Ask Question
26 December, 04:00

A damping force affects the vibration of a spring so that the displacement of the spring is given by y = e-4t (cos 2t + 3 sin 2t). Find the average val

+4
Answers (1)
  1. 26 December, 04:14
    0
    Average velocity = dy/dt = 4sin2t (2t+3) - 4cos2t (6t+1) e^-4t (cos 2t + 3 sin 2t).

    Explanation:

    Velocity is defined as the rate of change in displacement.

    Velocity = change in displacement/time taken

    Given the displacement of the string as;

    y = e^-4t (cos 2t + 3 sin 2t).

    To get the average velocity, we will find the derivative of the displacement with respect to time.

    Using function of a function to solve this;

    Let u = 4t (cos 2t + 3 sin 2t) ... (1)

    y = e^-u ... (2)

    Differentiating both functions with respect to their variables we have;

    dy/du = - e^-u

    du/dt is gotten using the product rule to have;

    du/dt = 4t (-2sin2t+6cos2t) + 4 (cos2t+3sin2t)

    Opening up the bracket we have;

    du/dt = - 8tsin2t+24tcos2t+4cos2t+12sin2t

    Collecting like terms;

    -8tsin2t+12sin2t+24tcos2t+4cos2t

    du/dt = - 4sin2t (2t-3) + 4cos2t (6t+1)

    dy/dt = dy/du * du/dt

    dy/dt = - e^-u * - 4sin2t (2t+3) + 4cos2t (6t+1)

    Substituting u = 4t (cos 2t + 3 sin 2t) into dy/dt, we will have;

    dy/dt = - e^-4t (cos 2t + 3 sin 2t) * - 4sin2t (2t+3) + 4cos2t (6t+1)

    dy/dt = 4sin2t (2t+3) - 4cos2t (6t+1) e^-4t (cos 2t + 3 sin 2t).

    The average velocity of the wave function therefore give us;

    dy/dt = 4sin2t (2t+3) - 4cos2t (6t+1) e^-4t (cos 2t + 3 sin 2t).
Know the Answer?
Not Sure About the Answer?
Find an answer to your question 👍 “A damping force affects the vibration of a spring so that the displacement of the spring is given by y = e-4t (cos 2t + 3 sin 2t). Find the ...” in 📗 Physics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers