Problems

Two blocks \(A\) and \(B\) of weight \(W_1\) and \(W_2\) respectively are connected by a string and placed on a rough table. The coefficient of friction between blocks and table is \(\mu\). A force \(P\), less than \(\mu(W_1+W_2)\), is applied to \(A\) in the direction \(BA\), and is gradually turned round in the horizontal plane. Show that the least value that \(P\) may have in order that \(A\) and \(B\) begin to slip simultaneously is \(\mu(W_1^2 + W_2^2)^{1/2}\), and that the slipping begins when \(P\) has turned through an angle \[ \cos^{-1}\frac{\mu^2(W_2^2-W_1^2)+P^2}{2\mu W_2 P}. \] Describe exactly what happens if \(\mu W_1 < P < \mu(W_1^2+W_2^2)^{1/2}\).

\(P, Q, R\) are points on a conic with focus \(S\) which vary in such a way that the angles \(PSQ, QSR\) remain constant. Shew that the locus of the point in which the tangent at \(P\) meets \(QR\) is a conic whose focus is also \(S\).

Sum, for any positive integer \(n\),

1. [(i)] \(\sin\theta + \sin 2\theta + \sin 3\theta + \dots + \sin n\theta + \frac{1}{2}\sin(n+1)\theta\),
2. [(ii)] \(\sin\frac{\theta}{2} + \sin\frac{3\theta}{2} + \sin\frac{5\theta}{2} + \dots + \sin\frac{2n+1}{2}\theta\);

Prove that the chords \(PQ\) of the rectangular hyperbola \(H \equiv xy-c^2=0\) which subtend a right angle at a fixed point \(A(h,k)\) envelop a conic \(\Sigma\) whose tangential equation is \[ c^2(l^2+m^2)+(h^2+k^2)lm+kl+hm=0. \] Shew that \(\Sigma\) is a parabola whose focus is \(A\) and whose directrix is the polar of \(A\) with respect to \(H\), and find the equation of the locus of the foot of the perpendicular from \(A\) to \(PQ\).

A weight \(W\) is suspended from a fixed point \(A\) by a uniform string of length \(l\) and weight \(wl\). The weight is drawn aside by a horizontal force \(P\). Show that in the equilibrium position the distance of \(W\) from the vertical through \(A\) is \[ \frac{P}{w}\left\{\sinh^{-1}\left(\frac{W+lw}{P}\right) - \sinh^{-1}\frac{W}{P}\right\}. \]

Describe briefly the process of reciprocation with respect to (a) a general conic, (b) a circle. A conic has a given focus \(S\), passes through a given point \(P\), and touches a given line \(l\). Shew that its directrix envelopes a conic which passes through \(S\).

Shew that, if \(\alpha, \beta, \theta, \phi\) lie between \(0\) and \(\pi\), and if \(\alpha+\beta=\theta+\phi\), and \(0 < \alpha - \beta < \theta - \phi\), then \[ \sin\alpha\sin\beta > \sin\theta\sin\phi. \] Deduce that, in any triangle \(ABC\), \[ 8 \sin A \sin B \sin C \le 3\sqrt{3}, \] except when \(ABC\) is equilateral; and hence, or otherwise, prove that of all the triangles which can be inscribed in a given circle, the equilateral has the largest area.

Shew that, if \(n\) straight lines are drawn in a plane in such a way that no two are parallel and no three meet in a point, then the plane is divided into \(\frac{1}{2}(n^2+n+2)\) regions. How many of these regions stretch to infinity?

Show that the path of a particle moving freely under gravity is a parabola, and that the velocity at any point is the same as would be acquired by the particle falling vertically from rest at the level of the directrix. A ball is thrown from a point on the ground so as to pass over a wall of height 25 feet situated at a distance of 60 feet from the thrower. The velocity is the least that will allow the ball to clear the wall. Show that the ball again reaches the ground in about 2.8 seconds.

Prove that the circumcircle of the triangle formed by the feet of the three normals from \((h, k)\) to the parabola \(y^2=4ax\) passes through the vertex of the parabola, and that the coordinates of its centre are \(\left( a+\frac{h}{2}, \frac{k}{4} \right)\).