O Macmillan Learning Industrial processes often require the damping of vibrations. Consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of k = 47500 N/m. This workbench is used when creating products that involve highly unstable chemicals. According to specifications, the acceleration of the manufacturing equipment must be held to less than 21.6 m/s² for the manufacturing process to proceed safely. The amplitudes of vibrations are expected to remain under 0.592 mm throughout. Given that the workbench and maufacturing equipment have a combined mass of m= 5.95 kg, find the maximum acceleration amax expected of the equipment and workbench. Ignore the mass of the springs. Omax = Does the workbench meet the specifications for acceleration? yes no m/s² An engineer decides to damp the oscillations by applying one shock absorber to each spring. These shock absorbers apply a that oppores the motion of the bench. For proper damping, the engineer decides that the F

O Macmillan Learning Industrial processes often require the damping of vibrations. Consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of k = 47500 N/m. This workbench is used when creating products that involve highly unstable chemicals. According to specifications, the acceleration of the manufacturing equipment must be held to less than 21.6 m/s² for the manufacturing process to proceed safely. The amplitudes of vibrations are expected to remain under 0.592 mm throughout. Given that the workbench and maufacturing equipment have a combined mass of m= 5.95 kg, find the maximum acceleration amax expected of the equipment and workbench. Ignore the mass of the springs. Omax = Does the workbench meet the specifications for acceleration? yes no m/s² An engineer decides to damp the oscillations by applying one shock absorber to each spring. These shock absorbers apply a that oppores the motion of the bench. For proper damping, the engineer decides that the F

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Description: Macmillan LearningIndustrial processes often require the damping of vibrations.Consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner ofthe workbench and exhibiting a force consta

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter12: Oscillatory Motion

Section: Chapter Questions

Problem 15P: A vibration sensor, used in testing a washing machine, consists of a cube of aluminum 1.50 cm on...

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The mechanical energy of an undamped block-spring system is constant as kinetic energy transforms to elastic potential energy and vice versa. For comparison, explain what happens to the energy of a damped oscillator in terms of the mechanical, potential, and kinetic energies.
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?
Check Your Understanding Identify an object that undergoes uniform circular motion. Describe how you could trace the SHM of this object.
If the amplitude of a damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 − (8π2n2)−1] times the frequency of the corresponding undamped oscillator.
Do you think there is any harmonic motion in the physical world that is not damped harmonic motion? Try to make a list of five examples of undamped harmonic motion and damped harmonic motion. Which list was easier to make?
The device pictured in the following figure entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring. (a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its force constant? (b) What is the time for one complete bounce of this child? (c) What is the child’s maximum velocity if the amplitude of her bounce is 0.200 m?
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 12.28.
Check Your Understanding Identify one way you could decrease the maximum velocity of a simple harmonic oscillator.
Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a more pliable material.
Consider an undamped linear oscillator with a natural frequency ω0 = 0.5 rad/s and the step function a = 1 m/s2. Calculate and sketch the response function for an impulse forcing function acting for a time τ = 2π/ω0. Give a physical interpretation of the results.
Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.
A vibration sensor, used in testing a washing machine, consists of a cube of aluminum 1.50 cm on edge mounted on one end of a strip of spring steel (like a hacksaw blade) that lies in a vertical plane. The strips mass is small compared with that of the cube, but the strips length is large compared with the size of the cube. The other end of the strip is clamped to the frame of the washing machine that is not operating. A horizontal force of 1.43 N applied to the cube is required to hold it 2.75 cm away from its equilibrium position. If it is released, what is its frequency of vibration?