A mass weigh 4 lb stretches a spring \'24 in. Assume that there is no damping. I

Question

A mass weigh 4 lb stretches a spring '24 in. Assume that there is no damping. If after this the mass Ls pushed 2 in down and then set in motion with upward velocity of 3 in/s, determine the position ii of the mass at any time t. Find the natural frequency, the period, the amplitude, and the phase of the motion (you can use calculator to determine the phase). Assume that in the case of the spring-mass system of item (b) there is also a damping and we can change the damping constant. What is the critical damping constant?

Explanation / Answer

a> here we'll take acceleration due to gravity , g = 32 ft/sec^2

the weight of the mass is , mg = 4 lb

and L = 24 inches = 24/12 =2 ft

damping constant = 0

=> the mass m = weight/g = mg/g = 4/32 = 1/8 lb

and the spring constant , k = mg/L = 4/2 = 2 lb/ft

=> the initial value describing this scenario would be :

1/8*u'' + 2u = 0 , u(0) = 3 in/s = 1/4 ft / sec and u'(0) = 0

The characteristic equation of the corresponding homogeneous equation r^2/8 +2 = 0 has roots

r^2 + 16 = 0

r = + 4i and r = - 4i

The general solution of the differential equation is :

u(t) = C1*cos(4t) + C2*sin(4t)

now we'll find C1 and C2 using the initial values

u'(t) = -4C1*sin(4t) + 4C2*cos(4t)

now u'(0) = 0

=> 0 = 0 + 4C2

=> C2 = 0

and u(0) = 1/4

=> 1/4 = C1*1 +0 , => C1 = 1/4

hence the solution is

u(t) = 1/4*cos(4t)

compare the above equation with u(t) = A*cos(wt + phi)

where A = amplitude , w = the angular frequency , phi = the phase , w = 2pi*f , where f is the ordinary frequency

=> amplitude = 1/4

angular frequency = 4 radians per second

f = w/2pi = 4/2pi = 2/pi per second

the time period is = 1/f = pi/2

the phase of the motion is = 0

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