A wheel of radius \(a\) rolls on a rough horizontal table so that the plane of the wheel is vertical and the point of contact describes a straight line. The centre of mass is a distance \(h\) from the axis of the wheel (which passes through its geometrical centre), and the radius of gyration of the wheel about its axis is \(k\). The wheel is started rolling from its stable equilibrium position so that the axis moves with velocity \(V\). Assuming that the wheel does not rise off the table, write down the equation of energy, and prove that when the wheel has rotated through an angle \(\theta\) its angular acceleration is \[ \frac{-h(g+V^2/a)(\frac{1}{2}(k^2+a^2)-ha)\sin\theta}{\{\frac{1}{2}(k^2+a^2)-ha\cos\theta\}^2}. \] Deduce the period of small oscillations about the equilibrium position.
Show that \[ \frac{2^{2n}}{2n} < \frac{(2n)!}{(n!)^2} < 2^{2n} \] and, by induction or otherwise, that \[ \frac{(2n)!}{(n!)^2} < 2^{2n-1} \] for \(n>5\). Deduce an inequality for \[ \prod_{\substack{m< p< 2m \\ p \text{ prime}}} p \] and hence, or otherwise, show that \[ \prod_{\substack{p \le m \\ p \text{ prime}}} p < 2^{2m} \] for all positive integers \(m\).
(i) Eight white discs numbered 1, 2, \dots, 8 and eight black discs are placed in a hat. A truthful man picks three discs at random and declares that one is white. Find the probability that at least two are white. (ii) Find also the probability if, instead of declaring that at least one of the three discs is white, he declares that one of the three is the white disc labelled 8.
The triangle \(ABC\) lies entirely inside the triangle \(DEF\). Show that the sum of the sides of \(ABC\) is less than the sum of the sides of \(DEF\). (You may assume without proof that any side of a triangle is less than the sum of the other two sides.)
The uniform scalene triangular lamina \(ABC\) is at rest in equilibrium freely suspended from a point \(K\) by three equal light inextensible strings \(KA, KB, KC\). Prove that the Euler line of the triangle \(ABC\) is a line of greatest slope of the plane \(ABC\). [The Euler line is the line containing the circumcentre, centroid, nine-point centre and orthocentre.]
Show that there are four normals to a central conic \(S\) through a general point \(P\). If \(P\) varies so that the line joining the feet of two of these normals passes through a fixed point, prove that the line joining the feet of the other two envelops a parabola which touches the principal axes of \(S\).
A bridge consists of a uniform plank of length \(2l\) and weight \(2w\), freely supported at each end at the same level. A man of weight \(10w\) walks across the bridge. Calculate, or show by diagrams, the shearing force and bending moment at each point of the plank when he is at a distance \(\lambda l\) (\(0 \le \lambda \le 1\)) from an end, and find the maximum numerical value of the bending moment, distinguishing between the cases \(\lambda < \frac{1}{6}\) and \(\lambda \ge \frac{1}{6}\). When during his passage does the bending moment reach its greatest value anywhere, and at what point of the plank? If the man is just able to cross the bridge without the plank breaking, show that he could not venture further than \(0.35l\) from an end had the plank been clamped at that end and free at the other.
Show that the radius of gyration of a triangular lamina about an axis perpendicular to its plane and passing through its centroid is \(\{\frac{1}{12}(a^2+b^2+c^2)\}^{\frac{1}{2}}\), where \(a,b\) and \(c\) are the lengths of the sides. A plane sheet of metal in the shape of a right-angled triangle is free to oscillate in its own plane about a horizontal axis through a vertex. Show that the length of the equivalent simple pendulum is equal to half the length of the hypotenuse if the vertex chosen is at the right angle, but is greater for either of the other vertices.
A uniform string of weight \(w\) is hung over a small rough cylindrical peg, the ends being allowed to dangle. If the point of suspension is chosen at random on the string, there is a probability \(\frac{1}{2}\) that the string will hang in equilibrium. Show that the coefficient of friction between the string and the peg is approximately 0.2206. The string has a weight \(\lambda w\) attached to one end and is again hung on the peg. Find the value of \(\lambda\) which makes the probability of its staying on the peg as large as possible.
A long straight wall of constant height \(2h\) is built on a horizontal piece of ground. A boy stands at a horizontal distance \(d\) from the wall, and is just able to throw a stone to hit the top of the wall at its nearest point. Assuming that he throws with equal velocity at any angle of projection, and that the stone leaves his hand at height \(h\) above the ground, show that he is able to hit a stretch of the bottom of the wall whose length is \[ 4h\{1+(1+d^2/h^2)^{\frac{1}{2}}\}^{\frac{1}{2}}. \]