A particle moves so that position x in meters is given as a function of time t i

Question

A particle moves so that position x in meters is given as a function of time t in seconds by the equation x(t) = c1 cos(c2t + c3), where c1 = .0208 m, c2 =305 ?s^?1,c3 = 1.18? . Give numerical values for the following: (a) Amplitude (m) (b) Angular frequency (rad/s) (c) Frequency (Hz) (d) Period (s) (e) Phase constant (rad) (f) What is the position (m) of the particle at t = 3.00 ms? (g) What is the velocity (m/s) of the particle at t = 3.00 ms? (h) What is the acceleration (m/s2) of the particle at t = 3.00 ms?

Explanation / Answer

the general equation for a mechanical wave is x = Asin(wt + Q) where Q is the phase constant , the given equation os similar thus a) A = c1 =.0208 m b) Angular frequency (rad/s) w = c2 =305 rad /s c)f = w/ (2 * pi) = 305 / (2 * 3.14) = 48.54 /s d)time period = 1/Frequency = 0.206 s e) Phase constant = c3 = 1.18 degrees f) x(t) = c1 * cos(c2t + c3) = x(t) = .0208 cos(305 t + 1.18?) putting the value of t =3 x(3)= .0208 * cos(305 *3 + 1.18) =-0.0199 m g) v(3)= w * sqrt (A^2- x^2) = 305 * sqrt(.0208^2- (-0.0199^2)) = 1.84 m/s h) a(3) = -w^2 * x = 1851.19 m/s^2

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