A particle moves in a straight line under the action of a given (variable) force. What physical quantity is represented by the area lying under the curve and bounded by two ordinates in the several cases when the abscissae and ordinates represent graphically (1) time and velocity, (2) time and force, (3) distance and force? A cable is used for raising loads, the greatest tension that the cable will bear being \(W\) tons weight. Shew, by consideration of the time-velocity graph, that the least time in which a load of \(W'\) tons can be raised through \(h\) feet from rest to rest by means of the cable is \(\sqrt{\left\{\dfrac{2h W}{g(W-W')}\right\}}\) seconds. The weight of the cable itself is negligible.
Establish the formulae \(dv/dt, v^2/\rho\), for the tangential and normal components of acceleration of a point moving on a given curve, where \(v\) denotes the velocity of the moving point, and \(\rho\) the radius of curvature of the curve. A light inextensible string \(AB\) of length \(l\) has the end \(A\) attached to a point of the surface of a fixed cylinder, whose cross-section is a simple closed oval curve whose intrinsic equation is \(s=f(\psi)\). Both \(\psi\) and \(s\) vanish at \(A\), and \(s\) increases always with \(\psi\). A particle of mass \(m\) is attached to \(B\), and is acted on by a constant force \(mc\) at right angles to the string, so that the string wraps itself round the cylinder, the whole motion being in a plane at right angles to the generators. Find the relation connecting the time with the inclination \(\psi\) of the straight part of the string to the tangent at \(A\), and shew that the tension of the string is \[ m \frac{u^2+2c\{l\psi - F(\psi)\}}{l-f(\psi)}, \] where \(u\) is the velocity of the particle when \(\psi=0\), and \(F(\psi)=\int_0^\psi f(x)dx\).
A particle of mass \(m\) moves in a plane, and is attracted towards a fixed origin \(O\) in the plane with a force \(mn^2r\), where \(r\) denotes distance from \(O\). It is projected from the point \((c,0)\), the axes being rectangular, with velocity \(nb\) and in a direction inclined at an angle \(\theta\) to the axis \(Ox\). Shew that the path of the particle is the ellipse \[ b^2(x\sin\theta-y\cos\theta)^2 + c^2y^2 = b^2c^2\sin^2\theta. \] Shew further that the points of the plane which are accessible by projection from the given point with the given velocity lie within the ellipse \[ \frac{x^2}{b^2+c^2} + \frac{y^2}{b^2} = 1. \]
Shew that four normals to an ellipse can be drawn through a general point of its plane. Shew that the normals at three points whose eccentric angles are \(\theta_1, \theta_2, \theta_3\) meet if \[ \sin(\theta_2+\theta_3) + \sin(\theta_3+\theta_1) + \sin(\theta_1+\theta_2) = 0, \] and that the fourth normal through their common point is the normal at the point whose eccentric angle is \(\theta_4\), where \(\theta_1+\theta_2+\theta_3+\theta_4\) is an odd multiple of \(\pi\).
If \(r\) denotes the distance of a point \(Q\) lying on a given curve from a fixed point \(S\) in the plane of the curve, and \(p\) is the perpendicular distance from \(S\) to the tangent at \(Q\) to the given curve, shew that the radius of curvature at \(Q\) is \(r\dfrac{dr}{dp}\). If the given curve is an ellipse of semi-axes \(a\) and \(b\) (\(a>b\)) and \(S\) is a focus, shew that \[ \frac{l}{p^2} = \frac{2}{r} - \frac{1}{a}, \quad \text{where } l=b^2/a, \] and hence determine its maximum and minimum radii of curvature.
Sketch the curve \[ x^3 = 3xy^2 + a^2x + y^2. \] Trace the inverse of the curve in the circle \[ x^2+y^2=1, \] and find the area of a loop of this inverse.
Explain what is meant by saying that a certain event has probability \(r\) (\(0 \le r \le 1\)). \(X\) and \(Y\) are partners at bridge against \(A\) and \(B\). \(X\) is dummy and when he puts his hand on the table \(Y\) sees that six trumps are held by the opponents. Shew that the probability that \(A\) and \(B\) each hold three trumps is \(\dfrac{286}{805}\).
Solution: We are interested whether \(A\) has 3 out of 6 trumps and 10 out of 20 non-trumps. This can be done in \(\binom{6}{3} \cdot \binom{20}{10}\) ways, so the probability is \begin{align*} && p &= \frac{\binom{6}{3} \cdot \binom{20}{10}}{\binom{26}{13}} \\ &&&= 20 \cdot \frac{20!}{26!} \cdot \frac{(13!)^2}{(10!)^2} \\ &&&= 2\frac{13 \cdot 11}{5\cdot 23 \cdot 7} \\ &&&= \frac{286}{805} \end{align*}
Functions \(u(x), v(x)\) are defined by the equations \begin{align*} u''+u=0, &\quad v''+v=0, \\ u(0)=0, &\quad u'(0)=1; \\ v(0)=1, &\quad v'(0)=0, \end{align*} where and \(u'=\dfrac{du}{dx}\), etc. Without using trigonometrical functions, prove that
If \(f(x)\) is a function defined in the interval \((a
A chain consists of two portions \(AC, CB\), each of length \(l\), and of uniform densities \(w, w'\) respectively. \(A\) and \(B\) are attached to two points and the chain hangs under gravity in such a way that \(C\) is at the lowest point. Prove that, if the heights of \(A\) and \(B\) above \(C\) are \(h\) and \(h'\), \[ w\left(h - \frac{l^2}{h}\right) = w'\left(h' - \frac{l^2}{h'}\right). \] Prove also that the curvature at \(C\) of the portions \(AC, CB\) are in the ratio \(w:w'\).