If an object suspended from a spring is displaced vertically
from its equilibrium position by a small amount and re-
leased, and if the air resistance and the mass of the spring
are ignored, then the resulting oscillation of the object is
called simple harmonic motion. Under appropriate condi-
tions the displacement y from equilibrium in terms of time
t is given by $$y=A\cos \omega t$$ where A is the initial displacement at time t = 0, and ω is
a constant that depends on the mass of the object and the
stiffness of the spring (see the accompanying figure). The
constant |A| is called the amplitude of the motion and ω the
angular frequency….. The period T is the time required to make one complete
oscillation. Show that T = 2π/ω.
If I take amplitude equal to $2\pi$ than, I found an equation which is nearly related to period. But, I noticed there's a $\cos$. I don't know how to remove it. How to solve it? I would request for hint.
Best Answer
Time period is defined as the time interval taken for the system to finish one complete oscillation. This means that the phase difference between the initial and final position is $2\pi$. This means that: $$\omega \Delta t=2\pi$$ so that $$\Delta t=\frac {2\pi}{\omega}$$ Remember that, amplitude is not a property of a system, it depends upon the external forces acting upon the system. Hence you cannot simply define it to be $2\pi$.