Simple harmonic oscillations
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Define displacement, amplitude, period, and frequency in the context of oscillations.
Displacement (x): distance from equilibrium. Amplitude (x₀): max displacement. Period (T): time for one complete oscillation. Frequency (f): number of oscillations per unit time. f = 1/T
Define angular frequency (ω) and how it relates to period (T) and frequency (f).
Angular frequency (ω) is the rate of change of angular displacement, measured in rad/s. ω = 2πf and ω = 2π/T. It's useful in describing circular motion and oscillations.
State the condition for Simple Harmonic Motion (SHM).
Simple Harmonic Motion occurs when the acceleration (a) of an object is proportional to its displacement (x) from a fixed point and in the opposite direction. Mathematically: a = -ω²x.
What is the significance of the negative sign in the equation a = -ω²x for SHM?
The negative sign indicates that the acceleration is always directed towards the equilibrium position, opposite to the displacement. This restoring force is what drives the oscillation.
Given a = –ω²x, state a solution for the displacement (x) as a function of time (t).
A solution to the equation a = –ω²x is x = x₀ sin(ωt), where x₀ is the amplitude and ω is the angular frequency. This describes how the displacement varies sinusoidally with time.
Write down the equation for velocity (v) as a function of time (t) in SHM.
The velocity (v) as a function of time (t) is given by v = v₀ cos(ωt), where v₀ is the maximum velocity (amplitude of velocity).
Write down the equation for velocity (v) as a function of displacement (x) in SHM.
The velocity (v) as a function of displacement (x) is given by v = ± ω√(x₀² - x²), where x₀ is the amplitude and ω is the angular frequency.
Describe the phase relationship between displacement, velocity, and acceleration in SHM.
In SHM, velocity leads displacement by π/2 (90°), and acceleration leads velocity by π/2 (90°). Therefore, acceleration and displacement are π (180°) out of phase.
Sketch graphs of displacement, velocity, and acceleration against time for SHM, highlighting key relationships.
Displacement (x) is a sine/cosine curve. Velocity (v) is the derivative of displacement (a cosine/sine curve, 90° ahead). Acceleration (a) is the derivative of velocity (negative sine/cosine, 180° out of phase with displacement). Note the max/min points.
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