A mass weighing 3 lb stretches a spring 3 in If the mass is

A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward, contracting the spring a distance of 1 in, and then set in motion with a downward velocity of 2 ft/s, and if there is no damping, find the position u of the mass at any time f. Determine the frequency, period, amplitude, and phase of the motion. (Note: 1 ft = 12 in).

Solution

Assuming the upward direction as positive .

=> u(0) = 1 and u\'(0) = -2

Balancing the forces in intial condition

=> mg = ku

=> 3(32.15) = k(3/12)

=> k = 385.8

Now from the equation of motion ,

=> mu\'\' = -ku

=> mu\'\' + ku = 0

=> u\'\' + (k/m)u = 0 ( k/m = 385.8/3 = 128.6 => 128.6 = 11.3402 )

=> u = c₁cos(11.3402t) + c₂sin(11.3402t)

=> u\' = 11.3402( -c₁sin(11.3402t) + c₂cos(11.3402t) )

Plugging initial conditions u(0) = 1 and u\'(0) = -2 => c₁ = 1 and c₂ = -0.176364

=> u = cos(11.3402t) - 0.176364sin(11.3402t)

2f = 11.3402 => f = 1.8048 s^-1

2/T = 11.3402 => T = 0.554 s

A = ( 1² + (-0.176364)² ) = 1.01543

=> u = 1.01543( sin(11.3402t) + 1.74537 )