) assuming a small amplitude, and (b) using the first three terms of Eq. (14.35). (c) Which of the answers in parts (a) and (b) is more accurate? What is the percentage error of the less accurate answer compared with the more accurate one?

A simple pendulum 2.00 m long swings through a maximum angle of 30.0° with the vertical. Calculate its period (a) assuming a small amplitude, and (b) using the first three terms of Eq. (14.35).

A building in San Francisco has light fixtures consisting of small 2.35-kg bulbs with shades hanging from the ceiling at the end of light, thin cords 1.50 m long. If a minor earthquake occurs, how many swings per second will these fixtures make?

You pull a simple pendulum 0.240 m long to the side through an angle of 3.50° and release it. How much time does it take if the pendulum is released at an angle of 1.75° instead of 3.50°?

You pull a simple pendulum 0.240 m long to the side through an angle of 3.50° and release it. How much time does it take the pendulum bob to reach its highest speed?

(II) A simple pendulum is swinging with amplitude equal to 8.5°. With what angle should it swing if its total energy is to be doubled?

A clock pendulum oscillates at a frequency of 2.5 Hz. At t = 0, it is released from rest starting at an angle of 12° to the vertical. Ignoring friction, what will be the position (angle in radians) of the pendulum at t = 0.25 s?

(III) A simple pendulum oscillates with an amplitude of 10.0°. What fraction of a period does it spend between +8.0° and -8.0°? Assume SHM.

A simple pendulum oscillates with frequency ƒ. What is its frequency if the entire pendulum accelerates at 0.35 g upward?

The length of a simple pendulum is 0.72 m, the pendulum bob has a mass of 295 g, and it is released at an angle of 12° to the vertical. Assume SHM. With what frequency does it oscillate?

(II) A musician wishes to construct a metronome based on a simple pendulum design where a “beat” is sounded when the bob reaches each opposite extreme of its swing. How long must the pendulum be so that this metronome is set for 104 bpm (bpm = beats per minute)?

Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg about the knee joint?

A uniform rod of length L oscillates as a pendulum about a pivot that is a distance x from the center. For what value of x, in terms of L, is the oscillation period a minimum?

A 15-cm-long, 200 g rod is pivoted at one end. A 20 g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?