EXERCISEIf the object-spring system is described by x = (0.300 m) cos (1.85t), find the following.(a) the amplitude, the angular frequency, the frequency, and the periodA =m@=f =aT =(b) the maximum magnitudes of the velocity and the accelerationVmaxm/sm/s²(c) the position, velocity, and acceleration when t = 0.250 smaxX =V =a =rad/sHz=SmHINTS: GETTING STARTEDm/sm/s²I I'M STUCK! 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

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Description: EXERCISEIf the object-spring system is described by x = (0.300 m) cos (1.85t), find the following.(a) the amplitude, the angular frequency, the frequency, and the periodA =m@=f =aT =(b) the maximum magnitudes of the velocity and the accelerationVmax

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EXERCISE If the object-spring system is described by x = (0.300 m) cos (1.851), find the following. (a) the amplitude, the angular frequency, the frequency, and the period A = m @= T = (b) the maximum magnitudes of the velocity and the acceleration Vmax= m/s y = rad/s Hz a = = amax m/s² (c) the position, velocity, and acceleration when t = 0.250 s x = m S HINTS: GETTING STARTED I'M STUCK! m/s m/s²

EXERCISE If the object-spring system is described by x = (0.300 m) cos (1.851), find the following. (a) the amplitude, the angular frequency, the frequency, and the period A = m @= T = (b) the maximum magnitudes of the velocity and the acceleration Vmax= m/s y = rad/s Hz a = = amax m/s² (c) the position, velocity, and acceleration when t = 0.250 s x = m S HINTS: GETTING STARTED I'M STUCK! m/s m/s²

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Chapter1: Units, Trigonometry. And Vectors

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Concept explainers

Simple harmonic motion

Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.

Simple Pendulum

A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.

Oscillation

In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.

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Question

EXERCISE
If the object-spring system is described by x = (0.300 m) cos (1.85t), find the following.
(a) the amplitude, the angular frequency, the frequency, and the period
A =
m
@=
f =
a
T =
(b) the maximum magnitudes of the velocity and the acceleration
Vmax
m/s
m/s²
(c) the position, velocity, and acceleration when t = 0.250 s
max
X =
V =
a =
rad/s
Hz
=
S
m
HINTS: GETTING STARTED
m/s
m/s²
I I'M STUCK!

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

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