- #1
Dustinsfl
- 2,281
- 5
Homework Statement
A weight of ##50## N is suspended from a spring of stiffness ##4000## N/m and is subjected to a harmonic force of amplitude ##60## N and frequency ##6## Hz.
Homework Equations
The Attempt at a Solution
Since ##W = mg = 50##, we have that the suspended mass, ##m = 5.10204##, and we know that ##f = \frac{\omega}{2\pi} = 6## so ##\omega = 12\pi##. The harmonic forcing term is then
$$
F(t) = 60\cos(12\pi t)
$$
and our equation of motion is
$$
\ddot{x} + \frac{4000}{5.10204}x = \frac{60}{5.10204}\cos(12\pi t).
$$
Solving the transient and steady solution, we obtain
$$
x(t) = A\cos(28t) + B\sin(28t) - 0.0184551\cos(12\pi t)
$$
How do I determine the extension of spring from the suspended mass? This value would then be ##x(0) = x_0##. Additionally, I will assume any motion starts from rest so ##\dot{x}(0) = 0## which leads to ##B = 0## and ##A## can be defined as ##x_0 - \frac{F_0}{k - m\omega^2}## where ##\omega = 12\pi## and ##F_0 = 60##
$$
x(t) = (x_0 + 0.0184551)\cos(28t) - 0.0184551\cos(12\pi t)
$$
Would the extension of the spring simply be, ##F = kx## where ##F = 50## so
$$
x_0 = \frac{F}{k} = \frac{1}{80}\mbox{?}
$$