Given that x=Acos(wt) is a sinusoidal function of time, show that v(velocity) and a(acceleration) are also sinusoidal functions of time. (HINT: Use v=square root [(k/m)(A^2-x^2)] and a=(-k/m)X
The hint tells you what to do: put Acoswt for X.
A second way which is easier is in calculus:
v= dx/dt
a= dv/dt= d"x/dt"
That is the most simple way.
To show that velocity (v) and acceleration (a) are also sinusoidal functions of time, we can substitute the expression for x into the given formulas for v and a.
Let's begin with the expression for velocity:
v = √[(k/m)(A^2 - x^2)]
Substituting x = Acos(wt) into this equation, we get:
v = √[(k/m)(A^2 - (Acos(wt))^2)]
Simplifying the expression inside the square root:
v = √[(k/m)(A^2 - A^2cos^2(wt))]
v = √[(k/m)(A^2 - A^2(1 - sin^2(wt)))]
v = √[(k/m)(A^2 - A^2 + A^2sin^2(wt))]
v = √[(k/m)(A^2sin^2(wt))]
Using the identity sin^2(a) = (1 - cos(2a))/2:
v = √[(k/m)(A^2(1 - cos(2wt))/2)]
v = √[(k/m)(A^2/2)(1 - cos(2wt))]
From this expression, we can see that v is a sinusoidal function of time because it contains the term 1 - cos(2wt). The amplitude of this sinusoidal function is √(k/m)(A^2/2) and its angular frequency is 2w.
Moving on to the expression for acceleration:
a = (-k/m)x
Substituting x = Acos(wt) into this equation:
a = (-k/m)(Acos(wt))
a = -kAcos(wt)/m
This shows that a is also a sinusoidal function of time, with an amplitude of kA/m and the same angular frequency w as the original function x.
Therefore, by substituting the given expression for x into the formulas for velocity and acceleration, we have shown that both v and a are indeed sinusoidal functions of time.