Question

With the use of a phase shift, the position of an object may be modeled as a cosine or sine function (as a function of time). If given the option, which function would you choose (Note: this is an open-ended question; there is no wrong answer; just be sure to explain your choice please)? Assuming zero phase shift, if you are using a sine function to model the position of an object, what are the initial conditions of the oscillator---that is, the initial position, velocity, and acceleration of the oscillator at t=0? How about if you use a cosine function to model the position?

Answer 1

Sine or Cosine — two functions that are essentially the same since each is just a phase shifted version of the other. They both have the same period i.e, .  When a trig function is phase shifted, it's derivative is also phase shifted. Nothing else is affected, so we could pick sine with a phase shift or cosine with a phase shift as well.

Personally, i would choose Sine and add a Phase shift which woild be Zero at t=0. Now, An oscillator can be represented as -

x = A sin(2πft + φ) ...............................(A)

where…

x =	position [m, cm, etc.]
A =	amplitude [m, cm, etc.]
f =	frequency [Hz]
t =	time [s]
φ =	phase [rad]

For initial conditions, = 0, t=0 , putting these values in equation (A) we get x=0.

If we choose cosine in place of sine wave, we would get

x = A cos(2πft + φ)

where…

x =	position [m, cm, etc.]
A =	amplitude [m, cm, etc.]
f =	frequency [Hz]
t =	time [s]
φ =	phase [rad]

Here, at t=0, phase will not be zero as expected.

At t=0 when the oscillation starts, we get x(0)= Acos⁡(ϕ). If ϕ=0 then we simply get x(0)=A. As if the motion starts at the maximum amplitude.

However if we have the motion starting at the centre of oscillation, with some negative velocity that would mean x(0)=0. This means cos⁡(ϕ)=0 and so ϕ=π/2 (or 3π/2), but think about what that would mean for the velocity.

Essentially the phase constant ϕ determines the initial position of the oscillation, at t=0. As ϕ goes from 0 to 2π, the initial position goes from A to −A and back to A, as the cosine of the phase.

To avoid these confusion i chose sine fuction to describe the oscillation.