Final answer:
The maximum velocity of an oscillator is directly proportional to the amplitude and the square root of the force constant, while being inversely proportional to the square root of the mass. Hence, the maximum velocity (V) of the oscillator is 0.
Step-by-step explanation:
The given equation represents an oscillator with the function xcos(yt). In this equation, x represents the displacement of the oscillator and yt represents the angular frequency.
To find the maximum velocity (V) of the oscillator, we need to differentiate the displacement equation with respect to time (t) and then find the maximum value of the derivative.
Let’s start by differentiating the equation:
dx/dt = -xysin(yt)
Now, to find the maximum value of dx/dt, we can set its derivative equal to zero and solve for t:
d²x/dt² = -xy²cos(yt) = 0
This implies that either x = 0 or cos(yt) = 0.
If x = 0, it means that the displacement is zero, which indicates that the oscillator is at its equilibrium position. At this point, the velocity is also zero.
If cos(yt) = 0, it means that yt = (2n + 1)π/2, where n is an integer. Solving for t gives:
yt (2n + 1)π/2 t = [(2n + 1)π/2] / y
Substituting this value of t back into dx/dt:
dx/dt = -xysin(y[(2n + 1)π/2] / y)
dx/dt = -xy(-1)ⁿ
Here, (-1)ⁿ alternates between -1 and 1 for different values of n. Therefore, the velocity alternates between positive and negative values but never reaches a maximum value.