Differential Equations Answers

Find the midspan deflection as well as the shear and moment at the support of the beam as shown. EI is constant.
W = sin x (kN/m)
L = pi (m)

A ball weighing 32 pounds is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (a) Find the equation of motion if the ball is driven by an external force equal to f(t) = 12 cos 2t + 3 sin 2t. (b) Determine the position and velocity of the ball at t = pi/4 sec.

A force of 2 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the
spring, and the system is then immersed in a medium that offers a damping force that is
numerically equal to 0.4 times the instantaneous velocity. (a) Find the equation of motion
if the mass is initially released from rest from a point 1 foot above the equilibrium position.
(b) Determine the position and velocity of the mass at t = pi/2 sec? (c) Find the first time at
which the mass passes through the equilibrium position heading upward.

A force of 2 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the
spring, and the system is then immersed in a medium that offers a damping force that is
numerically equal to 0.4 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. (b) Determine the position and velocity of the mass at t = sec? (c) Find the first time at which the mass passes through the equilibrium position heading upward.

A block weighing 8 pounds is attached to a spring. When set in motion, the spring/massvsystem exhibits simple harmonic motion. (a) Determine the equation of motion if the spring constant is 1 lb/ft and the mass is initially released from a point 6 inches below the equilibrium position with a downward velocity of 3/2 ft/sec. Express the equation of motion in alternative form. (b) What are the amplitude and period of motion? (c) What is the position and velocity of the mass at t = pi/ 4 sec?

A steel ball weighing 128 Ib is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length. The ball is started in motion with no initial velocity by displacing it 6 in above the equilibrium position. Assuming no air resistance, find (a) an expression for the position of the ball at any time t, and (b) the position of the ball at t=pi/4 sec

An LRC series circuit has L = 20 henries, R = 180 ohms, C = 1/280 farad, and an applied voltage E(t) = 10 sin t. Assuming no initial charge on the capacitor, but an initial current of 1 Ampere at t = 0 when the voltage is first applied, find the steady-state charge q(t) on the capacitor and the steady-state circuit current i(t).