Problems

An octopus propels itself horizontally from rest by jet propulsion: while at rest it sucks a volume \(V\) of water into an internal cavity. It then propels itself by squirting this water out at a constant rate of \(Q\) units of volume per unit time, through a nozzle of cross- sectional area \(A\). Let the mass of the octopus plus the water contained in the cavity at time \(t\) after an ejection begins, be \(m(t)\), let its speed be \(u(t)\) and let the drag force exerted on the octopus by the surrounding water be \(D(t)\). Show that, during ejection, \begin{align} m\frac{du}{dt} = \frac{\rho Q^2}{A} - D \end{align} where \(\rho\) is the density of water. Given that \(D = ku^2\) (\(k\) constant), show that the speed attained by the octopus at the end of ejection is \begin{align} u_1 = Q\left(\frac{\rho}{kA}\right)^{\frac{1}{2}}\frac{\alpha-1}{\alpha+1} \end{align} where \begin{align} \alpha = \left(1+\frac{\rho V}{m_0}\right)^{2\frac{k}{(\rho A)^{\frac{1}{2}}}} \end{align} and \(m_0\) is the value of \(m\) before intake of the volume \(v\). State a condition to be satisfied by \(\alpha\) for the drag to be negligible during water ejection. Find the time after the end of ejection at which \(u = u_1/10\).

A uniform block of ice of mass \(m\) has the form of a circular cylinder of radius \(a\) and moment of inertia \(\frac{1}{2}ma^2\) about its axis. It is contained in a close-fitting, thin-walled cylindrical drum with the same radius, mass \(M\) and moment of inertia \(\frac{1}{2}Ma^2\) about its axis. The surroundings are sufficiently warm that a very thin layer of water forms between the ice and the drum, so that slipping can occur between them, resisted by a couple equal to \(k\) times the relative angular velocity. Starting from a state in which both drum and ice are at rest, the drum rolls without slipping down the line of greatest slope of an inclined plane which makes an angle \(\alpha\) with the horizontal. Show that the difference, \(\omega\), between the angular velocities of the ice and the drum tends to the constant value \(\omega_0\), where \begin{align} \omega_0 = \frac{mga\sin\alpha (m + M)}{k(3m + 4M)} \end{align} and that \(\omega = (1 - e^{-t})\omega_0\) after a time \begin{align} \frac{ma^2(m + 2M)}{k(3m + 4M)} \end{align}

A heavy uniform circular cylinder of radius \(r\) rests on a rough horizontal plane. A heavy uniform rod of length \(l\) lies across it, touching the plane at its end \(A\) and touching the cylinder tangentially at a point \(B\). The rod lies in a vertical plane perpendicular to the axis of the cylinder, and its centre of gravity lies between \(A\) and \(B\). The coefficient of friction at both points of contact on the rod is \(\mu\) with \(0 < \mu < 1\). Friction is limiting at both \(A\) and \(B\), and the cylinder does not slip or roll on the plane. Show by a geometrical method or otherwise that \[l\mu^3 + r\mu^2 - l\mu + r = 0.\]

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A bead is free to slide on a curved rough wire which lies in a vertical plane. The coefficient of friction between the bead and the wire is \(\mu\). When the bead is projected down the wire with speed \(v\), it continues to move at a constant speed. Use a coordinate system \((x,y)\) as shown, in which the \(x\)-axis is inclined at an angle \(\alpha\) above the horizontal with \(\tan\alpha = \mu\), to show that the wire satisfies \[\frac{dx}{ds} = \tanh(c-as), \quad \frac{dy}{ds} = \text{sech}(c-as),\] where \(a = g/(v^2\sin\alpha)\), \(c\) is constant and \(s\) is the arc length along the wire. With the aid of the substitution \[\sin u = \tanh(c-as), \quad \cos u = \text{sech}(c-as),\] or otherwise, show that when the origin is suitably chosen, the wire must form part of the curve with equation \[ax = \ln\cos ay.\]

The sequence of real numbers \(x_n\) satisfies \[x_{n+1} = x_n + x_{n-1}, \quad x_0 = a, \quad x_1 = b, \quad a \neq 0, \quad b \neq 0.\] Find a solution of the form \(x_n = A\lambda^n + B\mu^n\), and hence prove that as \(n \to \infty\) the ratio of successive terms in the sequence tends to a certain number (to be found), unless the ratio \(a:b\) takes a certain value. What happens then?

(i) Guess an expression for \[\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right) \cdots \left(1-\frac{1}{n^2}\right),\] valid for \(n \geq 2\), and prove your guess by mathematical induction. (ii) Show that \[\sum_{r=0}^{k} (-1)^r \binom{n}{r} = (-1)^k \binom{n-1}{k},\] for all \(k = 0, 1, \ldots, n-1\), where \(\binom{n}{r}\) is the usual binomial coefficient.

Let \(p(x)\) be a polynomial of degree 4, with real coefficients, and satisfying the property that, for all rational numbers \(\alpha\), \(p(\alpha)\) is a rational number. Prove that \(p(x)\) has rational coefficients. If \(q(x)\) is a polynomial with rational coefficients, and \(q(n)\) is an integer for every integer \(n\), does it follow that \(q(x)\) has integer coefficients? Give either a proof or a counter-example. [A rational number is a number of the form \(p/q\) where \(p\), \(q\) are integers, \(q \neq 0\).]

A \(3 \times 3\) floor-tile comprises nine unit squares. The small squares are to be coloured red, white or blue in such a way that two squares with an edge in common must be of different colours. Two tiles are considered to have the same colouring if one can be rotated into the other. How many differently coloured floor-tiles can be produced? [Hint: consider the number of ways to colour the cross obtained by deleting the four corner squares.]

Prove that if \(p\) is a positive prime number and if \(k = 1, \ldots, p - 1\), then the binomial coefficient \(\binom{p}{k}\) is divisible by \(p\). Deduce, by induction or otherwise, that \(n^p - n\) is divisible by \(p\), for all positive integers \(n\) and prime numbers \(p\).

If \(p\) is a positive integer and \(n\) an integer in the range 1 to \(p\), describe the positions in the Argand diagram of the \(p\) points \[\left(\cos\frac{2n\pi}{p+1} + i \sin\frac{2n\pi}{p+1}\right)^m, \quad m=1,2,\ldots,p.\] Hence or otherwise prove that \[\sum_{m=1}^{p} \cos\frac{2nm\pi}{p+1} = -1\] for any \(n\) in the range specified.