This problem deals with a highly simplified model of a car of weight 2560 lb (mass m = 80 slugs in fps units). Assume that the suspension system acts like a single spring and its shock absorbers like a single dashpot, so that its vertical vibrations satisfy the equation mx" + cx' + kx=0 with appropriate values of the coefficients. (a) Find the stiffness coefficient k of the spring if the car undergoes free vibrations at 85 cycles per minute (cycles/min) when its shock absorbers are disconnected. (b) With the shock absorbers connected, the car is set into vibration by driving it over a bump, and the resulting damped vibrations have a frequency of 83 cycles/min. After how long will the time-varying amplitude be 3% of its initial value? (a) The stiffness coefficient of the spring is k = lb/ft. (Round to the nearest integer as needed.) (b) The time-varying amplitude will be 3% of its initial value after S. (Do not round until the final answer. Then round to the nearest hundredth as needed. Use the answer from part (a) to answer this part.) A mass weighing 140 lbs (mass m = 4.375 in fps) is attached to the end of a spring that is stretched 1 in. by a force of 140 lbs. A force Fo cos wt acts on the mass. At what frequency (in hertz) will resonance oscillations occur? Neglect damping. Resonance oscillations will occur at Hz. (Round to two decimal places as needed.) This problem deals with the RL circuit shown here, a series circuit containing an inductor of L henries, a resistor with a resistance of R ohms, and a source of electromotive force (emf), but no capacitor. In this case, the current in the circuit satisfies the linear first-order equation LI' + RI = E(t). Suppose that L = 9 H, R = 54 92, and the source E of emf is a battery supplying 162 V to the circuit. Suppose also that the switch has been in position 1 for a long time, so that a steady current of 3 A is flowing in the circuit. At time t = 0, the switch is thrown to position 2, so that I(0) = 3 and E = 0 for t 0. Find I(t). I(t)= E Switch 2 R w O 1 Solve the boundary value problems. x+4x=0, x(0)=1, x(x)= -2 A. Infinitely Many Solutions OB. x(t) = cos(2t) - 2 sin (2t) C. No solution Solve the boundary value problems. y" 2y', y(0) 1, y(1)=4 A. y(x)= e2-43e2x 2-1 + e -1 OB. Infinitely Many Solutions C. No solution