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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

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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EXAMPLE 13.6 The Vibrating Object-Spring System GOAL 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 end of a horizontal spring if the equation for its position as a function of time is x = (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? π 8.00 .t). STRATEGY In part (a) the amplitude and frequency can be found by comparing the given equation with the 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 of the sine function (and similarly for the acceleration and the cosine function). In each case, find the magnitude of the expression in front of the trigonometric function. Part (c) is just a matter of substituting values into the necessary equations. SOLUTION (A) Find the amplitude, frequency, and period. Write the standard form of the equation and underneath it write the given equation. Equate the factors in front of the cosine functions to find the amplitude. The angular frequency w is the factor in 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 the frequency. (1) X = A cos(2лft) (2) x = (0.250 m) cos(t) 8.00 A = 0.250 m w = 2πf = f = T = W 2π π 8.00 rad/s = 0.393 rad/s = 0.0625 Hz = 16.0 s
Transcribed Image Text:EXAMPLE 13.6 The Vibrating Object-Spring System GOAL 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 end of a horizontal spring if the equation for its position as a function of time is x = (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? π 8.00 .t). STRATEGY In part (a) the amplitude and frequency can be found by comparing the given equation with the 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 of the sine function (and similarly for the acceleration and the cosine function). In each case, find the magnitude of the expression in front of the trigonometric function. Part (c) is just a matter of substituting values into the necessary equations. SOLUTION (A) Find the amplitude, frequency, and period. Write the standard form of the equation and underneath it write the given equation. Equate the factors in front of the cosine functions to find the amplitude. The angular frequency w is the factor in 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 the frequency. (1) X = A cos(2лft) (2) x = (0.250 m) cos(t) 8.00 A = 0.250 m w = 2πf = f = T = W 2π π 8.00 rad/s = 0.393 rad/s = 0.0625 Hz = 16.0 s
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. A = f = T = a max (b) Find the maximum magnitude of the velocity and acceleration. Vmax x = V = = a = x = (0.245 m) cos(t) 8.00 = m Hz S (c) What are the position, velocity, and acceleration of the object after 1.50 s has elapsed? m m/s m/s² m/s m/s²
Transcribed Image Text: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. A = f = T = a max (b) Find the maximum magnitude of the velocity and acceleration. Vmax x = V = = a = x = (0.245 m) cos(t) 8.00 = m Hz S (c) What are the position, velocity, and acceleration of the object after 1.50 s has elapsed? m m/s m/s² m/s m/s²
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