Further Sheets on Applied Mathematics
Dynamics 1
- A particle, of mass $m$, has position vector $$ \mathbf{r}(t)=(x(t),y(t))=(3\sin 2t+4\cos 2t,3t+2) $$ at time $t$.
- Determine the particle's momentum $\displaystyle m \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}$ at time $t$.
- Determine the particle's kinetic energy $\displaystyle \frac{1}{2}m \left| \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \right|^2$ at time $t$.
- At what times is the particle's kinetic energy maximal?
- Determine the particle's acceleration $\displaystyle \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}$ at time $t$. Show that $$ \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}=-4x(t)\mathbf{i}. $$
- Consider a particle of mass $m$ moving in one vertical dimension with height $y(t)$. It moves under gravity, so that its acceleration always satisfies $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}t^2}=-g$. Initially the particle is projected from ground-level with speed $v$. That is, at $t=0$, we have $y=0$ and $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}=v$.
- Determine $y(t)$.
- What is the greatest height achieved by the particle?
- Find the time taken to return to ground-level.
- Show that the quantity $$ E=\frac{1}{2}m\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2+mgy$$ is constant throughout the motion.
- A particle of mass $m$ moves along the $x$-axis under the force $F(t)$, at time $t$, given below. $$ F(t)=\begin{cases} 3& 0\leq t <2,\\ 1& 2\leq t<3,\\ 2& 3\leq t \leq 5, \end{cases} $$and otherwise moves under no force. Initially, at $t=0$, the particle is at rest at $x=0$.
Newton's Second Law states that $$ F(t)=m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}. $$ Determine $x$ and $\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}$. On separate axes, sketch graphs of $x$ and $\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}$ against $t$. - A particle of mass $m$ moves along the $x$-axis under a force $F(x)$, when at position $x$, given below $$ F(x)=\begin{cases} -kx^3& -a<x<a\\ 0& |x|\geq a. \end{cases} $$ Initially, at $t=0$, we have $x=0$ and $\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}=u\geq 0$
- Let $\displaystyle v=\frac{\mathrm{d}x}{\mathrm{d}t}$. Show that $$ \frac{\mathrm{d}^2x}{\mathrm{d}t^2}=v\frac{\mathrm{d}v}{\mathrm{d}x}. $$
- From Newton's Second Law, show that $$ \frac{1}{2}mv^2+\frac{1}{4} kx^4=E $$ is constant throughout the motion.
- Find the minimum value $U$ of $u$ such that the particle moves outside of the interval $-a<x<a$. If $u<U$ what is the maximum value of $x$?
- Show that if $u=U$ then the time $T$ taken for the particle to reach $x=a$ equals $$ T=\sqrt{\frac{2m}{k}}\int_0^a \frac{\mathrm{d}x}{\sqrt{a^4-x^4}}. $$
Dynamics 2
- Say a projectile of mass $m$ is shot at time $t=0$ from the origin with speed $V$ at an angle $\alpha$ to the horizontal. Throughout the motion the particle is acted on by gravity so that $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}t^2}=-g$.
- Write down the initial conditions $x(0)$, $y(0)$, $x'(0)$, $y'(0)$.
- Determine the particle's position vector $(x(t),y(t))$ at time $t$.
- Determine where the projectile lands (returns to ground level $y=0$).
- What value of $\alpha$ maximises the distance travelled?
- Show that $$\frac{1}{2}m\left(\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2\right)+mgy=\frac{1}{2}mV^2$$ throughout the motion.
- If a spring, with spring constant $\alpha$, is stretch by an extension $x$, Hooke's Law states that the force on the particle has magnitude $\alpha |x|$ towards the equilibrium. Thus whether the extension $x$ is positive (an extension) or negative (a compression) Newton's Second Law gives $$ m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\alpha x $$ Show that the general solution of this equation is $$ x(t)=A\cos\omega t +B \sin \omega t, $$ for constants $A$ and $B$, and where $\omega^2=\alpha/m$. Show that $$ \frac{1}{2}m\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\frac{1}{2}\alpha x^2 = E $$ is constant throughout the motion. What does the quantity $\frac{1}{2}\alpha x^2$ represent?
- Say that the particle and spring in Question 2 lie on a rough table, so that there is a resistant frictional force of magnitude $\mu mg$ (coefficient of friction $\mu$) when the particle is in motion, and we have $$ m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\alpha x +\mu mg\quad\text{when $x\geq0$.} $$ Say that we have initially $x(0)=\varepsilon>0$ and $x'(0)=0$. What happens if $\mu \geq \alpha \varepsilon/(mg)$? Show that if $$ \frac{\varepsilon \alpha}{2mg}<\mu < \frac{\varepsilon \alpha}{mg} $$then the particle comes to rest before its normal equilibrium position, and find the value of $x$ where this occurs.
Show that $$ \frac{1}{2}m\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+\frac{1}{2}\alpha x^2=\frac{1}{2}\alpha \varepsilon^2 +\mu m g (x-\varepsilon) $$ throughout the motion and explain the significance of the terms in this identity. - Consider a mass $m$ at the end of a light inextensible rod of length $l$ making small swings under gravity; let $\theta$ denote the angle the rod makes with the vertical.
- Note that $\mathbf{r}=(l \sin \theta, -l \cos \theta)$. What do $\displaystyle \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}$ and $\displaystyle \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}$ equal?
- Use Newton's Second Law to show that $$ l\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}=-g \sin \theta, $$and find an expression for the tension in the rod.
- Show throughout the motion that $\displaystyle \frac{1}{2}ml^2\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^2-mgl\cos \theta=E$ is constant.
- Say that the pendulum's oscillations are small enough that the approximation $\sin\theta\approx\theta$ applies. Show that the pendulum's swings have period $2\pi\sqrt{l/g}$,
- More generally if the particle starts off with $\theta=\alpha$, $\displaystyle \frac{\mathrm{d}\theta}{\mathrm{d}t}=0$, show that the oscillations have exact period $$ 4\sqrt{\frac{l}{2g}}\int_0^\alpha \frac{\mathrm{d}\theta}{\sqrt{\cos\theta-\cos\alpha}}. $$