Assume that for all \(x\) such that \(|x| < 1\), \(\sin^{-1}x = \sum_{r=0}^{\infty} \frac{(2r)!}{2^{2r}(r!)^2}\frac{x^{2r+1}}{2r+1}\). Writing \(u_r\) for the coefficient of \(x^{2r+1}\) in the above expansion, show that \(\frac{u_r}{u_{r-1}} = \frac{(2r-1)^2}{2r(2r+1)} < 1\), for all \(r \geq 1\). By quoting this series with \(x = \frac{1}{2}\), express \(\pi\) as the sum of a series of positive terms; hence construct a flow diagram to calculate \(\pi\), accumulating terms up to and including the first whose value is less than \(10^{-10}\). Prove that the value of \(\pi\) computed is correct to within \(\frac{1}{3} \cdot 10^{-10}\).
A uniform solid, with total mass \(M\), occupies the volume obtained by rotating about the \(x\)-axis the area lying between the two parabolas \(y^2 = 4ax\) \((0 < x < b)\) and \(y^2 = 8ax - 4ab\) \((\frac{1}{2}b < x < b)\). Find the position of its mass centre and calculate its moment of inertia about the \(x\)-axis.
The surface of a lawn is a plane inclined to the horizontal at an angle \(\alpha\). A sprinkler is embedded in the surface, and emits droplets of water in all directions, the speed of projection being \(v\). Show that the region watered is an ellipse with area \(\pi v^4/g^2\cos^2\alpha\).
A circular disc rolls without slipping along a straight line, with uniform angular velocity. Show that the acceleration of each point of the disc is directed towards the centre. Discuss, without making detailed calculations, whether the same result holds if the disc rolls with non-uniform angular velocity.
Two equal uniform rods \(AB, BC\), each of length \(2a\) and weight \(W\), are freely jointed at \(B\). The angle \(ABC\) is maintained at a value \(2\alpha\) by means of a light string \(AC\). The rods are in equilibrium in a vertical plane with \(AB\) and \(BC\) resting on two small smooth pegs \(P, Q\), where \(PQ\) is horizontal and of length \(2c\) \((c > a\sin^3\alpha)\), and \(B\) is vertically above the midpoint of \(AC\). Show that the tension in \(AC\) is \(\frac{W\tan\alpha(c\textrm{cosec}^3\alpha-a)}{2a}\).
A particle of unit mass falls from a position of unstable equilibrium at the top of a rough sphere of radius \(a\). Show that the equations of motion may be written \(a\omega\frac{d\omega}{d\theta} = g\sin\theta - \mu R\), \(R = g\cos\theta - a\omega^2\), where \(\theta\) is the inclination to the upward vertical of the line from the particle to the centre of the sphere, \(\omega = \dot{\theta}\), and \(R\) is the reaction of the sphere on the particle. Show that if \(\mu = 0\), the particle leaves the sphere at \(\theta = \alpha\), where \(\cos\alpha = \frac{2}{3}\). Now suppose \(\mu\) is positive but small. Solve the first equation approximately by giving \(R(\theta)\) the value it has in the solution for \(\mu = 0\). Hence obtain an improved formula for \(R(\theta)\), and by regarding the required value of \(\theta\) as \(\alpha\) plus a small correction, show that the particle leaves the sphere where \(\theta = \alpha + \mu\left(2-\frac{4\alpha}{3\sin\alpha}\right)\) approximately. [Use the facts that, if \(x\) is small, \(\sin x\) and \(\cos x\) can be approximated by \(x\) and \(1\), respectively.]
Two uniform rough cylinders, each with radius \(a\), lie touching one another on a rough horizontal table. A third identical cylinder lies on these two. The end faces of all three cylinders are coplanar. The coefficient of friction for all pairs of surfaces in contact has the same value, \(\mu\). Find the least value of \(\mu\) for which the cylinders can be in equilibrium.
Prove that, if \(a\) and \(b\) are integers, then \(6a + 5b\) is divisible by 13 if and only if \(3a - 4b\) is. Determine all positive integers \(k\) such that if \(a\) and \(b\) are integers then \(6a + 5b\) is divisible by \(k\) if and only if \(3a - 4b\) is.
Suppose that of the 6 people at a party at least two out of every three know each other, and that all acquaintanceships are mutual. Prove that there are at least 3 people who all know each other. Does this assertion hold if the party consists of only 5 people?
Let \(n\) be an integer and let \(p\) be a prime. Prove that the exponent of \(p\) in the prime factorization of \(n!\) is given by \(\frac{n-s}{p-1}\), where \(s\) is the sum of the digits of \(n\) when written to the base \(p\). How many zeros are at the end of 1000!, when written to the base 60? [You are reminded that every integer \(n\) can be written as \(n = p_1^{a_1} \cdot \ldots \cdot p_k^{a_k}\), where \(p_1 < p_2 < \ldots < p_k\) are primes and \(a_1, \ldots, a_k\) are integers. The exponent of \(p\) in the prime factorization of \(n\) is \(a_i\) if \(p = p_i\) for some \(i\), \(1 \leq i \leq k\); otherwise it is zero.]