Problems

The perpendicular from the origin \(O\) on to the tangent at a point \(P\) of a plane curve \(C\) is of length \(p\) and is inclined to the axis of \(x\) at an angle \(\alpha\). Show that the coordinates of \(P\) are \[ (p \cos \alpha - p' \sin \alpha, \ p \sin \alpha + p' \cos \alpha), \] where the dash stands for \(d/d\alpha\). Show also that the radius of curvature of \(C\) at \(P\) is \(\pm (p+p'')\). \par If \(C\) is a simple closed curve containing \(O\) in its interior and everywhere convex (i.e. lying entirely to one side of each of its tangents), prove that the perimeter of \(C\) is of length \[ \int_0^{2\pi} p d\alpha, \] and encloses an area \[ \frac{1}{2} \int_0^{2\pi} (p^2 - p'^2) d\alpha. \]

A light rod \(AB\) of length \(2a\) can rotate freely about one end \(A\). A particle of mass \(m\) is attached to the other end \(B\), and a particle of mass \(2m\) is attached to the mid-point \(M\). The rod is released from rest in a horizontal position. Find its angular acceleration immediately afterwards, and show that the bending moment at \(M\) is then \(\frac{3}{7}mga\). \par Find the bending moment at \(M\) and the tension in the upper half of the rod when the inclination of the rod to the vertical is \(\theta\).

A flywheel in the form of a uniform disc of radius 9 in. and mass 250 lb. can rotate without friction about an axis through its centre perpendicular to the disc; a light cord is wound several times round the axle, which is rough and of radius 1 in., and this cord carries at one end a mass of 50 lb., which hangs freely from the axle. If the other end of the cord is pulled with a tension of 100 lb. wt., find the acceleration with which the 50 lb. mass rises. \par Find also what braking couple must be applied, in addition, if the wheel is brought to rest in 10 revolutions from an angular velocity of 5 revolutions per second. Specify the units in which this couple is evaluated. Take \(g\) to be 32 ft./sec.\(^2\)

Prove that the locus of a point in space which is at the same given distance from each of two intersecting straight lines consists of two ellipses with a common minor axis.

Evaluate: \[ \int \frac{(x+1)dx}{x\sqrt{(x^2-4)}}, \quad \int_0^\infty \frac{dx}{\cosh^3 x}, \quad \int_0^{\pi/2} \frac{\cos x + 2 \sin x + 2}{(1+2\cos x)^2} dx. \]

A homogeneous solid block, made of material weighing 112 lb. per cubic foot, is in the shape of a rectangular parallelepiped with a square base of side 5 feet and a height of \(h\) feet. The block rests on level horizontal ground, the coefficient of friction between the block and the ground being 0.6, and it is subjected to a thrust of total magnitude \(10\sqrt{2}\) tons weight, uniformly distributed along an upper edge and acting at an angle of 45\(^\circ\) with the vertical, downwards and towards the block. Determine the range of permissible values of \(h\) for equilibrium, and state how, for values of \(h\) outside this range, equilibrium would be broken when the thrust is applied.

Any number \(n\) of coplanar forces having components \((X_r, Y_r)\) act at the points whose rectangular Cartesian coordinates are \((x_r, y_r)\) respectively (\(r=1, 2, \dots, n\)). If the forces are all turned through the same angle in the same sense, prove that whatever the value of the angle their resultant always passes through the point whose coordinates are \[ \frac{MP+NQ}{P^2+Q^2}, \quad \frac{MQ-NP}{P^2+Q^2}, \] where \begin{align*} P &= \Sigma X_r, \quad Q = \Sigma Y_r, \\ M &= \Sigma(x_r X_r + y_r Y_r), \quad N = \Sigma(x_r Y_r - y_r X_r). \end{align*}

The upper ends of three equal similar light springs obeying Hooke's law are fastened to smooth rings which slide on a fixed horizontal rod, and the lower ends of the springs are fastened to three points \(A, B, C\) of a light rigid rod; \(B\) is between \(A\) and \(C\), and the lengths \(AB, BC\) are not equal. A weight is attached to the rod at the point \(D\) between \(A\) and \(B\), and the system is allowed to hang freely. If in the equilibrium position the spring through \(C\) is to be compressed, prove that the distance of \(D\) from \(B\) must be greater than \[ \frac{AB \cdot AC}{AC+BC}. \]

State the principle of virtual work for the equilibrium of a system of bodies subject to frictionless constraints. \par Four uniform rods \(AB, BC, CD, DA\), each of weight \(W\) and length \(2a\), are freely jointed together to form a rhombus, which is suspended from \(A\). A point \(E\) in \(AB\) and a point \(F\) in \(CD\), at equal distances \(l\) (not equal to \(a\)) from \(B\) and \(D\) respectively, are joined by a light rod of length \(2c\) which keeps the rhombus in the form in which the angle \(BAC\) is equal to \(\alpha\). Determine the condition that the rod \(EF\) should be in tension and find the magnitude of the tension.

Obtain the equations \[ y = c \cosh \frac{x}{c}, \quad s = c \sinh \frac{x}{c} \] for the form of a uniform heavy string fixed at its ends and hanging freely. \par A uniform string of length \(2l\) hangs symmetrically across two smooth pegs which are fixed at a distance \(2a\) apart at the same level. Shew that the parameter \(c\) of the catenary in which the middle portion hangs is given by \(l=ce^{a/c}\). \par By considering the variation of the tension at the peg for symmetrical displacements, shew that the equilibrium is stable if \(l>ae\).