Question

A mass of 1 slug is suspended from a spring whose spring constant is$\mathbf{9}\mathbf{lb}\mathbf{/}\mathbf{ft}$ . The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of$\sqrt{\mathbf{3}}\mathbf{ft}\mathbf{/}\mathbf{s}$ . Find the times at which the mass is heading downward at a velocity of $\mathbf{3}\mathbf{ft}\mathbf{/}\mathbf{s}$.

Short answer

$\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{7}\mathbf{\pi }}{\mathbf{18}}\mathbf{+}\frac{\mathbf{2}\mathbf{n\pi }}{\mathbf{3}}\mathbf{,}\mathit{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{\dots }$

Step-by-step solution

Definition

A differential equation is an equation with one or more derivatives of a function.

Form differential equation

A mass of$m=1$slug is suspended from a spring whose spring constant is$k=9\text{lb}/\text{ft}$.

The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of$3\text{ft}/\text{s}$.

Using the differential equation,$m\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}=-kx$we have

$\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}=-9x$

$\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}+9x=0$

Find solution

The auxiliary equation of the above differential equation is,

$$\begin{gathered} m^2+9=0 \\ {m}^2=-9 \\ m=\pm 3i \end{gathered}$$

Therefore, the general solution of the above differential equation is.$x\left(t\right)=c_{1}cos3t+c_{2}sin3t$

Then$x^{\text{'}}\left(t\right)=-3c_{1}sin3t+3c_{2}cos3t$

Apply initial condition

Use the initial condition, $x\left(0\right)=-1$to get,

$$\begin{gathered} c_1·1+c_2·0=-1 \\ {c}_1=-1 \end{gathered}$$

Use the initial condition,${\mathit{x}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\sqrt{\mathbf{3}}$to get,

localid="1668506294448" $$\begin{gathered} c_1·0+3c_2·1=-\sqrt{3} \\ 3c_2=-\sqrt{3} \\ {c}_2=-\frac{\sqrt{3}}{3} \end{gathered}$$

Thus, the solution of the differential equation islocalid="1668506302837" $x\left(t\right)=-cos3t-\frac{\sqrt{3}}{3}sin3t$

Find amplitude

Compare the solution with $x\left(t\right)=c_{1}cos\omega t+c_{2}sin\omega t$.

We have$\omega =3$and$c_1=-1,c_2=-\frac{\sqrt{3}}{3}$

So, the amplitude is,

$$\begin{gathered} A=\sqrt{{\mathrm{c}}_1^2+{\mathrm{c}}_2^2} \\ =\sqrt{1+\frac{3}{9}} \\ =\frac{2}{\sqrt{3}} \end{gathered}$$

And

$\begin{array}{rcl}tan\varphi & =& \frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}\\ & =& \frac{-1}{(-\sqrt{3}/3)}\\ & =& \sqrt{3}\\ \varphi & =& ta{n}^{-1}\left(\sqrt{3}\right)\\ \varphi & =& \frac{\mathrm{\pi }}{3}\\ & & \end{array}$

Find solution

Since$sin\phi =\frac{{\mathrm{c}}_1}{\mathrm{A}}<0$and$cos\varphi =\frac{{\mathrm{c}}_2}{\mathrm{A}}<0$, soshould lie in third quadrant.

Thus,$\varphi =\pi +\frac{\mathrm{\pi }}{3}=\frac{4\mathrm{\pi }}{3}$.

Therefore, the solution can be rewrite as,

$$\begin{gathered} x\left(t\right)=Asin(wt+\varphi ) \\ =\frac{2}{\sqrt{3}}sin(3\mathrm{t}+\frac{4\mathrm{\pi }}{3}) \end{gathered}$$

Then,$\begin{array}{rcl}\mathit{x}\mathbf{\left(}\mathit{t}\mathbf{\right)}& \mathbf{=}& \frac{\mathbf{2}}{\sqrt{\mathbf{3}}}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{3}\mathbf{t}\mathbf{+}\frac{\mathbf{4}\mathbf{\pi }}{\mathbf{3}}\mathbf{)}\mathbf{·}\mathbf{3}\\ & \mathbf{=}& \mathbf{2}\sqrt{\mathbf{3}}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{3}\mathbf{t}\mathbf{+}\frac{\mathbf{4}\mathbf{\pi }}{\mathbf{3}}\mathbf{)}\\ & & \end{array}$

Find time

When mass is heading downward at a velocity at$3\text{ft}/\text{s}$.

Then${\mathit{x}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{3}$, so,

$\mathbf{\backslash }\mathbf{[}\mathbf{\backslash }\mathit{b}\mathit{e}\mathit{g}\mathit{i}\mathit{n}\mathbf{\{}\mathit{a}\mathit{r}\mathit{r}\mathit{a}\mathit{y}\mathbf{\left\}}\mathbf{\right\{}\mathbf{*}\mathbf{\left\{}\mathbf{20}\mathbf{\right\}}\mathbf{\left\{}\mathit{c}\mathbf{\right\}}\mathbf{\}}\mathbf{}\mathbf{}\mathbf{}\mathbf{\{}\mathbf{2}\mathbf{\backslash }\mathit{s}\mathit{q}\mathit{r}\mathit{t}\mathbf{}\mathbf{3}\mathbf{}\mathbf{\backslash }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{e}\mathit{f}\mathit{t}\mathbf{(}\mathbf{}\mathbf{\{}\mathbf{3}\mathit{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{4}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{3}\mathbf{\right\}}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathit{r}\mathit{i}\mathit{g}\mathit{h}\mathit{t}\mathbf{)}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathbf{\backslash }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\{}\mathbf{\backslash }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{e}\mathit{f}\mathit{t}\mathbf{(}\mathbf{}\mathbf{\{}\mathbf{3}\mathit{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{4}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{3}\mathbf{\right\}}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathit{r}\mathit{i}\mathit{g}\mathit{h}\mathit{t}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{\backslash }\mathit{s}\mathit{q}\mathit{r}\mathit{t}\mathbf{}\mathbf{3}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{2}\mathbf{\right\}}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{d}\mathit{o}\mathit{t}\mathit{s}\mathbf{}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathbf{\backslash }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\{}\mathbf{\backslash }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{e}\mathit{f}\mathit{t}\mathbf{(}\mathbf{}\mathbf{\{}\mathbf{3}\mathit{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{4}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{3}\mathbf{\right\}}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathit{r}\mathit{i}\mathit{g}\mathit{h}\mathit{t}\mathbf{)}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\backslash }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{e}\mathit{f}\mathit{t}\mathbf{(}\mathbf{}\mathbf{\{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\{}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\}}\mathbf{\left\{}\mathbf{6}\mathbf{\right\}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathit{n}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathit{r}\mathit{i}\mathit{g}\mathit{h}\mathit{t}\mathbf{)}\mathbf{,}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{d}\mathit{o}\mathit{t}\mathit{s}\mathbf{}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathbf{\backslash }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\{}\mathbf{3}\mathit{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{4}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{3}\mathbf{\right\}}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\{}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\}}\mathbf{\left\{}\mathbf{6}\mathbf{\right\}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathit{n}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{,}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{d}\mathit{o}\mathit{t}\mathit{s}\mathbf{}\mathbf{\}}\mathbf{}\mathbf{\backslash }\mathbf{\backslash }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\{}\mathbf{3}\mathit{t}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}\mathbf{-}\mathbf{}\mathbf{\backslash }\mathit{f}\mathit{r}\mathit{a}\mathit{c}\mathbf{\left\{}\mathbf{\right\{}\mathbf{7}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{\left\}}\mathbf{\right\}}\mathbf{\left\{}\mathbf{6}\mathbf{\right\}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathit{n}\mathbf{\backslash }\mathit{p}\mathit{i}\mathbf{}\mathbf{,}\mathbf{\}}\mathbf{\&}\mathbf{\{}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{\backslash }\mathit{l}\mathit{d}\mathit{o}\mathit{t}\mathit{s}\mathbf{}\mathbf{\}}\mathbf{}\mathbf{}\mathbf{\backslash }\mathit{e}\mathit{n}\mathit{d}\mathbf{\{}\mathit{a}\mathit{r}\mathit{r}\mathit{a}\mathit{y}\mathbf{\}}\mathbf{\backslash }\mathbf{]}$localid="1668506395397" $\begin{array}{ccc}2\sqrt{3}\cos \left(3t+\frac{4\pi }{3}\right)& =3& \\ \cos \left(3t+\frac{4\pi }{3}\right)=\frac{\sqrt{3}}{2}& & n=0,1,2,\dots \\ \cos \left(3t+\frac{4\pi }{3}\right)& =\cos (\frac{\pi }{6}+2n\pi ),& n=0,1,2,\dots \\ 3t+\frac{4\pi }{3}& =\frac{\pi }{6}+2n\pi ,& n=0,1,2,\dots \\ 3t& =-\frac{7\pi }{6}+2n\pi ,& n=0,1,2,\dots \end{array}$

Thus, the times at which the mass is heading downward at a velocity oflocalid="1668506398627" $\mathbf{3}\mathbf{\text{ft}}\mathbf{/}\mathbf{\text{s}}$islocalid="1668506402264" $\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{7}\mathbf{\pi }}{\mathbf{18}}\mathbf{+}\frac{\mathbf{2}\mathbf{n\pi }}{\mathbf{3}}\mathbf{,}\mathit{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{\dots }$