Simple Harmonic Motion (displacement function)

[yoohyojinn]

Homework Statement

I learned that in simple harmonic motion,

the displacement fuction is

x(t) = A sin(ωt + Φ)

or

x(t) = A cos(ωt + Φ)

but when do you use sine function or cosine function?

can I use whatever I want whether a sine or cosine function?

And what's that Φ in the equation?

Is it a quantity called a phase angle??

Homework Equations

The Attempt at a Solution

 

[dougla]

Note that cos(x) = sin(x + pi/2)

 

[ehild]

You can use both the sine and cosine functions for the displacement during SHM. You can call Φ phase angle or rather phase constant, and you can determine it form given initial conditions.
If the SHM starts with maximum displacement the cosine function is convenient to use, as x=Acos(ωt) is maximum at t=0, so Φ=0.
In the case when the SHM starts from zero displacement by giving the object some velocity, it is easier to use x=Asin(ωt).

ehild

 

[yeshima]

A boby moves in S.H.M with a amplitude of 30mm and a frequency of 2.0Hz. Calulate the values of
(a) the acceleration at the centre and extremeties.
(b) the velocity at these positions
(c) the velocity and acceleration of a point between the centre and extremity of the oscillation

how would i approach a question like this?...

 

[asteriatic]

Yes, the displacement function for simple harmonic motion can be expressed as x(t) = A sin(ωt + Φ) or x(t) = A cos(ωt + Φ). The choice of using a sine or cosine function depends on the initial conditions of the system. If the object starts at its equilibrium position (x=0) at time t=0, then we can use either function. However, if the object starts at a position other than x=0 at time t=0, then we need to use the appropriate function to account for the phase shift. The phase angle Φ represents the initial phase of the motion and can be thought of as the starting position of the object on the oscillation cycle. It is a quantity that helps us determine the exact position of the object at any given time. So, in short, the choice of using sine or cosine function and the inclusion of the phase angle Φ depend on the initial conditions of the system.