The motion of a mass attached to a spring serves as a relativity simple example

Question

The motion of a mass attached to a spring serves as a relativity simple example of the viberations that occur in more complex mechanical systems. Fot many such systems, the analysis of these vibrations is a problem in the solution of linear differential equations with constant coefficients. We consider a body of mass m attached to one end of an ordinary spring that resists compression as well as stretching; the other end of the spring is attached to a fixed wall, as shown in image. assume that the body rests on a frictionless horizontal plane, so that it can move only back and forth as the spring compresses and stretches. Denote by x the distance of the body from its equalibrium position to its position when the spring is unstretched. We take x>0 when the spring is stretched, and thus x<0 when it is compressed. The image shows the mass attached to a dashpot device, like a shock absorber, that provides a force directed opposite to the instantaneous direction of motion of the mass m.

We derived the equation of the motion mx`` + cx` + kx = f(t), where c is the damping coefficient and k is the spring stiffness. The motion is free if f(t) = 0

Solve for x(t) and draw the graph of it for each of the following initial value problems and state whether the motion is underdamped, overdamped or critically damped:

(a) m=1, k=4, c=4 ; x(0) = 2 , v(0) = x`(0) = 0

(b) m=1, k=4, c=0.5 ; x(0) = 2 , v(0) = x`(0) = 0

(c) m=1, k=4, c=8 ; x(0) = 2 , v(0) = x`(0) = 0

if the motion is underdamped, express x(t) in the form x(t) = Ae^(?t) * cos(?t+?) and what are the physical interpretations of the constants.

Explanation / Answer

a) x(t)    (a) x(t)= 2cos(sqrt{5}t) +2/sqrt{5}* sin(sqrt{5}t))   

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.