10273 problems found
A circular sheet of metal (of negligible thickness) is cut into two sectors of angles \((1+t)\pi\) and \((1-t)\pi\) respectively, and each piece is bent into the form of a right circular cone by joining together its two bounding radii. If \(V(t)\) is the sum of the volumes of the two cones, prove that \(V(t)\) has a minimum when \(t=0\). Deduce, by general considerations, that \(V(t)\) is greatest when \(t=\pm t_0\), where \(t_0\) is a certain number satisfying \(0 < t_0 < 1\).
A uniform rod \(AB\) of mass \(m\) and length \(a\) can turn freely about a fixed point \(A\). A small ring of mass \(m'\) slides smoothly along the rod, and is attached by a light inelastic string of length \(b\) (\(b
A smooth cylinder, whose normal cross section is a semi-circle of radius \(a\), is fixed with its plane face horizontal and in contact with the ground. A uniform chain lies in a small heap at the top of the cylinder, except for a length \(\frac{1}{4}\pi a\) which hangs down one side of the cylinder, the end just reaching the ground. The chain is released from rest. Assuming that each link is suddenly jerked into motion as the chain runs, show that, so long as a length \(x\) of the chain moves in contact with the cylinder, the velocity \(v\) of the chain satisfies the equation \[ \pi a v \frac{dv}{dx} + 2v^2 = 2ga, \] where \(x\) is the length of the chain heaped upon the ground. Hence show that \[ v^2 = ga(1-e^{-4x/\pi a}). \]
Prove that the polars of a fixed point \(A\) with respect to a system of confocal conics envelop a parabola touching the axes of the conics. Prove that the directrix of the parabola is the line joining \(A\) to the centre \(O\) of the confocals and that, if \(S\) is the focus of the parabola, the axes are the bisectors of the angle \(AOS\).
A side \(a\) and the opposite angle \(A\) of a triangle \(ABC\) are measured and found to be 6 inches and 30 degrees respectively, and the radius \(R\) of the circumcircle is calculated from these measurements. If each measurement is liable to a maximum error of 1 per cent. (in either direction), prove that the calculated value of \(R\) may be in error to the extent of about 1.9 per cent.
Masses \(m_1, m_2, \dots m_n\) are attached to points of a light inextensible string which hangs in equilibrium, suspended by its two ends. If the lengths of the segments of the string are given, together with the relative positions of the two ends of the string, show how to obtain sufficient equations to determine the inclinations of the segments and the tensions in them. If the masses are each equal to \(m\) and are attached at equal horizontal intervals \(h\), show that the points of attachment lie on a parabola of latus rectum \(2hT_0/mg\), where \(T_0\) is the horizontal component of the tension. Show also that an equal parabola touches the segments of the strings at their middle points, and that the distance between the vertices of these two parabolas is \(mgh/8T_0\).
A smooth sphere of mass \(m\) collides with another smooth sphere of mass \(m'\) at rest, and after the collision the spheres move in perpendicular directions. Assuming only (i) the conservation of momentum, (ii) that the spheres remain free from rotation, and (iii) that no kinetic energy is gained in the collision, prove that \(m'\) cannot be less than \(m\). Prove further that, if the collision is of the usual type with a coefficient of restitution \(e\), then \(em' = m\). Deduce that \(e\) cannot exceed unity.
A point \(P\) of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] is joined to the points whose co-ordinates are \((\pm k, 0)\) and the joins meet the ellipse again in \(Q\) and \(R\). Prove that the pole of the line \(QR\) with respect to the ellipse lies on the ellipse \[ \frac{x^2}{a^4} + \frac{(a^2-k^2)^2}{(a^2+k^2)^2}\frac{y^2}{b^2} = 1. \]
A point \(Q\) is taken on the tangent at \(P\) to a plane curve \(\Gamma\) so that \(PQ\) is of fixed length. Prove that the normal at \(Q\) to the locus of \(Q\) when \(P\) moves along \(\Gamma\) passes through the centre of curvature of \(\Gamma\) at \(P\).
A boy of mass \(m\) stands on the horizontal floor of a truck of mass \(M\) that is free to move on level rails. He jumps, in a vertical plane parallel to the rails, so as just to clear a vertical tail-board, which is of height \(h\) and distant \(a\) horizontally. Show that his least initial velocity \(U\), relative to the truck, is given by \(U^2 = g\{(a^2+h^2)/h + h\}\). Find also the magnitude of the least impulsive reaction between the boy and the truck, and show that its direction \(\theta\) with the horizontal is given by \(\tan 2\theta = -Ma/(m+M)h\).