A heavy plane plate is dropped on to two identical parallel horizontal rough rollers whose axes are a distance \(a\) apart in the same horizontal plane. The rollers are rotating extremely rapidly and the coefficient of sliding friction \(\mu\) is constant. Discuss the motion of the plate according to the various senses of rotation of the rollers.
A batsman hits a cricket ball towards a fielder who is perfectly placed to catch it. Show that the rate of change of the tangent of the angle of elevation of the ball as seen by the fielder remains constant. The next batsman also hits the ball towards the fielder, but short so that the fielder must run forward to catch it. Show that if the fielder runs at a constant velocity so as to make the rate of change of the tangent of the elevation angle constant he will arrive in the right position to catch the ball.
In a cannery, peas of mass \(M\) come out of a pipe uniformly at a velocity \(V\) with a separation \(d\). A fly of mass \(m\) sitting on the end of the pipe hops with negligible speed on to a passing pea. How much energy must the fly's legs absorb in order to hang on to the pea? How far does the fly travel before the pea behind catches up? Just before the peas collide, the fly lets go of the first pea with no change of velocity. The peas then collide without loss of energy, after which the fly catches hold of the second pea. How fast is the first pea travelling after the collision? How far does the fly travel on the second pea before being caught up by the third pea?
A light inextensible string of length \(aL\) is attached at one end \(C\) to a smooth vertical wall and at the other end \(B\) to a uniform rigid straight rod \(AB\) of mass \(M\) and length \(L\). The end \(A\) rests against the wall; \(A\), \(B\) and \(C\) are not collinear, and the plane \(AB\) is vertical. Determine the inclination of the rod to the vertical and the limits on \(a\) between which equilibrium is possible. Show also that the tension in the string is \[\frac{3MgaL}{|2(a^2-1)|}.\]
Solve the vector equation \[\lambda \mathbf{x} + (\mathbf{x} \cdot \mathbf{a}) \mathbf{b} = \mathbf{c},\] where \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) are given non-zero vectors and \(\lambda\) is neither 0 nor \(-\mathbf{a} \cdot \mathbf{b}\). Derive solutions for the special cases \(\lambda = 0\) and \(\lambda = -\mathbf{a} \cdot \mathbf{b} (\neq 0)\), specifying any conditions needed on \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\).
The atmosphere at a height \(z\) above ground level is in equilibrium and has density \(\rho(z)\). By considering the force balance on a thin layer of the atmosphere and neglecting the curvature of the earth, show that the pressure \(p(z)\) is given by \[\frac{dp}{dz} = -\rho g,\] where \(g\) is the acceleration due to gravity (assumed constant). Hence derive an expression for the pressure in an isothermal atmosphere (in which \(p = k\rho\), where \(k\) is a constant) in terms of the pressure \(p_0\) at the surface of the earth. A large spherical balloon of radius \(a\) and total mass \(m\) floats with its centre at a height \(h\) above the surface of the earth. Show that \(h\) is given by \[e^{\alpha h/k} = \frac{2\pi p_0 k^2}{mg^3}\{e^{\alpha(a-1)}+e^{-\alpha(a+1)}\},\] where \(\alpha = ga/k\).
The real 6-dimensional vector space V consists of all homogeneous quadratics \begin{align*} p(x, y, z) \equiv ax^2 + by^2 + cz^2 + 2dyz + 2ezx + 2fxy \end{align*} in \(x, y\) and \(z\), under the usual definitions of addition and multiplication by scalars. Find the dimension of, and write down a basis for,
Positive rational 'weights' \(m_1, \ldots, m_n\) are attached to positive numbers \(a_1, \ldots, a_n\). Use the inequality connecting the arithmetic and geometric means to prove that \begin{align*} \frac{m_1a_1 + \ldots + m_na_n}{m_1 + \ldots + m_n} \geq (a_1^{m_1} a_2^{m_2} \ldots a_n^{m_n})^{1/(m_1 + \ldots + m_n)}. \end{align*} By attaching suitable weights to 1 and \(1 + x/n\), prove that, if \(x\) is positive, \begin{align*} \left(1 + \frac{x}{n+1}\right)^{n+1} \geq \left(1 + \frac{x}{n}\right)^n. \end{align*}
The real polynomial \(f(x)\) has degree 5. Prove that \begin{align*} \int_{-y}^{+y} f(x) dx = \frac{1}{2}y\{f(\lambda y) + f(\mu y) + f(-\lambda y) + f(-\mu y)\} \end{align*} for positive constants \(\lambda\) and \(\mu\) (independent of \(y\) and \(f\)) whose squares are the roots of a certain quadratic equation to be determined.
From the circumcentre \(S\) of a triangle \(ABC\), perpendiculars \(SD\), \(SE\) and \(SF\) are drawn to the sides \(BC\), \(CA\) and \(AB\) respectively, and produced to \(A'\), \(B'\) and \(C'\) so that \(D\), \(E\) and \(F\) are the mid-points of \(SA'\), \(SB'\) and \(SC'\). Prove that the triangles \(ABC\) and \(A'B'C'\) are congruent, that \(AA'\), \(BB'\) and \(CC'\) all have a common mid-point \(M\), and that a rotation about \(M\) moves one triangle to the other.