Download MATH 280: Applied Differential Equations Quiz 3

MATH 280: Applied Differential Equations
Quiz 3
Section 01, Class Number: 15302, Instructor: Dr. Jing Li
Due at the Beginning of the Class on Nov. 27, 2013
Student Name:
Student I.D.:
Question 1. [24 points] After a mass weighing 10 pounds is attached to a 5-foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire
system is placed in a medium that offers a damping force that is numerically equal to the instantaneous
velocity.
(a)[10 points] Find the equation of motion if the mass is initially released from a point 12 foot below
the equilibrium position with a downward velocity of 1 ft/s.
Solution: The equation is 14 x 00 + x 0 + 5x = 0, x(0) = 12 , and x 0 (0) = 1 =⇒ x = e −2t ( 12 cos 4t + 21 sin 4t ).
(b)[4 points] Express the equation of motion in the form given in
p
x(t ) = Ae −λt sin( ω2 − λ2 t + φ).
Solution: x =
p1 e −2t
2
sin(4t + π4 ).
(c)[6 points] Find the times at which the mass passes through the equilibrium position heading
downward.
π
Solution: x = p1 sin(4t + π4 ) = 0, and x 0 > 0 =⇒ 4t + π4 = 2kπ, k is an positive integer. =⇒ t = k2 − 16
,k =
2
1, 2, 3, . . . .
(d)[4 points] Graph the equation of motion.
Solution:
1
Question 2. [24 points] A mass of 1 slug 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)[16 points] Find the equation of motion if the mass is driven by an external force equal to f (t ) =
12 cos 2t + 3 sin 2t .
Solution: If x 00 + 2x 0 + 5x = 12 cos 2t + 3 sin 2t , x(0) = 1 and x 0 (0) = 5, then
x c = e −t (c 1 cos 2t + c 2 sin 2t ) and x p = 3 sin 2t ,
so that the equation of motion is
x = e −t cos 2t + 3 sin 2t .
(b)[4 points] Graph the transient and steady-state solutions on the same coordinate axes.
Solution:
(c)[4 points] Graph the equation of the motion.
Solution:
2