Problems

A uniform circular disc of radius \(a\) and mass \(M\) can turn in its own plane about a fixed horizontal axis through the centre. A light inextensible string lies in a rough groove in the edge of the disc; the end portions of the string are vertical and masses \(M, 2M\) are carried at the ends. If these masses move vertically, show that the angular acceleration of the disc is \(2g/7a\). Find also the tensions in the vertical portions of the string. Find the magnitude of the frictional couple that must be applied to the disc in order that its angular acceleration may be \(g/7a\).

\(ABC\) is a given triangle and \(P\) is a general point in its plane. The lines \(PA, PB, PC\) meet \(BC, CA, AB\) in \(X, Y, Z\), respectively, and \(YZ, ZX, XY\) meet \(BC, CA, AB\) in \(L, M, N\), respectively. Show that \(L, M, N\) lie on a line \(p\). Show also that, in general, a given line \(p\) may be derived in this way from one and only one point \(P\). Calling \(P\) and \(p\) pole and polar with respect to the triangle \(ABC\), prove that, if a point \(P\) describes a straight line \(q\), its polar \(p\) envelops in general a conic \(\Sigma\). Show also that \(q\) has the same pole with respect to \(ABC\) and with respect to \(\Sigma\).

If \(B(p,q) = \int_0^1 x^{p-1} (1-x)^{q-1} dx\) for \(p>0, q>0\), prove that \begin{align*} B(p,q) &= B(p+1,q) + B(p,q+1), \\ B(p,q) &= B(q,p). \end{align*} Prove that, if \(p\) and \(q\) are integers, \[ B(p,q) = \frac{(p-1)!(q-1)!}{(p+q-1)!}. \]

A uniform rectangular block of wood, of weight \(W\), lies on a rough horizontal floor, and \(ABCD\) is a section by a vertical plane passing through the mass centre and meeting four edges at right angles, so that \(AB\) is vertically above \(DC\). A force \(P\) is applied at \(B\) in a direction \(BE\) in the plane \(ABCD\) so that the angles \(ABE\) and \(CBE\) are respectively \(\frac{\pi}{2}-\alpha\) and \(\frac{\pi}{2}+\alpha\), where \(\alpha\) is acute; \(AB = 2l\) and \(BC = h\). The lower face is slightly hollowed out, so that contact with the floor takes place along the edges through \(D\) and \(C\), where the coefficient of friction is \(\mu\). Show that, if equilibrium is on the point of being broken by slipping without tilting, \(P = \mu W/(\cos\alpha + \mu\sin\alpha)\), subject to the inequalities \(\mu h + l > \mu l \tan\alpha > \mu h - l\).

Show that the mass centre of a wedge-shaped portion cut from a uniform solid sphere of radius \(a\) by two planes inclined at an angle \(2\alpha\) and intersecting in a diameter is distant \(3\pi a \sin\alpha/16\alpha\) from that diameter. Find the corresponding result for a wedge cut from a sphere whose density is \(\rho\) from \(r=0\) to \(r=a\) and \(\sigma\) from \(r=a\) to \(r=b\). Deduce that the mass centre of a lune, of angle \(2\alpha\), cut from a thin spherical shell of radius \(a\), is distant \(\pi a \sin\alpha/4\alpha\) from the centre of the sphere.

Two ladders \(AB, BC\), each of length \(2l\), have their centres of gravity at their mid-points. They are freely hinged at \(B\) and stand on a smooth horizontal floor, with \(A, C\) joined by a cord. If the weights of \(AB, BC\) are \(W_1, W_2\) respectively and the angle \(ABC\) is equal to \(2\beta\), find the magnitude and direction of the reaction at \(B\). Find the magnitude and sense of the couple that must be applied to \(AB\) in order that the reaction at \(B\) may be (a) horizontal, (b) vertical.

A closed loop of uniform string of length \(2l\) hangs in equilibrium across a smooth horizontal rail whose cross-section is a circle of radius \(a (< l/\pi)\). Show that the inclination \(\alpha\) of the string to the horizontal at the points where the part that hangs freely in a catenary touches the rail is given by \[ \frac{\sin\alpha \tan\alpha}{\log(\sec\alpha + \tan\alpha)} + \pi - \alpha = \frac{l}{a}. \]

A smooth plane is inclined at an angle \(\alpha\) to the horizontal, and \(AO\) is a rod fixed perpendicularly to the plane at a point \(O\), so that \(A\) is above the plane. A light rod \(AB\) is freely hinged at \(A\) to the fixed rod, and carries at \(B\) a particle of weight \(W\), which rests on the plane at a lower level than \(O\); \(OB\) makes an acute angle \(\beta\) with the line of greatest slope drawn downwards through \(O\). Equilibrium is maintained by a horizontal force \(F\) acting at \(B\) in the plane. Show that the magnitude of \(F\) is \(W \sin\alpha \tan\beta\). Find also the magnitude of the normal reaction of the plane on the particle.

A smooth horizontal rod is rigidly attached at one end to a thin vertical spindle, which is constrained to rotate with a constant angular velocity \(\omega\) about its axis, so that the rod describes a plane. A light spiral spring, of modulus \(\lambda\) and unstretched length \(l\), is coiled round the rod, and is attached at one end to the spindle and at the other end to a mass \(M\) which can slide along the rod. If \(n^2 > \omega^2\), where \(n^2 = \lambda/lM\), and at the instant \(t=0\) the mass is at rest relative to the rod and distant \(l\) from the spindle, find the distance of \(M\) from the spindle at any subsequent time. Show that the greatest value of the horizontal component of the reaction of the rod on \(M\) is \(2lM\omega^3/(n^2-\omega^2)^{\frac{3}{2}}\).

A smooth straight rigid wire is fixed at an angle \(\beta\) with the horizontal, and a bead of mass \(m\) can slide on the wire; \(P\) is a point in the same vertical plane as the wire and \(PQ (=l)\) is the perpendicular drawn to the wire, with \(P\) above \(Q\). A light elastic string of unstretched length \(l\) joins \(P\) to the bead. If the bead is at rest on the wire at a point lower than \(Q\) and distant \(l\) from it, show that the modulus of elasticity of the string is \((2+\sqrt{2})mg \sin\beta\). If the bead is slightly displaced along the wire, show that it will oscillate with period \(2\pi/n\), where \(n^2 = (3+\sqrt{2}) g \sin\beta/2l\).