Question

Write y(t)=2sin 4 pi t + 5 cos 4 pi t) in the form y(t) A sin (wt + Ø) and identify the amplitude, angular frequency, and the phase shift of the spring motion.

Answer 1

The amplitude of the spring motion is √29, the angular frequency is 4π, and the phase shift is arccos(-5/√29).

Step-by-step explanation:

The given wave function, y(t) = 2sin(4πt) + 5cos(4πt), can be written in the form y(t) = A sin (wt + Ø) by using trigonometric identities. The amplitude, A, is the coefficient of the sine or cosine function with the highest coefficient. In this case, A = √(2² + 5²) = √29. The angular frequency, w, is equal to 4π. The phase shift, Ø, can be determined by finding the angle whose cosine is the ratio of the coefficients of sine and cosine. In this case, Ø = arccos(-5/√29).