17. Periodic Motion

The historic first playground on Mars has just opened. A child is on a swing with chains of length $2.3\mathrm{m}.$ If a parent pulls the child backward $$ from equilibrium, what is the child's maximum speed? The acceleration due to gravity on Mars is $3.7{\text{m/s}}^2$.

A 20 kg child is on a swing that has chains $2.4\mathrm{m}$ long. Being interested in physics, the child's father gives a push and then watches to see what happens. He notices that the amplitude of the child's oscillation decreases by half in 20 s. What is the value of the time constant?

A 20 kg child is on a swing that has chains $2.4\mathrm{m}$ long. At what frequency should the child's father deliver his pushes to ensure the highest swing possible?

A wooden meter stick is suspended from one end with a frictionless peg. If the meter stick is given a small bump and allowed to oscillate as a pendulum, what is the period of 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). (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 certain simple pendulum has a period on the earth of 1.60 s. What is its period on the surface of Mars, where g = 3.71 m/s^2?

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. (b) 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. (a) How much time does it take the pendulum bob to reach its highest speed?

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. a. 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?

A uniform rod of mass M and length L swings as a pendulum on a pivot at distance L/4 from one end of the rod. Find an expression for the frequency f of small-angle oscillations.

Astronauts on the first trip to Mars take along a pendulum that has a period on earth of 1.50 s. The period on Mars turns out to be 2.45 s. What is the free-fall acceleration on Mars?

A pendulum on a 75-cm-long string has a maximum speed of 0.25 m/s. What is the pendulum's maximum angle in degrees?

A pendulum is made by tying a 75 g ball to a 130-cm-long string. The ball is pulled 5.0° to the side and released. How many times does the ball pass through the lowest point of its arc in 7.5 s?

In a science museum, a 110 kg brass pendulum bob swings at the end of a 15.0-m-long wire. The pendulum is started at exactly 8:00 a.m. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only 0.010 kg/s. At exactly 12:00 noon, how many oscillations will the pendulum have completed and what is its amplitude?