Consider the spring-mass system whose motion is governed by...

Consider the spring-mass system whose motion is governed by the initial-value problem $\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=0, \quad y(0)=1, \quad \frac{d y}{d t}(0)=-3$ (a) Determine the position of the mass at time $t$ (b) Determine the time when the mass passes through the equilibrium position. (c) Make a sketch depicting the general motion of the system.

Consider the spring-mass system whose motion is governed by the initial-value problem
$\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=0, \quad y(0)=1, \quad \frac{d y}{d t}(0)=-3$
(a) Determine the position of the mass at time $t$
(b) Determine the time when the mass passes through the equilibrium position.
(c) Make a sketch depicting the general motion of the system.

Key Concepts

Equilibrium Position

The equilibrium position is where the net forces acting on the system cancel out and the system is at rest. In the context of a spring-mass system, it is the position where the mass would remain if no external disturbances were applied. Determining when the mass passes through this position can give insights into the oscillatory behavior of the system.

Second Order Linear Differential Equations

These are differential equations that involve the second derivative of a function, and they are used to model dynamic systems such as mechanical oscillators. Their solutions often involve finding a general solution composed of homogeneous and particular parts, and they can describe a wide range of physical phenomena including vibrations, circuits, and mechanical systems.

Characteristic Equation

In solving homogeneous linear differential equations with constant coefficients, the characteristic equation is derived by assuming solutions of an exponential form. The roots of this equation determine the nature of the system's response—whether the motion is oscillatory, overdamped, or critically damped.

Damped Oscillations

Damping refers to the presence of a term in the differential equation that causes the amplitude of oscillations to decrease over time, due to effects such as friction or resistance. Damped oscillations are typical in realistic physical systems where energy is dissipated, leading to motion that gradually dies out.

Initial Value Problems

An initial value problem involves not only a differential equation but also specified initial conditions that determine a unique solution. This approach is crucial in physical applications where the state of the system at a starting time is known, allowing for precise prediction of the system's future behavior.

Transcript

00:01 Okay, so in this problem, to start finding the displacement, we are going to first find the auxiliary equation that's p of r is going to be equal to r squared plus 3r plus 2.

00:13 We said that equal to 0.

00:14 If you notice, we can factor this, so the factors of 2 are 2 and 1, which does add up to 3.

00:20 So our factors are going to be r plus 2, r plus 1 is equal to 0, so that our roots are going to be negative 1 or negative 2 and negative 2 and negative.

00:30 1.

00:31 So our general solution, y of t, is going to look like c1, e to the negative 2t, plus c2e to the negative t.

00:45 Now we're going to use our initial conditions to solve for c1 and c2.

00:49 So to solve for c1 at c2, now we're going to first find y prime of t.

00:56 So it's going to be negative 2, c1, e to the negative 2t, and then minus c2e to the negative t.

01:04 Then we're going to plug in 0.

01:07 So y of 0 is equal to c1 plus c2, which is equal to 1.

01:13 Then my prime of 0 is going to be, and then again, sorry, e to the 0 is 1.

01:21 So we get negative 2, c1 minus c2 is equal to negative 3.

01:29 So if you notice we can add these two, and the c2s will cancel out.

01:34 So we'll get negative c1 is going to be equal to negative 2.

01:38 So c1 is equal to 2.