DIFFERENTIAL EQUATIONS An 8-lb weight is attached to the lower end of a coil spring suspended...

DIFFERENTIAL EQUATIONS

An 8-lb weight is attached to the lower end of a coil spring suspended from a fixed support. The weight comes to rest in its equilibrium position, thereby stretching the spring 6 in. The weight is then pulled down 9 in. below its equilibrium position and released at t=0. The medium offers a resistance in pounds numerically equal to 4x', where x' is the instantaneous velocity in feet per second. Determine the displacement of the weight as a function of the time.

Please be clear with how you arrive to the answer, I have a quiz coming up and want to make sure I do this correctly! Thank you very much

Expert Solution

in this problem the damping frequency will be B=~8 hz. Do not get confused the value of B in Second Order differential equation. Here you can put the any other constant like 'w'.

genius_generous answered 2 years ago

A 6 lb weight is placed upon the lower end of a coil spring suspended from...

A 6 lb weight is placed upon the lower end of a coil spring suspended from a fixed beam. The weight comes to rest in its equilibrium position, thereby stretching the spring 4 inches. Then beginning at t = 0 an external force given by F (t) = 27 sin(4t) − 3 cos(4t) is applied to the system. The resistance of the medium is three times the velocity (in ft/sec). Find the displacement of the weight as a function of...

A 12-lb weight is placed upon the lower end of a suspended spring. The weight comes...

A 12-lb weight is placed upon the lower end of a suspended spring. The weight comes to rest in its equilibrium position, stretching the spring by 1.5 inches. The weight is then pushed upward 2 inches above the equilibrium and released at t=0 with an initial velocity downward of 2ft/sec. Assume there is no damping caused by an external force a) Find the displacement of the weight function x(t) as a function of time, b) Determine the amplitude, period, and...

A 64 pound weight is attached to the lower end of a coiled spring suspended from...

A 64 pound weight is attached to the lower end of a coiled spring suspended from the ceiling. The weight comes to rest at its equilibrium position, thereby stretching the spring 1/2 foot. At time t = 0, the weight is positioned 1 /√3 feet below equilibrium and given a downward velocity of 8 feet per second. (a) Determine the equation of motion of the weight as a function of time. (b) Find the amplitude of the motion. (c) Find...

2. A 6-lb weight is attached to a vertically suspended spring that it stretches 4 in....

2. A 6-lb weight is attached to a vertically suspended spring that it stretches 4 in. and to adashpot that provides 1.5 lb of resistance for every foot per second of velocity.(a) If the weight is pulled down 1 ft below its static equilibrium position and then released from rest at time t = 0, find its position function .(b) Find the frequency, time-varying amplitude, and phase angle of the motion.(Give exact answers for both parts.

Differential Equations (FREE, UNDAMPED MOTION) A 12-lb weight is placed upon the lower end of a...

Differential Equations (FREE, UNDAMPED MOTION) A 12-lb weight is placed upon the lower end of a coil spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring 1.5 in. The weight is then pulled down 2 in. Below its equilibrium position and released from rest at t = 0. Find the displacement of the weight as a function of time; determine the amplitude, period, and the frequency of the resulting motion. Assume...

Differential Equations. A 10kg weight is attached to a wall horizontally via a spring. The rest...

Differential Equations. A 10kg weight is attached to a wall horizontally via a spring. The rest (equilibrium) length of the spring is 0.2m. The spring has stiffness coefficient k = 50. The weight drags along the ground, giving an effective damping coefficient of b = 40. We will ignore gravity and all other outside forces on the spring. If the weight is pulled to a position 0.5m away from the wall (thus +0.3m past equilibrium) and then flicked toward the...

A 128-lb weight is attached to a spring having a spring constant of 64 lb/ft. The...

A 128-lb weight is attached to a spring having a spring constant of 64 lb/ft. The weight is started in motion with no initial velocity by displacing it 6 in below the equilibrium position and by simultaneously applying an external force F(t) = 8sin(4t). a. Assuming no air resistance, find the equation of motion. b. What is the long term behavior of the motion? Please use differential equations to solve this

A weight is attached to a spring suspended from a beam. At time t = 0,...

A weight is attached to a spring suspended from a beam. At time t = 0, it is pulled down to a point 7 cm above the ground and released. After that, it bounces up and down between its minimum height of 7 cm and a maximum height of 25 cm, and its height h(t) is a sinusoidal function of time t. It first reaches a maximum height 1.4 seconds after starting. (b) What are the mean, amplitude, phase shift...

An 8-pound weight is attached to a spring hanging from a ceiling. The weight stretches the...

An 8-pound weight is attached to a spring hanging from a ceiling. The weight stretches the spring 9.6 inches, coming to rest at its stretched equilibrium position. The weight is displaced 8 inches below this stretched equilibrium position, and at time t = 0 the mass is released. At the same instant, an external force of F(t)= 2cos(2t) pounds is applied to the system. The damping constant is 1 lb/(ft/sec). Find the function that gives the displacement of the mass...

A mass weighing 8 lb is attached to a spring hanging from the ceiling, and comes...

A mass weighing 8 lb is attached to a spring hanging from the ceiling, and comes to rest at its equilibrium position. The spring constant is 4 lb/ft and there is no damping. A. How far (in feet) does the mass stretch the spring from its natural length? L= B. What is the resonance frequency for the system? ω0= C. At time t=0 seconds, an external force F(t)=3cos(ω0t) is applied to the system (where ω0 is the resonance frequency from...

Question