Part IA, 2001, Paper 4

# Part IA, 2001, Paper 4

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4.I.3A

commentDerive the equation

$\frac{d^{2} u}{d \theta^{2}}+u=\frac{f(u)}{m h^{2} u^{2}}$

for the motion of a particle of mass $m$ under an attractive central force $f$, where $u=1 / r$ and $r$ is the distance of the particle from the centre of force, and where $m h$ is the angular momentum of the particle about the centre of force.

[Hint: you may assume the expressions for the radial and transverse accelerations in the form $\ddot{r}-r \dot{\theta}^{2}, 2 \dot{r} \dot{\theta}+r \ddot{\theta}$.]

4.I.4A

commentTwo particles of masses $m_{1}$ and $m_{2}$ at positions $\mathbf{x}_{1}(t)$ and $\mathbf{x}_{2}(t)$ are subject to forces $\mathbf{F}_{1}=-\mathbf{F}_{2}=\mathbf{f}\left(\mathbf{x}_{1}-\mathbf{x}_{2}\right)$. Show that the centre of mass moves at a constant velocity. Obtain the equation of motion for the relative position of the particles. How does the reduced mass

$\mu=\frac{m_{1} m_{2}}{m_{1}+m_{2}}$

of the system enter?

4.II.10A

commentA spherical raindrop of radius $a(t)>0$ and density $\rho$ falls down at a velocity $v(t)>0$ through a fine stationary mist. As the raindrop falls its volume grows at the rate $c \pi a^{2} v$ with constant $c$. The raindrop is subject to the gravitational force and a resistive force $-k \rho \pi a^{2} v^{2}$ with $k$ a positive constant. Show $a$ and $v$ satisfy

$\begin{aligned} &\dot{a}=\frac{1}{4} c v, \\ &\dot{v}=g-\frac{3}{4}(c+k) \frac{v^{2}}{a} . \end{aligned}$

Find an expression for $\frac{d}{d t}\left(v^{2} / a\right)$, and deduce that as time increases $v^{2} / a$ tends to the constant value $g /\left(\frac{7}{8} c+\frac{3}{4} k\right)$, and thence the raindrop tends to a constant acceleration which is less than $\frac{1}{7} g$.

4.II.11A

commentA spacecraft of mass $m$ moves under the gravitational influence of the Sun of mass $M$ and with universal gravitation constant $G$. After a disastrous manoeuvre, the unfortunate spacecraft finds itself exactly in a parabolic orbit about the Sun: the orbit with zero total energy. Using the conservation of energy and angular momentum, or otherwise, show that in the subsequent motion the distance of the spacecraft from the Sun $r(t)$ satisfies

$\left(r-r_{0}\right)\left(r+2 r_{0}\right)^{2}=\frac{9}{2} G M\left(t-t_{0}\right)^{2},$

with constants $r_{0}$ and $t_{0}$.

4.II.12A

commentFind the moment of inertia of a uniform solid cylinder of radius $a$, length $l$ and total mass $M$ about its axis.

The cylinder is released from rest at the top of an inclined plane of length $L$ and inclination $\theta$ to the horizontal. The first time the plane is perfectly smooth and the cylinder slips down the plane without rotating. The experiment is then repeated after the plane has been roughened, so that the cylinder now rolls without slipping at the point of contact. Show that the time taken to roll down the roughened plane is $\sqrt{\frac{3}{2}}$ times the time taken to slip down the smooth plane.

4.II.9A

commentThe position $\mathbf{x}$ and velocity $\dot{\mathbf{x}}$ of a particle of mass $m$ are measured in a frame which rotates at constant angular velocity $\boldsymbol{\omega}$ with respect to an inertial frame. Write down the equation of motion of the particle under a force $\mathbf{F}=-4 m \omega^{2} \mathbf{x}$.

Find the motion of the particle in $(x, y, z)$ coordinates with initial condition

$\mathbf{x}=(1,0,0) \quad \text { and } \quad \dot{\mathbf{x}}=(0,0,0) \quad \text { at } t=0$

where $\boldsymbol{\omega}=(0,0, \omega)$. Show that the particle has a maximum speed at $t=(2 n+1) \pi / 4 \omega$, and find this speed.

[Hint: you may find it useful to consider the combination $\zeta=x+i y$.]

4.I.1E

comment(a) Show that, given a set $X$, there is no bijection between $X$ and its power set.

(b) Does there exist a set whose members are precisely those sets that are not members of themselves? Justify your answer.

4.I.2E

commentProve, by induction or otherwise, that

$\left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{c} n+1 \\ 1 \end{array}\right)+\cdots+\left(\begin{array}{c} n+m \\ m \end{array}\right)=\left(\begin{array}{c} n+m+1 \\ m \end{array}\right)$

Find the number of sequences consisting of zeroes and ones that contain exactly $n$ zeroes and at most $m$ ones.

4.II.5E

comment(a) Prove Wilson's theorem, that $(p-1) ! \equiv-1(\bmod p)$, where $p$ is prime.

(b) Suppose that $p$ is an odd prime. Express $1^{2} .3^{2} .5^{2} \ldots(p-2)^{2}(\bmod p)$ as a power of $-1$.

[Hint: $k \equiv-(p-k)(\bmod p)$.]

4.II.6E

commentState and prove the principle of inclusion-exclusion. Use it to calculate $\phi(4199)$, where $\phi$ is Euler's $\phi$-function.

In a certain large college, a survey revealed that $90 \%$ of the fellows detest at least one of the pop stars Hairy, Dirty and Screamer. $45 \%$ detest Hairy, $28 \%$ detest Dirty and $46 \%$ detest Screamer. If $27 \%$ detest only Screamer and $6 \%$ detest all three, what proportion detest Hairy and Dirty but not Screamer?

4.II.7E

comment(a) Prove that, if $p$ is prime and $a$ is not a multiple of $p$, then $a^{p-1} \equiv 1(\bmod p)$.

(b) The order of $a(\bmod p)$ is the least positive integer $d$ such that $a^{d} \equiv 1(\bmod p)$. Suppose now that $a^{x} \equiv 1(\bmod p)$; what can you say about $x$ in terms of $d$ ? Show that $p \equiv 1(\bmod d)$.

(c) Suppose that $p$ is an odd prime. What is the order of $x(\bmod p)$ if $x^{2} \equiv-1(\bmod p)$ ? Find a condition on $p(\bmod 4)$ that is equivalent to the existence of an integer $x$ with $x^{2} \equiv-1(\bmod p)$.

4.II.8E

commentWhat is the Principle of Mathematical Induction? Derive it from the statement that every non-empty set of positive integers has a least element.

Prove, by induction on $n$, that $9^{n} \equiv 2^{n}(\bmod 7)$ for all $n \geq 1$.

What is wrong with the following argument?

"Theorem: $\sum_{i=1}^{n} i=n(n+1) / 2+126$.

Proof: Assume that $m \geq 1$ and $\sum_{i=1}^{m} i=m(m+1) / 2+126$. Add $m+1$ to both sides to get

$\sum_{i=1}^{m+1} i=m(m+1) / 2+m+1+126=(m+1)(m+2) / 2+126$

So, by induction, the theorem is proved."