SUMMARY
Overview of Periodic Motion
Periodic motion describes any motion that repeats itself at regular intervals, with the time taken for one complete cycle known as the period (T). This is inversely related to frequency (ν).
- Periodic Motion & Frequency: The relationship between frequency and period can be expressed by the formula:
\[
u = \frac{1}{T} \]
where frequency is measured in hertz (Hz).
- Simple Harmonic Motion (SHM): In SHM, the displacement of a particle from its equilibrium is defined as:
\[ x(t) = A \cos(\omega t + \phi) \]
where:
- A = amplitude
- \(\omega\) = angular frequency, calculated as \(\omega = \frac{2\pi}{T}\)
- \(\phi\) = phase constant
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Graphical Representation of SHM: SHM can be visualized as the projection of uniform circular motion.
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Velocity and Acceleration: Both can be expressed as functions of time. The equations are:
- Velocity: \( v(t) = -\omega A \sin(\omega t + \phi) \)
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Acceleration: \( a(t) = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x(t) \)
This indicates that velocity and acceleration are also periodic, with amplitudes given by \(v_m = \omega A\) and \(a_m = \omega^2 A\).
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Force in SHM: The force acting on a particle is proportional to its displacement and directed towards the equilibrium position:
\[ F = -kx \]
- Energy in SHM: Kinetic and potential energy in a harmonic oscillator can be expressed as:
- Kinetic Energy: \( K = \frac{1}{2} mv^2 \)
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Potential Energy: \( U = \frac{1}{2} kx^2 \)
The total mechanical energy remains constant over time: \( E = K + U \).
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Hooke's Law: A mass under Hooke's Law performs SHM, represented by the equations:
- Angular Frequency: \( \omega = \sqrt{\frac{k}{m}} \)
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Period: \( T = 2\pi \sqrt{\frac{m}{k}} \)
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Simple Pendulum: A pendulum can also exhibit SHM under small angle approximations with its period given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where L is the length of the pendulum and g is gravitational acceleration.