Page 136 - Schaum's Outline of Differential Equations
P. 136
CHAP. 14] SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 119
whereupon 0 = v(0) = 2 sin 0 + 4c 2 cos 0 = 4c 2
Thus, c 2 = 0, and (2) simplifies to
as the equation of motion of the steel ball at any time t.
14.2. A mass of 2 kg is suspended from a spring with a known spring constant of 10 N/m and allowed to come
to rest. It is then set in motion by giving it an initial velocity of 150 cm/sec. Find an expression for the
motion of the mass, assuming no air resistance.
The equation of motion is governed by Eq. (14.1) and represents free undamped motion because there is no
externally applied force on the mass, F(t) = 0, and no resistance from the surrounding medium, a = 0. The mass and
the spring constant are given as m = 2 kg and k = 10 N/m, respectively, so Eq. (14.1) becomes x + 5x = 0. The roots
of its characteristic equation are purely imaginary, so its solution is
At t=0, the position of the ball is at the equilibrium position x 0 = 0m. Applying this initial condition to (_/),
we find that
0 = x(0) = c l cos 0 + c 2 sin 0 = c l
whereupon (1) becomes
The initial velocity is given as v 0 = 150 cm/sec = 1.5 m/sec. Differentiating (2), we obtain
whereupon,
and (2) simplifies to
as the position of the mass at any time t.
14.3. Determine the circular frequency, natural frequency, and period for the simple harmonic motion
described in Problem 14.2.
Circular frequency:
Natural frequency:
Period:
