A spring with a mass of 4 kg has damping constant 28, and a force of 12 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring is stretched 1 m beyond its natural length and then released with zero velocity. Find the position of the mass at any time t and mass that would produce critical damping
To find the position of the mass at any time t after the spring is released, we can use the equation of motion for a damped harmonic oscillator:
m * x'' + c * x' + k * x = 0
where:
m is the mass of the system,
x is the displacement of the mass from its equilibrium position,
c is the damping constant, and
k is the spring constant.
In this case, the mass of the spring (m) is given as 4 kg, and the damping constant (c) is given as 28. We need to find the spring constant (k) to solve the equation.
The force required to keep the spring stretched 0.5 m beyond its natural length is given as 12 N. This force is equal to the spring constant multiplied by the displacement:
F = k * x
12 N = k * 0.5 m
k = 12 N / 0.5 m
k = 24 N/m
Now we have all the necessary values to solve the equation of motion:
4 kg * x'' + 28 * x' + 24 * x = 0
To solve this second-order differential equation, we need to find the roots of the characteristic equation:
4 * r^2 + 28 * r + 24 = 0
Solving this equation will give us two complex roots: r1 and r2.
The general solution for the displacement of the mass (x) with respect to time (t) is given by:
x(t) = e^(r1*t) * (A * cos(b1*t) + B * sin(b1*t)) + e^(r2*t) * (C * cos(b2*t) + D * sin(b2*t))
where A, B, C, and D are arbitrary constants determined by the initial conditions (e.g., initial displacement and velocity), and b1 and b2 are the imaginary parts of r1 and r2, respectively.
To find the position of the mass at any time t, we need to determine the values of A, B, C, and D using the initial conditions. Since the mass is initially stretched 1 m beyond its natural length and released with zero velocity, we have:
x(0) = 1 m (initial displacement)
x'(0) = 0 m/s (initial velocity)
Now we can use these initial conditions to find the values of A, B, C, and D and substitute them into the general solution to get the position of the mass at any time t.
To find the mass that would produce critical damping, we need to set the damping constant (c) equal to the critical damping constant (cc) and solve for the mass (m) using the formula:
cc = 2 * √(m * k)
Given that cc is equal to 28:
28 = 2 * √(m * 24)
Solving this equation will give us the mass that would produce critical damping (m).