A mass weighing 16 pound is attached to a spring whose spring constant is 25 lb/ft. Find the equation of motion. (use 32 ft/s^2 for acceleration due to gravity. Assume t is measured in seconds. ) What is the period of simple harmonic motion (in seconds)?
A mass weighing 24 pounds, attached to the end of a spring, stretches it 4 inches. Initially, the mass is released from rest from a point 8 inches above the equilibrium position. Give the initial conditions. Find the equation of motion. (use g = 32 ft/s^2 for acceleration due to gravity. Assume t is measured in seconds. ) Find the equation of motion?
A mass weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the mass is initially released from a point 1 foot above the equilibrium position with an upward velocity 5 ft/s. How many complete cycle will the mass have completed at the end of 3*pi seconds?
A mass weighing 4 pound is attached to a spring whose spring constant is 2 lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from 1 foot above the equilibrium position with a downward velocity of 14 ft/s. Determine the time at which mass passes through the equilibrium position. (Use g = 32 ft/s^2 for acceleration due to gravity.) Find the time (in s) after mass passes through the equilibrium position at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?
A force of 5 pounds stretches a spring 1 foot. A mass weighing 6.4 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 1.2 times the instantaneous velocity. (a) Find the equation of motion if mass is initially from the rest from a point 1 foot above the equilibrium position. (b) Express the equation of motion in the form x(t) = Ae^(-lambda*t) sin(sqrt(w^2 - lambda^2 )t + phi) (c) Find the first time at which mass passes through the equilibrium position heading upward.
A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position, Starting at t = 0, an external force equal to f(t) = 6sin(4t) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. (Use g = 32 ft/s^2 for the acceleration due to gravity.)
Find the charge q(t) on the capacitor and the current i(t) in the given LRC-series circuit. L = 1 h, R = 100 ohm, C = 0.0004 f, E(t) = 40 V, q(0) = 0 C, i(0) = 2A. Find the maximum charge on the capacitor. (Round your answer to four decimal places.)
The given vectors are solutions of a system X' = AX. Determine whether the vectors form a fundamental set on the interval (-inf, inf). X1 = ( 1, -1 )e^t, X2 = (3 5)e^t + (8 -8)te^t Yes, since the set X1, X2 is linearly independent for -inf < t < inf. Yes, since the set X1, X2 is linearly dependent for -inf < t < inf. No, since the set X1, X2 is linearly independent for -inf < t < inf. No, since the set X1, X2 is linearly dependent for -inf < t < inf. There is not enough information.
X1 = (1 -2 4) + t(1 2 2), X2 = (1 -2 4), X3 = (5 -10 20) + t(4 8 8) The given vectors are solutions of a system X' = AX. Determine whether the vectors form a fundamental set on the interval (-inf, inf). Yes, since the set X1, X2, X3 is linearly independent for -inf < t < inf. Yes, since the set X1, X2, X3 is linearly dependent for -inf < t < inf. No, since the set X1, X2, X3 is linearly independent for -inf < t < inf. No, since the set X1, X2, X3 is linearly dependent for -inf < t < inf. There is not enough information.
Verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. dx/dt = x + 4y + 2t - 4 dy/dt = 3x + 2y - 4t - 9; XP = (2 -1)t + (2 1) Writing the system in the form X' = AX + F for some coefficient matrix A and vector F, one obtains the following. XP = (2 -1)t + (2 1) one has Since the above expressions are equal XP = (2 -1)t + (2 1) is a particular solution of the given system.
Consider the following system. dx/dt = 4x + 5y dy/dt = 10x + 9y Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenvalue.) Find the general solution of the given system.
Consider the following system. dx/dt = x + y - z dy/dt = 5y dz/dt = y - z Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenvalue.) Find the general solution of the given system.