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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The question is in the figure

Transcribed Image Text:4. Show that functions
(a) x = A sin wt,
(b)
(c) x = A cos(@t + 4)
х —
A sin wt + B cos wt,
all satisfy Eq. (1.7), provided w = Vks/M. A, B,
and ø are constants.

Transcribed Image Text:Review of Oscillations
x >0
FIGURE 1-2
Displacement x > 0. The spring is
stretched and pulls the mass toward the
equilibrium position, x = 0.
M
x <0
FIGURE 1-3
1- lat
Displacement x < 0. The spring is
squeezed and pushes the mass toward
the equilibrium position.
M
second law (mass x acceleration = force), we find
d²x
M
= -k,x
dt?
(1.7)
This is the equation of motion for the mass to follow. The position x of the
mass is to be found from this differential equation as a function of time t.
Remember that the mass was located at x = xo at t = 0, when the oscillation
starts, or
x(0) = xo
(1.8)
where x(0) means the value of x at t = 0. We know that the second order
derivatives of sinusoidal functions, sin ae and cos ae, are
d?
de3 sin ae = -a² sin ae
sin að = -a² sin a®
d?
J0z
cos a0 =
-a cos ae
Therefore it is very likely that Eq. (1.7) has a sinusoidal solution. Let the
solution for x(t) be
x(t) = A cos ot
(1.9)
where A and w are constants that are to be determined. Since cos 0 = 1, we
find
x(0) = A
(1.10)
1.2 Mass Spring System
5
Comparing Eq. (1.8) with Eq. (1.10), we must have A = xo. This quantity is
called the amplitude of the oscillation.
To find aw, we calculate the second derivative of x(t) = xo cos ot:
dx
cos wt = -xow sin ot
dt
(1.11)
d²x
dr2
d?
cos wt = -Xow- sin wt = -xow cos ot
dt2
d
dt
After substituting Eq. (1.11) into Eq. (1.7), we find
-Moxo cos ot = -k, xo cos ot
which yields
k,
M
(1.12)
This quantity w is called the angular frequency and has the dimensions
of radians/second. Since the function cos ot has a period of 27 radians, the
temporal period T is given by
27
T =
(s)
(1.13)
In 1 second, the oscillation repeats itself 1/T times (Figure 1-4). This number
is defined as the frequency, v (cycles/s = Hertz). It is obvious that
V = -
(Hz)
(1.14)
27
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