This problem is an example of critically damped harmonic motion. A hollow steel

Question

This problem is an example of critically damped harmonic motion.
A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring 1/8 feet. The ball is started in motion from the equilibrium position with a downward velocity of 5 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second) . Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t.

Take as the gravitational acceleration 32 feet per second per second. (Note that the positive y direction is down in this problem.)

y= ?

Explanation / Answer

m = (1/8) slugs c = 4 pounds-s/feet k = 4/(1/8) = 32 pounds/feet The differential equation si given as: Therefore, the general solution of this equation is y(t) = (A + Bt) exp(st) Or y(t) = (A + Bt) e-16t But, at t = 0, y(t) = 1/8 Therefore, 1/8 = (A + B*0) e-16*0 = A Therefore, y(t) = (1/8 + Bt)e-16t At t = 0, dy/dt = 8 Therefore, 8 = B*e-16*0 – 16(1/8 + B*0)e-16*0 = B – 16/8 = B – 2 Therefore, B = 8 + 2 = 10 The complete solution is: y(t) = (1/8 + 10t)e-16t

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.