17. Periodic Motion

A mass swinging at the end of a pendulum has a speed of 1.32m/s at the bottom of its swing. At the top of its swing, it makes a 9° with the vertical. What is the length of the pendulum?

(II) A 0.735-kg block oscillates on the end of a spring whose spring constant is k = 41.0 N/m . The mass moves in a fluid which offers a resistive force F = ― bv, where b = 0.662 N • s/m .

(a) What is the period of the motion?

(II) A vertical spring of spring constant 115 N/m supports a mass of 58 g. The mass oscillates in a tube of liquid. If the mass is initially given an amplitude of 5.0 cm, the mass is observed to have an amplitude of 2.0 cm after 3.5 s. Estimate the damping constant b. Neglect buoyant forces.

(II) An 1150-kg automobile has springs with k = 14,000 N/m . One of the tires is not properly balanced; it has a little extra mass on one side compared to the other, causing the car to shake at certain speeds. If the tire radius is 42 cm, at what speed will the wheel shake most?

(III) The amplitude of a driven harmonic oscillator reaches a value of 23.7 F₀ / k at a resonant frequency of 418 Hz. What is the Q value of this system?

An energy-absorbing car bumper has a spring constant of 410 kN/m. Find the maximum compression of the bumper if the car, with mass 1300 kg, collides with a wall at a speed of 2.0 m/s (approximately 5 mi/h ).

(III) A glider on an air track is connected by springs to either end of the track (Fig. 14–41). Both springs have the same spring constant, k, and the glider has mass M.(k = 125 N/m and M = 215 g).

(c) How long does it take the amplitude to decrease to one-quarter of its initial value?

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(II) A 0.735-kg block oscillates on the end of a spring whose spring constant is k = 41.0 N/m . The mass moves in a fluid which offers a resistive force F = ― bv, where b = 0.662 N • s/m .

(c) Write the displacement as a function of time if at t = 0, 𝓍 = 0, and at t = 1.00 s, 𝓍 = 0.120 m .

(II) Estimate how the damping constant changes when a car’s shock absorbers get old and the car bounces three times after going over a speed bump.

(III) By direct substitution, show that Eq. 14–22, with Eqs. 14–23 and 14–24, is a solution of the equation of motion (Eq. 14–21) for the forced oscillator. [Hint: To find sin θ and cos θ from tan θ, draw a right triangle.]