Rewriting SHM equation from a function of sin(wt) and cos(wt) to just cos(wt-∅)

[jammydav93]

Homework Statement

Rewrite the equation of motion x(t) = 2cos(wt) - 2sin(wt) in the form x(t) = C cos(wt-∅)

Homework Equations

The Attempt at a Solution

I am able to work out the constant C as it is just the magnitude of the sin and cos function,

C = (2^2 + 2^2)^1/2 = √8

However I am unable to find ∅, can anyone point me in the direction of how to do this?

Many thanks

James

 

[LawrenceC]

Hint:

General equation of motion is: x=Asin wt+Bcos wt. If started from position x0 with velocity v0

x = v0/w sin wt + x0cos wt

Make substitution:

x0 = X sin theta
v0/w = X cos theta

 

[jammydav93]

Hint:

General equation of motion is: x=Asin wt+Bcos wt. If started from position x0 with velocity v0

x = v0/w sin wt + x0cos wt

Make substitution:

x0 = X sin theta
v0/w = X cos theta

Thanks that was all I needed!

Essentially ∅ = arctan(b/a)

Thanks for the help!