Betty Hayward: X(t)=acos(ωt+φ)

Kamis, 28 Juli 2022

X(t)=acos(ωt+φ)

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X(t) = acos(ωt+φ) has a maximum when cos(ωt+φ) = 1. The latter implies that the maximum of the displacement closest to time t = 0 occurs when ωt + φ = 0 or t = −φ/ω. If φ < 0, then this maximum occurs for positive t, as shown in ﬁg. 14. 7 of giancoli.

An example of a typical problem would be, you're given some initial conditions (say: Period and amplitude of the oscillator) and asked to find the equation x(t)= acos(ωt+φ) using what you already know. For the most part it's okay, just using formulas, but when it comes to solving for φ i just can't do it! The object oscillates about the equilibrium position x 0. if we choose the origin of our coordinate system such that x 0 = 0, then the displacement x from the equilibrium position as a function of time is given by V =aejφ↔v()t =acos(ωt +φ) • in words we can say that: O the magnitude of the phasor is the amplitude of the cosine. O the angle of the phasor is the phase angle of the cosine. O in mathematical terms we have defined a mapping from “sinusoidal time functions” v()t =acos(ωt +φ) into complex numbers call phasors of the form v =aejφ. Then the general expression for x(t) is x(t) = acos(ωt+φ) (4. 1) a is called the amplitude of the motion.