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Question

The period of oscillation of the body is 3.14 s, and the amplitude is 2 cm. What is the velocity if the phase angle is π radians?

Answer 1

Simple harmonic motion mathematically could be described by the use of sin or cosine function. Let’s use cosine function. Then we can write that at any given time displacement

                                          x (t) = A cos ωt,

where A – amplitude,

 φ  = ωt – phase of the oscillations.

We will get velocity v, if take a derivative with respect to time, So

                                            v = dx dt = - Aω∙sin ωt

or

                                              v = - A∙ω∙sin φ  

Because angular frequency ω = 2π/T, we can write:

                                          v = - (A2π/T )∙sin φ  or

                                            v = - (2πA/T)∙sin φ

Now we use the fact that amplitude A = 2 cm, period T = 3.14 s and phase  φ = π.

Substituting their values to the equation, we obtain velocity

                                          v = - (2∙3.14∙2/3.14)∙sin π

sin π is zero, hence v = 0.

                                                                                    Answer: 0

BTW, if from the very beginning we would use sin function, the result will be different, although all logic will be the same. I would like more strict initial conditions.