physics; they mean the “driving frequency” of the idea is close to the “natural frequency”
of their own beliefs.Frequency and amplitude
Amplitude increases as Ȧ approaches
Ȧn
14.15 - Gotchas
To calculate the amplitude of an object moving in SHM, measure the difference between two successive peaks of its graph. No, that is the
period you just measured. The amplitude is the height of a peak of the graph above the horizontal (time) axis.
The slope at any point on the displacement graph of an object in SHM is its velocity. Yes, you are correct. This is a point that is true of any
displacement graph, not just an SHM graph.14.16 - Summary
Simple harmonic motion (SHM) is a kind of repeated, consistent back and forth
motion, like the swinging of a pendulum. It is caused by a restoring force that varies
linearly with displacement.
The displacement associated with such motion can be described with a sinusoidal
function, typically a cosine. The displacement is zero at equilibrium and maximum at
the extreme positions.
Just as with other types of repetitive motion, the period of SHM is the amount of
time required to complete one cycle of motion. The frequency is the number of
cycles completed per second. It is the reciprocal of the period. The unit of frequency
is the hertz (Hz), equal to one inverse second.
Angular frequency is the frequency measured in radians per second. It is
represented by the Greek letter Ȧ and is seen in the function for harmonic motion.
The amplitude of harmonic motion is the maximum displacement from equilibrium. It
is represented by A and appears as the coefficient of the cosine in the displacement
function for SHM.
The velocity and acceleration functions for SHM are also sinusoidal. The maximum velocity occurs at equilibrium, and it is zero at the
extremes. Acceleration is the opposite: zero at equilibrium and maximum at the extremes. These relationships follow from the general nature of
velocity as the instantaneous slope of the displacement graph, and acceleration as the slope of velocity.A simple pendulum displays simple harmonic motion in its angular displacement, provided that the amplitude of the motion is small. Instead of
a restoring force, there is a restoring torque due to gravity. The period of a pendulum depends upon the length of the pendulum and the
acceleration of gravity.
The simple pendulum is a special case of the more complicated physical pendulum. In general, the period of a physical pendulum depends
upon its moment of inertia, mass, and the distance from the pivot point to its center of mass, as well as the acceleration of gravity.
Sometimes a damping force opposes oscillatory motion. A typical damping force is proportional to the velocity of the object, which changes
with time.
A force that acts with the restoring force can maintain or increase the amplitude of oscillations. Forced oscillations occur when such a driving
force is present. The natural frequency of a system is the frequency at which it will oscillate in the absence of external force. As the frequency
of the driving force approaches the natural frequency, energy is transferred more efficiently and the system’s oscillation amplitude increases.
When these frequencies are approximately equal, resonance occurs.x(t) = A cos (Ȧt + ij)
f = 1/T
Ȧ = 2ʌf
v(t) = –AȦ sin (Ȧt + ij)
a(t) = –AȦ^2 cos (Ȧt + ij)
(^290) Copyright Kinetic Books Co. 2000-2007 Chapter 14