Question

A 1-kilogram mass is attached to a spring whose constant is 18 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 11 times the instantaneous velocity. Determine the equations of motion if the following is true?
a. the mass is initially released from rest from a point 1 meter below the equilibrium position
b. the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 11 m/s

Answer 1

Answer:

Let $x(t)$ denote the position (in meters, with respect to the equilibrium position of the spring) of this mass at time $t$ (in seconds.) Note that this question did not specify the direction of this motion. Hence, assume that the gravity on this mass can be ignored.

a. $\displaystyle x(t) = -\frac{9}{7}\, e^{-2 t} + \frac{2}{7}\, e^{-9 t}$.

b. $\displaystyle x(t) = \frac{2}{7}\, e^{-2 t} - \frac{9}{7}\, e^{-9 t}$.

Explanation:

Let $x$ denote the position of this mass (in meters, with respect to the equilibrium position of the spring) at time $t$ (in seconds.) Let $x^\prime$ and $x^{\prime\prime}$ denote the first and second derivatives of  $x$, respectively (with respect to time $t$.)

• $x^\prime$ would thus represent the velocity of this mass.
• $x^{\prime\prime}$ would represent the acceleration of this mass.

Constructing the ODE

Construct an equation using $x$, $x^\prime$, and $x^{\prime\prime}$, with both sides equal the net force on this mass.

The first equation for the net force on this mass can be found with Newton's Second Law of motion. Let $m$ denote the size of this mass. By Newton's Second Law of motion, the net force on this mass would thus be equal to:

$F(\text{net}) = m\, a = m\, x^{\prime\prime}$.

The question described another equation for the net force on this mass. This equation is the sum of two parts:

• The restoring force of the spring: $F(\text{spring}) = -k\, x$, where $k$ denotes the constant of this spring.
• The damping force: $F(\text{damping}) = - 11\,x^\prime$ according to the question. Note the negative sign in this expression- the damping force should always oppose the direction of motion.

Assume that there's no other force on this mass. Combine the restoring force and the damping force obtain an expression for the net force on this mass:

$F(\text{net}) = -k\, x - 11\, x^\prime$.

Combine the two equations for the net force on this mass to obtain:

$m\, x^{\prime\prime} = -k\, x - 11\, x^\prime$.

From the question:

• Size of this mass: $m = 1\; \rm kg$.
• Spring constant: $k = 18\; \rm N \cdot m^{-1}$.

Hence, the equation will become:

$x^{\prime\prime} = -18\, x - 11\, x^\prime$.

Rearrange to obtain:

$x^{\prime\prime} + 11\, x^\prime + 18\; x = 0$.

Finding the general solution to this ODE

$x^{\prime\prime} + 11\, x^\prime + 18\; x = 0$ fits the pattern of a second-order homogeneous ODE with constant coefficients. Its auxiliary equation is:

$m^2 + 11\, m + 18 = 0$.

The two roots are:

• $m_1 = -2$, and
• $m_2 = -9$.

Let $c_1$ and $c_2$ denote two arbitrary real constants. The general solution of a second-order homogeneous ODE with two distinct real roots $m_1$ and $m_2$ is:

$x = c_1\, e^{m_1\cdot t} + c_2\, e^{m_2\cdot t}$.

For this particular ODE, that general solution would be:

$x = c_1\, e^{-2 t} + c_2\, e^{-9 t}$.

Finding the particular solutions to this ODE

Note, that if $x(t) = c_1\, e^{-2 t} + c_2\, e^{-9 t}$ denotes the position of this mass at time $t$, then $x^\prime(t) = -2\,c_1\, e^{-2 t} -9\, c_2\, e^{-9 t}$ would denote the velocity of this mass at time

• The position at time $t = 0$ would be $x(0) = c_1 + c_2$.
• The velocity at time $t = 0$ would be $x^\prime(0) = -2\, c_1 - 9\, c_2$.

For section $\rm a.$:

$\left\lbrace\begin{aligned}& x(0) = -1 \\ &x^\prime(0) = 0\end{aligned}\right. \implies \left\lbrace\begin{aligned} &c_1 + c_2 = -1 \\ &-2\, c_1 - 9\, c_2 = 0\end{aligned}\right. \implies \left\lbrace\begin{aligned} &c_1 = -\frac{9}{7} \\ &c_2 = \frac{2}{7}\end{aligned}\right.$.

Hence, the particular solution for section $\rm a.$ will be:

$\displaystyle x(t) = -\frac{9}{7}\, e^{-2 t} + \frac{2}{7}\, e^{-9 t}$.

Similarly, for section $\rm b.$:

$\left\lbrace\begin{aligned}& x(0) = -1 \\ &x^\prime(0) = 11\end{aligned}\right. \implies \left\lbrace\begin{aligned} &c_1 + c_2 = -1 \\ &-2\, c_1 - 9\, c_2 = 11\end{aligned}\right. \implies \left\lbrace\begin{aligned} &c_1 = \frac{2}{7} \\ &c_2 = -\frac{9}{7}\end{aligned}\right.$.

Hence, the particular solution for section $\rm b.$ will be:

$\displaystyle x(t) = \frac{2}{7}\, e^{-2 t} - \frac{9}{7}\, e^{-9 t}$.