To derive the expression for velocity in simple harmonic motion (SHM) in terms of displacement, amplitude, and angular speed, we can start with the equation of motion for SHM:
x = A * cos(ωt)
Where: x is the displacement from the equilibrium position, A is the amplitude of motion, ω (omega) is the angular speed, given by ω = 2πf or ω = √(k/m).
To find the velocity, we need to differentiate the displacement equation with respect to time:
v = dx/dt
Taking the derivative of x = A * cos(ωt) with respect to time, we get:
v = d(A * cos(ωt))/dt
The derivative of cos(ωt) with respect to t is given by:
d(cos(ωt))/dt = -ω * sin(ωt)
Therefore, the velocity equation becomes:
v = -A * ω * sin(ωt)
In terms of amplitude (A), angular speed (ω), and displacement (x), we can substitute x = A * cos(ωt) into the velocity equation:
v = -ω * A * sin(ωt)
This is the expression for velocity in simple harmonic motion in terms of displacement, amplitude, and angular speed.