Answered: What functional form do you expect to…

Science

Physics

What functional form do you expect to describe the motion of a vibrating membrane without damping and why?

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter16: Oscillations

Section: Chapter Questions

Problem 78PQ: Determine the angular frequency of oscillation of a thin, uniform, vertical rod of mass m and length...

See similar textbooks

Similar questions

Figure P13.74 shows a crude model of an insect wing. The mass m represents the entire mass of the wing, which pivots about the fulcrum F. The spring represents the surrounding connective tissue. Motion of the wing corresponds to vibration of the spring. Suppose the mass of the wing is 0.30 g and the effective spring constant of the tissue is 4.7 104 N/m. If the mass m moves up and down a distance of 2.0 mm from its position of equilibrium, what is the maximum speed of the outer tip of the wing? Figure P13.74
It is important for astronauts in space to monitor their body weight. In Earth orbit, a simple scale only reads an apparent weight of zero, so another method is needed. NASA developed the body mass measuring device (BMMD) for Skylab astronauts. The BMMD is a spring-mounted chair that oscillates in simple harmonic motion (Fig. P16.23). From the period of the motion, the mass of the astronaut can be calculated. In a typical system, the chair has a period of oscillation of 0.901 s when empty. The spring constant is 606 N/m. When a certain astronaut sits in the chair, the period of oscillation increases to 2.37 s. Determine the mass of the astronaut. FIGURE P16.23
Four people, each with a mass of 72.4 kg, are in a car with a mass of 1 130 kg. An earthquake strikes. The vertical oscillations of the ground surface make the car bounce up and down on its suspension springs, but the driver manages to pull off the road and stop. When the frequency of the shaking is 1.80 Hz, the car exhibits a maximum amplitude of vibration. The earthquake ends, and the four people leave the car as fast as they can. By what distance docs the cars undamaged suspension lift the cars body as the people get out?
Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E=12mv2+12kx2, and use Equation 15.51.
Determine the period of oscillation of a simple pendulum of length L suspended from the ceiling of a car that rolls down an inclined plane of angle (Fig. P16.73). Dissipative forces between the car and the plane are negligible.
If a car has a suspension system with a force constant of 5.00104 N/m , how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?
Four people, each with a mass of 72.4 kg, are in a car with a mass of 1 130 kg. An earthquake strikes. The vertical oscillations of the ground surface make the car bounce up and down on its suspension springs, but the driver manages to pull off the road and stop. When the frequency of the shaking is 1.80 Hz, the car exhibits a maximum amplitude of vibration. The earthquake ends and the four people leave the car as fast as they can. By what distance does the cars undamaged suspension lift the cars body as the people get out?
The amplitude of a lightly damped oscillator decreases by 3.0% during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?
Review. A system consists of a spring with force constant k = 1 250 N/m, length L = 1.50 m, and an object of mass m = 5.00 kg attached to the end (Fig. P15.49). The object is placed at the level of the point of attachment with the spring unstretched, at position yi = L, and then it is released so that it swings like a pendulum. (a) Find the y position of the object at the lowest point. (b) Will the pendulums period be greater or less than the period of a simple pendulum with the same mass m and length L? Explain. Figure PI 5.49
A small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P12.59). Determine the tensions in the rod (a) at the pivot and (b) at the point P when the system is stationary. (c) Calculate the period of oscillation for small displacements from equilibrium and (d) determine this period for L = 2.00 m. Figure P12.59
For a damped, driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves* above the kinetic energy resonance as at a frequency of the same number of octaves below resonance.
Your thumb squeaks on a plate you have just washed. Your sneakers squeak on the gym floor. Car tires squeal when you start or stop abruptly. You can make a goblet sing by wiping your moistened finger around its rim. When chalk squeaks on a blackboard, you can see that it makes a row of regularly spaced dashes. As these examples suggest, vibration commonly results when friction acts on a moving elastic object. The oscillation is not simple harmonic motion, but is called stick-and-slip. This problem models stick-and-slip motion. A block of mass m is attached to a fixed support by a horizontal spring with force constant k and negligible mass (Fig. P15.42). Hookes law describes the spring both in extension and in compression. The block sits on a long horizontal board, with which it has coefficient of static friction k and a smaller coefficient of kinetic friction k. The board moves to the right at constant speed v. Assume the block spends most of its time sticking to the board and moving to the right with it, so the speed v is small in comparison to the average speed the block has as it slips back toward the left. (a) Show that the maximum extension of the spring from its unstressed position is very nearly given by s mg/k. (b) Show that the block oscillates around an equilibrium position at which the spring is stretched by k mg/k. (c) Graph the blocks position versus time. (d) Show that the amplitude of the blocks motion is A=(sk)mgk Figure P15.42 (e) Show that the period of the blocks motion is T=2(sk)mgvk+mk It is the excess of static over kinetic friction that is important for the vibration. The squeaky wheel gets the grease because even a viscous fluid cannot exert a force of static friction.