An object oscillates with an angular frequency = 4 rad/s. At t = 0, the object i

Question

An object oscillates with an angular frequency = 4 rad/s. At t = 0, the object is at x0 = 5.5 cm. It is moving with velocity vx0 = 2 cm/s in the positive x-direction. The position of the object can be described through the equation x(t) = A cos(t + ).   

Part (a)  What is the the phase constant of the oscillation in radians? (Caution: If you are using the trig functions in the palette below, be careful to adjust the setting between degrees and radians as needed.)  

Part (b)  Write an equation for the amplitude A of the oscillation in terms of x0 and . Use the phase shift as a system parameter.   

Part (c)  Calculate the value of the amplitude A of the oscillation in cm.

Explanation / Answer

If t = 0, x0 = A cos and v0 = - A sin

Since we don’t know A, and we do know of a relationship between sin and cos, if we divide v0 by x0,

we’ll get v0 / x0 = - A sin / Acos = - tan

If we cross-multiply-solve (or just divide by - ), we get

tan = -v0 / x0

so = tan -1 (- v0 / x0) = tan -1[ (2 cm/s) / (4 rad/s)*(5.5cm)] = 0.0909 rad

You can find A using either the x0 or v0 eq’s.

part (b)

A = x0 / cos

(c) A= 5.5cm / cos 0.0909

= 5.5 cm

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