LawrenceC said: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
The difference between sin(wt) and cos(wt) is that they represent different trigonometric functions. Sin(wt) represents the sine function, which is the ratio of the opposite side to the hypotenuse in a right triangle. Cos(wt) represents the cosine function, which is the ratio of the adjacent side to the hypotenuse in a right triangle.
The SHM (simple harmonic motion) equation can be rewritten from a function of sin(wt) and cos(wt) to just cos(wt-∅) by using the trigonometric identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b) and substituting a=wt and b=∅. This results in the equation x = A cos(wt-∅), where A is the amplitude and ∅ is the phase difference.
The phase difference (∅) in the SHM equation represents the difference in the starting points of the sine and cosine functions. It determines the starting position of the oscillation and can affect the amplitude and period of the motion.
Yes, the SHM equation can be rewritten using only the sine function by using the trigonometric identity sin(a-b) = sin(a)cos(b) - cos(a)sin(b) and substituting a=wt and b=∅. This results in the equation x = A sin(wt-∅). However, the cosine function is often used because it represents the initial position of the oscillation.
Changing the phase difference (∅) in the SHM equation can affect the motion by altering the starting position of the oscillation. This can result in changes in the amplitude and period of the motion. For example, a phase difference of 0 will result in a motion with maximum amplitude, while a phase difference of π/2 will result in a motion with no amplitude.