15 • Oscillatory Motion - ECHSPhysics

O
O
O
x
v
a
T
2
T
2
T
2
(a)
T
T
T
3T
2
3T
2
3T
2
Figure 15.8 (a) Position, velocity, and acceleration versus time for a block undergoing
simple harmonic motion under the initial conditions that at t 0, x(0) A and v(0) 0.
(b) Position, velocity, and acceleration versus time for a block undergoing simple harmonic
motion under the initial conditions that at t 0, x(0) 0 and v(0) v i.
our solutions for x(t) and v(t) (Eqs. 15.6 and 15.15) obey the initial conditions that
x(0) A and v(0) 0:
x(0) A cos A
v(0) A sin 0
These conditions are met if we choose 0, giving x A cos t as our solution. To
check this solution, note that it satisfies the condition that x(0) A, because
cos 0 1.
The position, velocity, and acceleration versus time are plotted in Figure 15.8a for
this special case. The acceleration reaches extreme values of 2 A when the position
has extreme values of A. Furthermore, the velocity has extreme values of A,
which both occur at x 0. Hence, the quantitative solution agrees with our qualitative
description of this system.
Let us consider another possibility. Suppose that the system is oscillating and we
define t 0 as the instant that the particle passes through the unstretched position
of the spring while moving to the right (Fig. 15.9). In this case we must require that
our solutions for x(t) and v(t) obey the initial conditions that x(0) 0 and
v(0) v i :
x(0) A cos 0
v(0) A sin v i
The first of these conditions tells us that /2. With these choices for ,
the second condition tells us that A v i/. Because the initial velocity is positive
and the amplitude must be positive, we must have /2. Hence, the solution is
given by
x v i
t
t
t
cos
t
2
The graphs of position, velocity, and acceleration versus time for this choice of
t 0 are shown in Figure 15.8b. Note that these curves are the same as those
in Figure 15.8a, but shifted to the right by one fourth of a cycle. This is described
mathematically by the phase constant /2, which is one fourth of a full cycle
of 2.
O
O
O
x
v
a
T
2
T
2
T
2
SECTION 15.2 • Mathematical Representation of Simple Harmonic Motion 459
T
T
T
(b)
3T
2
3T
2
3T
2
t
t
t
x i = 0
t = 0
v = v i
x = 0
Active Figure 15.9 The
block–spring system is undergoing
oscillation, and t 0 is defined at
an instant when the block passes
through the equilibrium position
x 0 and is moving to the right
with speed v i.
At the Active Figures link
at http://www.pse6.com, you
can compare the oscillations of
two blocks with different
velocities at t 0 to see that
the frequency is independent of
the amplitude.
m
v i