Suppose a mass of 1 kg is attached to a spring with spring constant k = 2, and rests at equilibrium position. Starting at t = 0, an external force of f(t) = e t is applied to the system. Suppose the surrounding medium offers a damping force numerically equal to β times the instantaneous velocity, where β > 0 is some given number.
(a) What is the IVP governing this harmonic motion.
(b) For what value(s) of β will the spring experience over-damped, under-damped, and critically damped motion, respectively.
(c) Using variation of parameters, solve for the general solution of the ODE given in (a). In particular, do not take the initial conditions into consideration; do not solve for c1, c2. (
d) Using (c), describe the behavior of motion as β → ∞. Why does this physically make sense?
Suppose a mass of 1 kg is attached to a spring with spring constant k =...
plz print your result -1 points МУ Not A mass weighing 3V 10 N stretches a spring 2 m. The mass is attached to a dashpot device that offers a damping force numerically equal to β (B > 0) times the instantaneous velocity Determine the values of the damping constant B so that the subsequent motion is overdamped, critically damped, and underdamped. (If an answer is an interval, use interval notation. Use g 9.8 m/s2 for the acceleration due to...
5. (20 points) A mass weighing 8 pounds stretches a spring 1.6 feet. The entire system is placed in a medium that offers a damping force numerically equivalent to twice the instantaneous velocity. The mass is initially released from a point 1/2 foot above the equilibrium point with a downward velocity of 5 ft/sec. (a) (6 points) Write the differential equation for the mass/spring system and identify the initial conditions. 7 5. (b) (12 points) Solve the IVP in part...
5. (20 points) A mass weighing 8 pounds stretches a spring 1.6 fort. The entire system is placed in a medium that offers a damping force numerically equivalent to twice the instantaneous velocity. The mass is initially released from a point 1/2 foot above the equilibrium point with a downward velocity of 5 sec (a) (6 points) Write the differential equation for the mass spring system and identify the initial conditions 7 5. (b) (12 points) Solve the IVP in...
Q4. A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting att 0, an external force equal to f(t) is applied to the system. Given that the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity, solve the resulting initial value problem when (a) f(t) 0 (b) s) e sin 4t. Also determine the limit lim r()
A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position X, and initial velocity vo Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e-pt cos (0,t-a). Also, find the undamped position function u(t) = Cocos (0,0+ - )...
ONLY attempt to solve if you know what you are doing. A mass of 1 kg is attached to a spring whose constant is 5 N/m. Initially, the mass is released 1 m below the equilibrium position with a downward velocity of 5 m/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocit y. a) Find the equation of motion if the mass is driven...
3a. [10pts) A 32 lb weight is attached to a spring whose constant is 25 lb/ft. Initially the mass is released I ft below the equilibrium position with a downward velocity of 1 ft/sec. Find the equation of motion 3b. 10pts) Determine the equation of motion in part(a) if the surrounding medium offers a damping force numerically equal to 10 times the instantaneous velocity. 3c. [14pts) Determine the equation of motion in parts(a)-(b) if the weight is driven by an...
6. A mass of 2 kilogram is attached to a spring whose constant is 4 N/m, and the entire system is then submerged in a liquid that inparts a damping force equal to 4 tines the instantansous velocity. At t = 0 the mass is released from the equilibrium position with no initial velocity. An external force t)4t-3) is applied. (a) Write (t), the external force, as a piecewise function and sketch its graph b) Write the initial-value problem (c)Solve...
Use Laplace's method to solve A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 3 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Find the equation of motion if the mass is driven by an external force equal to f(t)...
(7 points) 14. A mass weighing 4 pounds stretches a spring 2 feet. The system is submerged in a medium which offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 8 ft/s. Find the equation of motion, ä(t). What type of damped motion is this system?