Problems

The diagram shows a light framework made of freely-jointed uniform rods, all of the same material and cross-section. It is supported in a vertical plane, with \(BD\) and \(ACE\) horizontal, on smooth piers at \(A, E\). The rod \(BC\) is vertical, and \(AC=BC=BD=\frac{1}{2}CE = a\) ft. If a load of 300 lb. hangs from \(C\), calculate the tensions or thrusts in the bars. Name the bars that are in compression. [The diagram shows a symmetrical truss. Points A and E are supports at the same horizontal level. Point C is midway between A and E. Point B is directly above C. Point D is horizontally level with B. The truss consists of rods AB, BC, CD, DE, AC, CE, BD.] If the bars can expand or contract very slightly under stress in such a way that the extension produced by a tension \(T\) lb. wt. is \(kT\) ft. per unit length of the bar, find the energy stored in the rods, and by applying the principle of virtual work (or otherwise) show that the sag at \(C\) due to loading is \[ \tfrac{1}{4} ak \{1400 + 1200\sqrt{2}\}. \]

Show that an infinity of straight lines can be drawn to meet three given straight lines \(a, b, c\) in space, of which no two intersect or are parallel. If a variable line meets \(a, b, c\) in \(X, Y, Z\), respectively, and if \(L\) and \(M\) are two fixed points on \(a\), prove that the lines \(LY\) and \(MZ\) meet, and that their point of intersection lies on a fixed straight line. Prove also that \(X, Y, Z\) describe homographic ranges on \(a, b, c\).

What do you understand by \(z^{p/q}\), where \(z\) is a complex number and \(p,q\) are positive integers? Does \(z^{p/q}\) mean the same thing as \((z^p)^{1/q}\), or as \((z^{1/q})^p\)? Find all the fourth roots of \(28+96i\).

Trace the curve \[ x^5 + y^5 = 5ax^2y^2 \quad (a>0). \] By writing \(y=tx\), or otherwise, prove that the area of the loop is \(\frac{5}{2}a^2\).

A tractor of mass \(M\) is moving against a constant frictional resistance \(R\) up a hillside inclined at an angle \(\alpha\) to the horizontal. If the power \(P\) developed is constant, show that the speed changes from \(V_1\) to \(V_2\) in a time \[ \frac{P}{MB^2} \log\left(\frac{P-MBV_1}{P-MBV_2}\right) - \frac{V_2-V_1}{B}, \] where \(B=g \sin\alpha + R/M\). Find an expression for the distance travelled during this change of speed.

Prove that in general three normals (real or imaginary) can be drawn to a parabola from an arbitrary point in its plane. If \(M\) is a fixed point on the parabola and \(P\) a variable point on the normal at \(M\), show that the line joining the feet of the other two normals from \(P\) is parallel to a fixed direction.

The coordinates \((x,y)\) of a point on a curve are given in terms of a parameter \(t\) by the equations \[ x=at^2, \quad y=bt^3. \] Prove that, if \(t_0 \ne 0\), the tangent at the point \(t=t_0\) cuts the curve again at the point \(t = -\frac{1}{2}t_0\). Prove that there are two lines, each of which is both a tangent and a normal to the curve, and obtain their equations.

Taking as your starting point the triangle of forces, develop the theory of the composition of parallel coplanar forces (proving all the theorems you need) so as to lead up to the following theorem, the proof of which should be given: \(P, Q, \dots\) are fixed points in a plane. Parallel forces of given magnitudes \(p, q, \dots\) act in the plane through \(P, Q, \dots\) respectively. Then in general the forces have a resultant of magnitude \(p+q+\dots\) which passes through a point \(G\) which is independent of the direction of the forces.

Obtain the expressions \[ \frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2, \quad \frac{1}{r}\frac{d}{dt}\left(r^2 \frac{d\theta}{dt}\right) \] for the components of the acceleration of point with the polar coordinates \((r, \theta)\). A bead of mass \(m\) can slide smoothly along a straight rod that is made to rotate in a horizontal plane with constant angular velocity \(\omega\) about one end \(O\). Initially \(r=a\) and \(\frac{dr}{dt}=0\), where \(r\) is the distance of the bead from \(O\). Express in terms of \(t\) the reaction of the rod on the bead. Verify that the work done in rotating the rod is equal to the increase of the kinetic energy of the bead.

If \(A\) and \(B\) are two fixed points, and \(P\) is a variable point, lying in a fixed plane through \(AB\) and on one side of \(AB\), such that \(\angle PAB - \angle PBA\) has a constant positive value \(\alpha\), prove that the locus of \(P\) is a part of a rectangular hyperbola. Specify which part of the hyperbola is involved. Show that, as \(\alpha\) varies, the centre of curvature at \(A\) of the locus lies on a fixed straight line perpendicular to \(AB\).