An 8 lb weight stretches a spring 8 feet. The weight is moving through a surrounding medium that has a damping effect equal to the instantaneous velocity of the weight. Initially the weight is displaced 4 feet below the equilibrium position with a downward velocity of 2 ft/sec. Find the maximum vertical displacement.
To find the maximum vertical displacement, we need to analyze the motion of the weight under the damping effect.
First, let's identify the known quantities:
- Mass of the weight (m): Since we are given an 8 lb weight, we can convert it to mass using the conversion factor of 1 lb = 0.45359237 kg. Therefore, the mass of the weight is m = 8 lb * 0.45359237 kg/lb = 3.629 kg.
- Spring constant (k): Since the weight stretches the spring by 8 feet, we can use Hooke's Law to find the spring constant. F = kx, where F is the force applied by the spring, k is the spring constant, and x is the displacement. Thus, k = F / x = mg / x = 3.629 kg * 9.8 m/s^2 / 8 ft * 0.3048 m/ft = 14.71 N/m.
- Initial displacement (y0): The weight is initially displaced 4 feet below the equilibrium position, which is y0 = -4 ft = -1.2192 m.
- Initial velocity (v0): The weight has a downward velocity of 2 ft/sec, which is v0 = -2 ft/sec = -0.6096 m/s.
Next, let's determine the equation of motion for the weight. Considering the damping effect, the equation of motion for a damped harmonic oscillator is given by:
m * y'' + c * y' + k * y = 0,
where y is the vertical displacement, y' is the velocity, and y'' is the acceleration. c is the damping constant, which is equal to the instantaneous velocity of the weight in this case.
Taking the derivatives of y with respect to time, we have:
y' = v,
y'' = v',
where v is the velocity and v' is the acceleration. Substituting these expressions into the equation of motion, we get:
m * v' + c * v + k * y = 0.
Since c is equal to the instantaneous velocity of the weight, c = v. The equation of motion then becomes:
m * v' + v^2 + k * y = 0.
Substituting the known values of m, k, y0, and v0 into the equation, we have:
3.629 * v' + v^2 + 14.71 * y = 0.
Now, we can solve this second-order nonlinear ordinary differential equation using appropriate numerical methods or software.
Once we have the solution to the differential equation, we can find the maximum vertical displacement by examining the peak values of the oscillatory motion.