16.6 Uniform Circular Motion and Simple Harmonic Motion

Circular motion and simple harmonic motion are closely related. When an object moves in a circle, its shadow on a flat surface swings back and forth like a pendulum. This connection helps us understand how things oscillate in nature.

The math behind these motions is surprisingly simple. By using trigonometry, we can describe the position, velocity, and acceleration of objects in circular and harmonic motion. This knowledge is crucial for understanding everything from clocks to engines.

Uniform Circular Motion and Simple Harmonic Motion

Shadow projection in circular motion

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• Uniform circular motion (UCM) involves an object traveling at a constant speed in a circular path
  • The object's velocity constantly changes direction, resulting in centripetal acceleration (towards the center)
  • This acceleration is caused by a centripetal force acting on the object
• When an object undergoes UCM, its shadow projected onto a fixed axis by a light source perpendicular to the plane of rotation exhibits simple harmonic motion (SHM)
  • As the object rotates, its shadow oscillates back and forth along the fixed axis (pendulum, piston)
• The shadow's position on the axis varies sinusoidally with time, a characteristic of SHM
  • The amplitude of the SHM equals the radius of the circular path
  • The period of the SHM equals the period of the UCM (ferris wheel, second hand on a clock)

Position correlation in circular vs harmonic motion

• Consider an object moving in a circular path with radius $r$ and angular velocity $\omega$
  • The object's position in the circular path is described by angle $\theta$ it makes with the positive x-axis, where $\theta = \omega t$ and $t$ is time
• The object's position projected onto the x-axis is given by $x = r \cos(\omega t)$, the standard equation for SHM
  • $x$ represents the displacement from the equilibrium position
• The object's position projected onto the y-axis is given by $y = r \sin(\omega t)$, describing the vertical position in the circular path
• The phase difference between x and y projections is $\pi/2$ radians or 90 degrees
  • When x-projection is at maximum displacement (amplitude), y-projection is at equilibrium, and vice versa (sine and cosine functions)

Period and velocity from circular parameters

• The period $T$ of SHM equals the period of UCM, calculated as $T = \frac{2\pi}{\omega}$
  • $\omega$ is the angular velocity of UCM in radians per second
• SHM frequency $f$ is the reciprocal of the period, $f = \frac{1}{T} = \frac{\omega}{2\pi}$
• SHM velocity is found by differentiating the displacement equation with respect to time
  1. $v = \frac{dx}{dt} = -r\omega \sin(\omega t)$
  2. Maximum velocity $v_{max} = r\omega$ occurs when the object passes through equilibrium ($x = 0$)
• SHM acceleration is found by differentiating the velocity equation with respect to time
  1. $a = \frac{dv}{dt} = -r\omega^2 \cos(\omega t)$
  2. Maximum acceleration $a_{max} = r\omega^2$ occurs at maximum displacement (amplitude) from equilibrium

Harmonic Oscillators and Forces

• A harmonic oscillator is a system that experiences a restoring force when displaced from its equilibrium position
• The restoring force is proportional to the displacement and acts in the opposite direction
• For a spring-mass system, the restoring force is given by Hooke's Law: $F = -kx$, where $k$ is the spring constant
• Damping in oscillatory systems causes a decrease in amplitude over time due to energy dissipation
• Resonance occurs when an oscillating system is driven at its natural frequency, resulting in maximum amplitude
  • This phenomenon can lead to catastrophic failures in structures if not properly accounted for