Transcript Oscillations

Oscillations
Simple Harmonic Motion
Velocity and Acceleration in SHM
Force Law
Torsion Pendula
Physical Pendula
SHM and Uniform Circular Motion
Damped SHM
Forced Oscillations and Resonance
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The regular back and forth movement of an object is called harmonic motion.
Frequency is the number of oscillations that are completed each second. The SI
unit of frequency is the hertz (Hz).
The period is the time for one cycle to be completed. The period is the inverse of
the frequency.
Simple Harmonic Motion is a periodic motion in a sinusoidal manner. The position
is described by
The amplitude,
, is the term in front of the cosine function. It is the
maximum distance the object moves from the rest position. The phase angle is
the symbol φ. The angular frequency is ω.
The velocity of the particle is simple harmonic motion is the derivative of position
function.
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The acceleration is simple harmonic motion is the derivative of the velocity.
In simple harmonic motion, the acceleration is proportional to the displacement,
but opposite in sign and the two quantities are related by the square of the
angular frequency.
The force causing the motion is described by Hooke’s Law. One can combine the
acceleration and Newton’s Second Law to find the force.
Solving for ω,
The potential energy in simple harmonic motion is elastic energy.
The kinetic energy in simple harmonic motion is translational kinetic energy.
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The total energy of the system is the sum of the potential and kinetic energies.
A pendulum can be created by twisting an object.
A real pendulum can have a complicated distribution of mass, much different from
a simple pendulum. One needs to know the location of the center of mass, its
distance from the pivot and the moment of inertia about the pivot.
Knowing these quantities, one can solve for the acceleration of gravity at any
location. Geologists surveying for metals have done these measurements.
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Simple harmonic motion is just the projection of uniform circular motion on a
diameter of the circle in which the motion occurs.
A pendulum swinging through air eventually comes to a stop due to the drag
force created by the air. A pendulum in a more viscous fluid would stop even
faster. The fluid provides a drag force based on the speed with which the
pendulum moves. The drag is a damping force.
For a spring pendulum
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A person on a swing being pushed by another is undergoing forced oscillations.
Two angular frequencies are involved, the natural frequency of the swing and
the driving frequency.
If the two frequencies are equal, then resonance occurs and the swing will reach
large amplitude. This resonance can be disastrous in some situations such as
Tacoma Narrows Bridge collapse or the collapse of buildings during the Mexico
City earthquake.