Three uniform spheres, \(A, B, C\), of masses \(2m, m, 2m\) respectively, lie in a straight line on a horizontal table, with \(B\) between \(A\) and \(C\). Initially \(B\) and \(C\) are at rest, and \(A\) is projected along the line of centres towards \(B\); the coefficient of restitution for any pair of the spheres is \(e\). Show that, if \(e>0\), there will be at least three collisions, and that there will be only three provided that \(e \ge \frac{1}{3}\).
A compound pendulum consists of a plane lamina which can swing about a horizontal axis perpendicular to the plane of the lamina. The axis can be made to pass through either of two points \(A_1, A_2\) in the lamina. The distances of the centre of gravity of the lamina from \(A_1\) and \(A_2\) are \(h_1\) and \(h_2\). The period of small oscillations when swinging about \(A_1\) is \(T_1\), and when swinging about \(A_2\) is \(T_2\). Prove that \[ \frac{8\pi^2}{g} = \frac{T_1^2+T_2^2}{h_1+h_2} + \frac{T_1^2-T_2^2}{h_1-h_2}, \] if \(h_1 \ne h_2\).
A uniform circular disc, of radius \(a\) and mass \(2m\), is freely jointed at a point \(A\) of its circumference to a uniform rod \(AB\) of length \(2a\) and mass \(m\). The system is at rest on a horizontal table, the points \(B, A\) and \(O\) (the centre of the disc) lying in a straight line. An impulse \(J\) is applied at the point \(B\) along a horizontal line perpendicular to \(AB\). Show that the impulsive reaction at \(A\) is parallel to the impulse at \(B\) and is of magnitude \(4J/11\). Find the initial values of the angular velocity and the velocity of the centre of mass of both the rod and the disc, and verify that the system acquires kinetic energy \(\frac{1}{2}JV\), where \(V\) is the initial velocity of the point \(B\).
A thin uniform rod of mass \(m\) is welded inside a uniform hollow cylinder of equal length, and lies along a generator; the cylinder is of mass \(3m\) and radius \(a\). The system rolls without slipping on a rough horizontal plane. The angle between the downward vertical and the plane containing the axis of the cylinder and the rod is \(\theta\). If the cylinder is instantaneously at rest when \(\theta=\alpha\), show that the angular velocity \(\dot{\theta}\) in a general position is given by \[ \dot{\theta}^2 = \frac{g(\cos\theta-\cos\alpha)}{a(4-\cos\theta)}. \] Find the horizontal and vertical components of the reaction of the plane on the cylinder when the cylinder is instantaneously at rest.
A plane polygon of \(n\) sides has vertices \(A_1, A_2, \dots, A_n\). Forces acting along the sides in the same sense round the polygon have magnitudes \(\lambda \cdot A_1A_2, 2\lambda \cdot A_2A_3, \dots, r\lambda \cdot A_rA_{r+1}, \dots, n\lambda \cdot A_nA_1\), respectively, where \(\lambda\) is a constant. Show that the magnitude and direction of the resultant force is the same if the forces act in the opposite sense round the polygon and have magnitudes \(n\lambda \cdot A_2A_1, (n-1)\lambda \cdot A_3A_2, \dots, \lambda \cdot A_1A_n\) respectively. Show that for a regular polygon the magnitude and direction of the resultant is given by \(n\lambda \cdot OA_1\), where \(O\) is the centre of the polygon. In the case of a regular pentagon find the perpendicular distance from \(O\) of the line of action of the resultant in the former case.
A non-uniform sphere of radius \(a\) rests in equilibrium on top of a fixed sphere of radius \(b\). The surfaces are perfectly rough and the centre of mass of the first sphere is at a distance \(d\) above the point of contact. Find the vertical distance through which the centre of mass rises when the first sphere is made to roll in contact with the other until the radius to the original point of contact is inclined at an angle \(\theta\) to the vertical. Hence, or otherwise, prove that the equilibrium is stable if \[ \frac{1}{d} > \frac{1}{a} + \frac{1}{b}. \]
A rod of uniform material but of variable section is held with one end horizontally in a clamp. The shape of the rod is designed so that at any point the value of the bending moment is proportional to the area of cross-section. Show that if the total weight of the rod is \(W\), the value of the bending moment at a point of the rod distant \(x\) from the clamped end is given by \(M = W\sinh n(l-x)\{n(\cosh nl-1)\}^{-1}\), where \(l\) is the length of the rod and \(n^2\) is the constant ratio of the weight per unit length to the bending moment at any point of the rod.
A uniform flexible chain of given total weight \(W\) is suspended between two points on the same horizontal level and at a variable distance apart. The chain will break if the tension at any point exceeds the value \(R\). Prove that the greatest possible ratio of the distance between the points of suspension to the total length of the chain is given by \[ \frac{\log_e(\lambda+\sqrt{\lambda^2-1})}{\sqrt{\lambda^2-1}}, \quad \text{where} \quad \lambda^2 = 1 - \frac{W^2}{4R^2} \quad \text{and} \quad \lambda>0. \]
A particle is allowed to fall from rest under gravity in a medium offering resistance per unit mass \(\kappa v+\lambda v^2\), where \(v\) is the velocity of the particle, and \(\kappa\) and \(\lambda\) are positive constants. Show that at any time \(t\) after release the particle moves with velocity \(v\) given by \(v+\mu=(u+\mu)\tanh\{\lambda(u+\mu)t+\alpha\}\), where \(\mu=\kappa/2\lambda\), \(\alpha=\tanh^{-1}\mu/(u+\mu)\), and where \(u\) is the positive quantity defined by the equation \(\lambda u^2+\kappa u-g=0\). Deduce that there is a limiting velocity and state its value.
Particles are emitted with fixed velocity \(V\) from a point \(O\) and move under gravity in a vertical plane containing \(O\). Prove that any point in space lying on one path of a particle will in general also lie on one other path of a particle. If the coordinates of the point are \(x,y\) referred to origin \(O\) and rectangular axes \(Ox\) and \(Oy\) through \(O\) horizontally and vertically respectively, show that the ranges on the line \(Ox\) of the two paths are given by the roots \(R\) of the equation \[ R^2(x^2+y^2)-2Rx\left(x^2+y^2+\frac{V^2}{g}y\right)+x^2\left(x^2+y^2+2y\frac{V^2}{g}\right)=0. \]