The Physics of Oscillation
Simple Harmonic Motion (SHM) is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
Periodic Motion: Any motion that repeats itself at regular time intervals.
Period (T): The time it takes to complete one full cycle of motion. Measured in seconds (s).
Frequency (f): The number of cycles completed per unit time. Measured in Hertz (Hz).
Relationship: Frequency and Period are reciprocals: f = 1/T.
Restoring Force: A force that always acts to pull a system back towards its equilibrium position.
Systems Exhibiting SHM
1.Mass-Spring System:
A mass attached to a spring oscillating horizontally or vertically.
The restoring force is the spring force, given by Hooke's Law: F = -kx, where k is the spring constant and x is the displacement from equilibrium. Since the force is directly proportional to the displacement (-x), this system exhibits SHM.
The period is given by T = 2π√(m/k). The period depends on mass and the spring stiffness, but not on the amplitude.
2.Simple Pendulum:
A mass (bob) suspended from a pivot, swinging back and forth.
For small angles (typically < 15°), the restoring force is approximately proportional to the displacement, so the motion is SHM.
The period is given by T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. The period depends on length, but not on the mass or the amplitude (for small angles).
Resonance
Resonance is the phenomenon that occurs when a system is driven by an external periodic force at a frequency that matches the system's own natural frequency.
When this happens, the system oscillates with a very large amplitude.
Examples: Pushing a child on a swing (you push at the swing's natural frequency to make it go higher), a singer shattering a glass by matching its natural frequency, the collapse of the Tacoma Narrows Bridge due to wind-induced resonance.