A rough cylinder rests in equilibrium on a fixed cylinder, in contact with it along its highest generator, which is horizontal. Shew, by consideration of the potential energy, that the equilibrium is stable for rocking displacements if \(1/h > 1/\rho_1 + 1/\rho_2\), where \(h\) is the height of the centre of gravity above the point of contact, \(\rho_1\) and \(\rho_2\) the radii of curvature of the two surfaces. If the cylinder be weighted so that the equilibrium is apparently neutral, shew that it is really unstable unless the cylinders are in contact at vertices (points of stationary curvature). Shew that if a cylinder rest on a horizontal plane in neutral equilibrium and touch the plane at a vertex, the equilibrium is really stable if the curvature be a maximum at the point of contact. If it be in contact with a fixed cylinder, exactly similar to it, the apparently neutral equilibrium is really stable if at the vertex \(\rho \frac{d^2\rho}{ds^2} > 3\); that is, if \[ \rho^2 \frac{d^2\rho}{d\psi^2} - 2\rho\left(\frac{d\rho}{d\psi}\right)^2 - 3\rho > 0. \]
A wedge of mass \(M\) and angle \(\alpha\) is placed on a rough horizontal plane whose coefficient of friction is \(\mu\). A smooth particle of mass \(m\) is placed gently on the inclined face of the wedge. Shew that if the wedge moves, it does so with an acceleration \[ \frac{\{m \cos\alpha (\sin\alpha - \mu\cos\alpha) - \mu M\}g}{m\sin\alpha(\sin\alpha - \mu\cos\alpha) + M}. \]
Shew that the lengths of the three shortest lines that bisect the area \(\Delta\) of a triangle \(ABC\) are \[ \sqrt{2\Delta \tan\frac{A}{2}}, \quad \sqrt{2\Delta \tan\frac{B}{2}}, \quad \sqrt{2\Delta \tan\frac{C}{2}}, \] assuming that no side is more than double another.
State Newton's laws concerning the direct impact of two uniform spheres. Deduce that the impulse during the period of compression bears to the total impulse a ratio which is independent of the masses and velocities of the spheres. Two smooth spheres of masses \(M\) and \(M'\) impinge obliquely; lines \(OP\) and \(OQ\) represent the velocities of \(M\) before and after the impact, and lines \(OP'\) and \(OQ'\) represent the velocities of \(M'\) before and after the impact. Prove that \(PQ\) and \(P'Q'\) are parallel and of lengths inversely proportional to \(M\) and \(M'\); and that, if \(p, p', q, q'\) are the projections of \(P, P', Q, Q'\) on any line parallel to \(PQ\), the ratio \(qq':pp'\) is the coefficient of elasticity. Shew that these results are applicable to the case when the velocities of the centres of the spheres before and after impact are not coplanar, and deduce a graphical construction for the velocities after impact, the masses, the velocities before impact, the coefficient of elasticity and the direction of the line of centres at the moment of impact being known.
The figure represents a series of cylindrical rollers rotating about fixed horizontal axes as used in rolling mills for moving heavy plates. If a constant peripheral speed of 10 feet per sec. is maintained in each roller and a plate weighing 1 ton is placed without velocity on the rollers, find how long it takes to acquire its full speed and the energy expended in giving it this motion. The coefficient of friction between the plate and the rollers is 0.12. [Diagram showing a plate on a series of rollers.] If another plate weighing 4 tons is then placed without initial velocity on the top of the first plate, the latter having its full velocity of 10 feet per sec., find how far the two plates will move relatively to one another before they acquire the same velocity, and calculate the magnitude of this common velocity. The coefficient of friction between the two plates is 0.18.
From the top of a hill the depression of a point on the plain below is 12\(^\circ\) and from a place three-quarters the way down the depression of the same point is 6\(^\circ\); find to the nearest minute the constant inclination of the hill.
If a bullet of mass \(m\) moving with velocity \(v\) is found to penetrate a distance \(a\) into a fixed body, the resistance being supposed uniform, prove that if it meet directly a body of mass \(M\) and thickness \(b\) moving with velocity \(V\), it will be imbedded if \[ b/a > (1+V/v)^2 M/(M+m), \] the uniform resistance being assumed the same in both cases. If however the bullet penetrates the body, shew that it will emerge with velocity \[ \frac{mv-MV+Mv\sqrt{(1+V/v)^2 - (1+m/M)b/a}}{M+m}, \] and that the time of penetration will be \[ \frac{2a}{v}\frac{M}{M+m}\left[1+\frac{V}{v} - \sqrt{\left(1+\frac{V}{v}\right)^2 - \left(1+\frac{m}{M}\right)\frac{b}{a}}\right]. \] Determine also the final velocity of the body in both these cases.
A point \(P\) moves in a straight line with an acceleration which is directed to a fixed point \(O\) and which is equal to \(\mu \cdot OP\), where \(\mu\) is constant; when the distance of \(P\) from \(O\) is equal to \(a\), the velocity of \(P\) is \(v\). Determine the amplitude and periodic time of the motion. One end of an elastic string of natural length \(l\) and modulus of elasticity \(4mg\) is fixed at \(A\), the other end is attached to a particle of mass \(m\). The particle is held at \(A\) and is then let fall. Prove that the time which elapses before the particle returns to \(A\) is \[ 2\sqrt{\frac{l}{g}}\left\{\tan^{-1}\sqrt{2} + \sqrt{2}\right\}. \] If the length of the string is 2 feet, evaluate this time in seconds, correct to two places of decimals, taking \(g=32\).
At a point \(P\) on the circumference of the auxiliary circle of an ellipse whose major axis is \(AA'\), the tangent drawn to the circle meets the major axis in \(T\), and the lines \(PA, PA'\) meet the ellipse in \(Q\) and \(R\) respectively. Shew that \(T, Q, R\) are collinear.
A heavy particle hangs by a string of length \(a\) from a fixed point \(O\) and is projected horizontally with the velocity due to falling freely under gravity through a distance \(h\); prove that if the particle makes complete revolutions \(h \geq \frac{5}{2}a\), that if the string becomes slack \(\frac{5}{2}a > h > a\), and that in this latter case the greatest height reached above the point of projection is \((4a-h)(a+2h)^2/27a^2\).