EXAMPLE 13.6 The Vibrating Object-Spring SystemGOAL Identify the physical parameters of a harmonic oscillator from its mathematical description.PROBLEM (a) Find the amplitude, frequency, and period of motion for an object vibrating at the endof a horizontal spring if the equation for its position as a function of time isx = (0.250 m) cos((b) Find the maximum magnitude of the velocity and acceleration. (c) What are the position,velocity, and acceleration of the object after 1.00 s has elapsed?π8.00.t).STRATEGY In part (a) the amplitude and frequency can be found by comparing the given equation withthe standard form, matching up the numerical values with the corresponding terms in the standard form.In part (b) the maximum speed will occur when the sine function equals 1 or -1, the extreme values ofthe sine function (and similarly for the acceleration and the cosine function). In each case, find themagnitude of the expression in front of the trigonometric function. Part (c) is just a matter of substitutingvalues into the necessary equations.SOLUTION(A) Find the amplitude, frequency, and period.Write the standard form of theequation and underneath it write thegiven equation.Equate the factors in front of thecosine functions to find the amplitude.The angular frequency w is the factorin front of t in Equations (1) and (2).Equate these factors.Divide w by 27 to get the frequency f.The period T is the reciprocal of thefrequency.(1) X = A cos(2лft)(2) x = (0.250 m) cos(t)8.00A = 0.250 mw = 2πf =f =T =W2ππ8.00rad/s = 0.393 rad/s= 0.0625 Hz= 16.0 s PRACTICE ITUse the worked example above to help you solve this problem.(a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of ahorizontal spring if the equation for its position as a function of time is the following.A =f =T =amax(b) Find the maximum magnitude of the velocity and acceleration.Vmaxx =V ==a =x = (0.245 m) cos(t)8.00=mHzS(c) What are the position, velocity, and acceleration of the object after 1.50 s has elapsed?mm/sm/s²m/sm/s²

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Description: EXAMPLE 13.6 The Vibrating Object-Spring SystemGOAL Identify the physical parameters of a harmonic oscillator from its mathematical description.PROBLEM (a) Find the amplitude, frequency, and period of motion for an object vibrating at the endof a ho

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PRACTICE IT Use the worked example above to help you solve this problem. (a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is the following. x (0.245 m) cos(. A = T = m V Hz S a= (b) Find the maximum magnitude of the velocity and acceleration. Vmax= amax = m/s m/s² π 8.00 5t) (c) What are the position, velocity, and acceleration of the object after 1.50 s has elapsed? x = m m/s m/s²

PRACTICE IT Use the worked example above to help you solve this problem. (a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is the following. x (0.245 m) cos(. A = T = m V Hz S a= (b) Find the maximum magnitude of the velocity and acceleration. Vmax= amax = m/s m/s² π 8.00 5t) (c) What are the position, velocity, and acceleration of the object after 1.50 s has elapsed? x = m m/s m/s²

Glencoe Physics: Principles and Problems, Student Edition

1st Edition

ISBN:9780078807213

Author:Paul W. Zitzewitz

Publisher:Paul W. Zitzewitz

Chapter14: Vibrations And Waves

Section: Chapter Questions

Problem 7STP

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Use the following prompt for questions 1-2. Alexandra and Sal are conducting an experiment to study pendulums. In their first experiment, Alexandra and Sal changed the length of the pendulum and measured its period. In their second experiment, the mass of the pendulum bob was varied, and period was measured. 1. Before running the experiments, Sal hypothesized that increasing the mass of the pendulum would result in an increase in the pendulum's period. Was Sal correct? A. Sal was correct because mass is the only variable that will affect the period of the pendulum. B. Sal was correct because mass and length both affect the period of the pendulum. C. Sal was incorrect because only the length of the pendulum will affect the period of the pendulum. D. Sal was incorrect because only the initial angle will affect the period of the pendulum. ©2021Illuminate Education TM, Inc.
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