91Ó°ÊÓ

The spring constant, denoted as \( k \), is a crucial factor that quantifies a spring's stiffness. In simple terms, it measures the amount of force needed to stretch or compress a spring by a unit length. The higher the spring constant, the stiffer the spring is. It is derived using Hooke's Law, which is given by the formula:
\[ F = k imes x \]where:

• \( F \) is the force exerted on the spring,
• \( k \) is the spring constant,
• \( x \) is the displacement of the spring from its rest position.

In the given problem, a force of 8 pounds causes an 8-foot spring to stretch to 4 feet beyond its natural length, making the displacement \( x = 4 \) feet. Solving for \( k \), we have:
\[ k = \frac{8}{4} = 2 \, \text{lb/ft} \]Thus, the spring constant is 2 lb/ft, indicating the amount of force needed per foot to stretch or compress the spring.

Damping force is a resisting force that acts opposite to the direction of motion, usually proportional to the velocity of the object moving through a medium. It is introduced into a system to reduce vibrations and oscillations. The damping force is critical in bringing a vibrating system to rest in an efficient manner.
In the mass-spring-damper system, the damping force is described as:
\[ F_d = c \times v \]where:

• \( F_d \) is the damping force,
• \( c \) is the damping coefficient,
• \( v \) is the instantaneous velocity of the moving mass.

In this scenario, the damping coefficient \( c \) is \( \sqrt{2} \), signifying that the damping force is \( \sqrt{2} \) times the velocity. This value helps in counteracting the energy in the system, eventually leading the mass to return to its equilibrium position smoothly.

The mass-spring-damper system is a classical model used in physics and engineering to explain and analyze the motion of a mass attached to a spring with a damping component. This model is used to represent the balance of forces acting on the mass.
The differential equation that governs this system is as follows:
\[ m \frac{d^2y}{dt^2} + c \frac{dy}{dt} + ky = 0 \]where:

• \( m \) is the mass,
• \( c \) is the damping coefficient,
• \( k \) is the spring constant,
• \( y \) is the displacement from the equilibrium position.

This equation takes into account the forces due to the spring, damping, and mass. It is a second-order linear differential equation that describes how the mass moves over time, responding to forces from the spring and damper.
In the problem, the mass calculation yields \( m = \frac{1}{4} \) slugs. Combining this with our other known values, the equation becomes simplified to:
\[ \frac{d^2y}{dt^2} + 4\sqrt{2} \frac{dy}{dt} + 8y = 0 \]This form reflects how the particular system will respond over time, with its damping and spring constant influencing the nature of the motion.

Initial conditions are critical parameters in solving differential equations, defining the system's state at the beginning of observation, typically at time \( t = 0 \). For dynamic systems like the mass-spring-damper system, they specify the object's initial displacement and velocity.
In this case study, we have:

• Initial position \( y(0) = 0 \)
• Initial velocity \( y'(0) = -5 \, \text{ft/s} \)

These initial conditions are used to determine specific constants in the general solution of the differential equation. They allow the formulation of a solution that precisely matches the actual behavior of the system right from the start.
Applying these conditions yields the constants \( C_1 = 0 \) and \( C_2 = -\frac{5}{2} \), leading to a particular solution:
\[ y(t) = e^{-2\sqrt{2}t} \left(-\frac{5}{2} \sin 2t\right) \] This solution reflects the system's dynamics with its initial velocity and position integrated into the expression for \( y(t) \).