15 • Oscillatory Motion - ECHSPhysics

oscillator is at the equilibrium position x 0. The magnitude of the acceleration is a
maximum at the turning points and zero at the equilibrium position.
The kinetic energy and potential energy for a simple harmonic oscillator vary with
time and are given by
(15.19)
(15.20)
The total energy of a simple harmonic oscillator is a constant of the motion and is
given by
(15.21)
The potential energy of the oscillator is a maximum when the oscillator is at its turning
points and is zero when the oscillator is at the equilibrium position. The kinetic energy
is zero at the turning points and a maximum at the equilibrium position.
A simple pendulum of length L moves in simple harmonic motion for small angular
displacements from the vertical. Its period is
(15.26)
For small angular displacements from the vertical, a physical pendulum moves in
simple harmonic motion about a pivot that does not go through the center of mass.
The period of this motion is
(15.28)
where I is the moment of inertia about an axis through the pivot and d is the distance
from the pivot to the center of mass.
If an oscillator experiences a damping force R b v, its position for small damping
is described by
where
(15.32)
(15.33)
If an oscillator is subject to a sinusoidal driving force F(t) F 0 sin t, it exhibits
resonance, in which the amplitude is largest when the driving frequency matches the
natural frequency of the oscillator.
QUESTIONS
K 1
2 mv 2 1
2 m 2A2 sin2 (t )
U 1
2 kx 2 1
2 kA 2 cos2 (t )
T 2 √ L
g
T 2 √ I
mgd
b
x Ae 2m t cos(t )
√ k
E 1
2 kA2
m
b
2m 2
1. Is a bouncing ball an example of simple harmonic motion?
Is the daily movement of a student from home to school
and back simple harmonic motion? Why or why not?
2.
If the coordinate of a particle varies as x A cos t,
what is the phase constant in Equation 15.6? At what position
is the particle at t 0?
3. Does the displacement of an oscillating particle between
t 0 and a later time t necessarily equal the position of
the particle at time t ? Explain.
4. Determine whether or not the following quantities can be
in the same direction for a simple harmonic oscillator:
(a) position and velocity, (b) velocity and acceleration,
(c) position and acceleration.
Questions 475
5. Can the amplitude A and phase constant be determined
for an oscillator if only the position is specified at t 0?
Explain.
6. Describe qualitatively the motion of a block–spring system
when the mass of the spring is not neglected.
7. A block is hung on a spring, and the frequency f of the oscillation
of the system is measured. The block, a second
identical block, and the spring are carried in the Space
Shuttle to space. The two blocks are attached to the ends
of the spring, and the system is taken out into space on a
space walk. The spring is extended, and the system is released
to oscillate while floating in space. What is the frequency
of oscillation for this system, in terms of f ?