A circle of radius \(a\) rolls round the outside of a closed oval curve whose total perimeter is \(s\) and whose area is \(S\). Show that the locus of the centre of the circle is an oval curve of perimeter \(s+2\pi a\) enclosing an area \(S+as+\pi a^2\). \newline If the circle rolls on the inside of the oval curve and is sufficiently small to be always entirely within it, show that the locus of the centre is another oval of perimeter \(s-2\pi a\) enclosing an area \(S-as+\pi a^2\).
A particle of mass \(m\) moves under gravity in a medium that opposes the motion with a resisting force \(kmv\), where \(k\) is a constant and \(v\) is the speed. If it is projected up vertically with speed \(v_0\), show that after time \(t\) its height above the point of projection is \[ \left(v_0 + \frac{g}{k}\right) \frac{(1-e^{-kt})}{k} - \frac{gt}{k}. \] Hence find the greatest height reached by the particle. \newline Show that subsequently the speed cannot exceed a certain finite value however far the particle falls.
If \(\theta\) is the angular displacement of a simple pendulum of length \(l\) from the vertical, prove that \[ l \left(\frac{d\theta}{dt}\right)^2 = 2g (\cos \theta - \cos \alpha), \] where \(\alpha\) is the greatest value of \(\theta\). Deduce that the period is given by \[ 2 \left(\frac{l}{g}\right)^{\frac{1}{2}} \int_0^{\alpha} (\sin^2 \frac{1}{2}\alpha - \sin^2 \frac{1}{2}\theta)^{-\frac{1}{2}} d\theta. \] By making the substitution \(\sin \frac{1}{2}\theta = \sin \frac{1}{2}\alpha \sin \phi\) and expanding the integrand in powers of \(\sin \frac{1}{2}\alpha\), prove that the period is approximately \[ 2\pi \left(\frac{l}{g}\right)^{\frac{1}{2}} (1 + \frac{1}{16}\alpha^2) \] for small values of \(\alpha\).
Two particles \(A\) and \(B\), of masses \(\alpha\) and \(\beta\) respectively, lie on a smooth horizontal plane and are joined by a light spring of modulus \(k\) and natural length \(l\). At first the spring is unstretched. The particle \(B\) is then set in motion with speed \(u\) in the direction \(AB\). Calculate the maximum extension of the spring in the subsequent motion.
A circular disc, of mass \(M\) and radius \(a\), rests on a rough horizontal table; the coefficient of friction is \(\mu\). If the normal reaction between the disc and the table is uniformly distributed over the disc, show that the greatest couple that can be applied to the disc without causing rotation has moment \(\frac{2}{3}\mu Mga\). \newline If the disc is set spinning in the plane of the table about its centre with angular velocity \(\omega\), find the angle through which it turns before coming to rest.
A particle is projected from a point \(O\) so as to return to \(O\) after rebounding from a smooth vertical wall situated at a distance \(d\) from \(O\). Prove that \[ u_0 v_0 = \frac{(1+e)gd}{2e}, \] where \(u_0\) and \(v_0\) are the horizontal and vertical components of the velocity of projection, and \(e\) is the coefficient of restitution between the particle and the wall. \newline Show that the particle reaches its greatest height during the motion at a point whose distance from the wall depends only upon \(d\) and \(e\). Find the least possible speed of projection.
A vessel steams at given speeds on two given courses, and the direction of the trail of smoke is observed in each of the two cases. Devise a geometrical construction for the direction of the wind. \newline Find graphically or otherwise the wind direction if the smoke-trail is in the direction N. 140\(^{\circ}\) E. when the vessel steams due N., and in the direction N. 65\(^{\circ}\) E. when the vessel steams due W. at the same speed.
A uniform rod \(AB\) of weight \(w\) and length \(2l\) is supported by a smooth hinge at \(A\), and an equal rod \(BC\) is smoothly hinged to the first at \(B\). The hinges allow the rods to rotate freely in a vertical plane. A couple of moment \(L\) is applied to \(AB\) and a couple of moment \(M\) is applied to \(BC\) (the two couples lying in this vertical plane) and the system is maintained in a position of equilibrium with the rods inclined at angles \(\theta\) and \(\phi\) to the vertical. Prove that \[ 3wl \sin \theta = L, \quad wl \sin \phi = M. \]
A uniform beam is supported horizontally at its two points of trisection. Calculate the shearing force and the bending moment at any point of the beam, stating carefully your conventions of sign, and show your results in diagrammatic form.
Prove that the centre of gravity of a uniform solid hemisphere of radius \(a\) is at distance \(\frac{3}{8}a\) from the plane face. \newline A solid hemisphere of radius \(a\) rests symmetrically upon a fixed sphere of radius \(b\); the plane face of the hemisphere is horizontal and uppermost, and the centre of this face is vertically above the centre of the sphere. Discuss the stability of the position of equilibrium for all possible values of \(a/b\), assuming that no slipping occurs.