The equation of transverse wave on a rope is
y (x, t) = 5 sin (4.0t −0.02x)
where y and x are measured in cm and t is expressed in second. Calculate the maximum
speed of a particle on the rope.
To calculate the maximum speed of a particle on the rope, we'll need to find the derivative of the displacement equation with respect to time (t), and then find the maximum value of the resulting expression.
Given equation of the transverse wave on the rope:
y(x, t) = 5 sin(4.0t - 0.02x)
To find the derivative, let's differentiate the equation with respect to time (t) while treating x as a constant:
dy/dt = 5 * d(sin(4.0t - 0.02x))/dt
The derivative of sine function is cosine function, so we obtain:
dy/dt = 5 * cos(4.0t - 0.02x) * d(4.0t - 0.02x)/dt
Differentiating (4.0t - 0.02x) with respect to t, we get:
dy/dt = 5 * cos(4.0t - 0.02x) * (4.0 - 0)
Simplifying further, the expression becomes:
dy/dt = 20 * cos(4.0t - 0.02x)
The maximum speed occurs when the absolute value of the derivative is at its maximum. Hence, we need to find the maximum value of cos(4.0t - 0.02x).
Since the cosine function oscillates between -1 and 1, the maximum value of cos(4.0t - 0.02x) is 1.
Therefore, the maximum speed of a particle on the rope is:
dy/dt_max = 20 * 1 = 20 cm/s
So, the maximum speed of a particle on the rope is 20 cm/s.
To calculate the maximum speed of a particle on the rope, we need to find the maximum value of the derivative of the equation with respect to time.
Given: y(x, t) = 5 sin (4.0t − 0.02x)
To find the maximum value of the derivative, we need to differentiate the equation with respect to time (t).
dy/dt = 5 * d/dt [sin (4.0t − 0.02x)]
Using the chain rule, we have:
dy/dt = 5 * (4.0) * cos (4.0t − 0.02x) * (d/dt [4.0t − 0.02x])
Since we are finding the maximum speed, we need to find the maximum value of dy/dt. The maximum value of cos (4.0t − 0.02x) is 1, so we can ignore it.
Therefore, we need to find the maximum value of d/dt (4.0t − 0.02x). The derivative of 4.0t − 0.02x with respect to t is simply 4.0, so we have:
dy/dt = 5 * (4.0)
dy/dt = 20
So, the maximum speed of a particle on the rope is 20 cm/s.