Question

A 288-kg mass, when attached to the end of a spring hanging vertically, stretches the spring 8 m. The mass is in a medium that exerts a viscous resistance of 576 N when the mass has a velocity of 4 m/sec. Assume the mass is given an initial velocity of 18 m/s from the equilibrium position. a) Determine the spring constant k. Use g = 10 m/sec. k b) Determine the damping coeffient 7. 7= c) If the initial value problem for y(t), where y(t) denotes the displacement (in meters) of the mass from its equilibrium rest position, can be written in the form my' + yy' + ky = 0 where y(0) = yo and y' (O) = yo what are the values of yo and yo? Yo = yo d) Solve it only far enough to describe the motion of the system with the terminology of this course. The system is e) What value of the damping coefficient y is required for the system to be critically damped? No decimals. Use "sqrt(foo)", without quotes, if you need the square root of foo. 7=

Answer 1

(a)

a mas verghs of 288 kg is attached to the end of the spring. it stretches the spring 8m a) To determine the spring Constanti m. Giren m= 288 kg. g= 10 m/sec. & Lz of m. Constant = k - Spring mg L 288x10 8 kg 182 K 2 360 kg/82

(b)

(c)

e) To determine the initial value problem for y(t), where, y (l) displacement (in meters) of the mass from its equilibrium position my" y (o)- J. + sy + ky where, - O. and y'(o)=y! whene m= 288 kg. 2 144 N-Sec/m. Kz 360 kg 18² y (o)=0 y (o)= 18m18

(d)

d). The coresponding auxiliary equation of the homogeneous system 18 288 p2 + 1440+ 360 20 on 24 p²+ 120 + 30 -0. or 4 p² + 25 + 5 20. -2 ± 14-80 2x4 an -2 I J - 76 N 2x9 2 - 27 2019 i 2x4. 19 + is, in The System 1 / 4 general form, [n cos (en) * y (A) = fB Sin ( 4.

(e)

is to be Su when the e) let the damping coefficient Critically damped. 82 4K m. m on. 2 an 5, 2 = 4 km. 4 X 360x288 - 1800B 28875. 8 on