Answer:
t =1.294
Explanation:
Let m be the mass attached, let k be the spring constant and let p be the positive damping constant. The Newton's second law for the system is
where x(t) is the displacement from the equilibrium position. The equation can be transformed into
(a) Let's determine the equation of motion. We must convert units of weight into units of mass
From Hooke's law we can calculate the spring constant k.
If we put m = 1/10 slugs, k = 2 lb/ft and = 0.4 into the DE, we get
The auxiliary equation is m^2+4*m+20 = 0 and its solutions are m_1 = -2-4_i and m_2 = -2 + 4i. The general solution is then
From the initial conditions
we can find the equation of motion
(b) We need to express the equation of motion in the alternative form
first attachment
The amplitude is
and the phase angle is
sin∅=
cos∅=
tan∅=2
∅=
=4.24
(c) lb find the wanted time, we will need the derivative of the equation of motion
x'(t)=
The mass passes through the equilibrium when
x(t) =0
t =-0.2786+k/4
The velocity at those times is
second attachment
The first (positive) time at which the mass passes through the equilibrium heading upward is for k = 2 (when x' < 0).
t =1.294