the equation of transverse wave on a rope is y(x,t)=5sin(4.0t-0.02x) where (y) and(x) are measured in cm and (t) in second.calculate the maximum speed of particle on the rope
To calculate the maximum speed of a particle on the rope, we need to find the derivative of the transverse wave equation with respect to time (t), and then determine the maximum value of that derivative.
The equation of the transverse wave given is:
y(x,t) = 5sin(4.0t - 0.02x)
To find the derivative of this equation with respect to time, we differentiate the sine function with respect to its argument, which in this case is (4.0t - 0.02x).
dy/dt = 5 * cos(4.0t - 0.02x) * (d/dt)(4.0t - 0.02x)
Since we are interested in finding the maximum speed, we need to find the maximum value of dy/dt. To do this, we can set the cosine function to its maximum value of 1.
Therefore, we have:
dy/dt = 5 * (d/dt)(4.0t - 0.02x)
The derivative of (4.0t - 0.02x) with respect to t is simply 4.0.
So, dy/dt = 5 * 4.0 = 20
The maximum speed of the particle on the rope is given by the absolute value of the derivative, which is 20 cm/s.