Two intersecting forces act on a rigid body along the lines \(OP, OQ\) respectively and are of magnitude \(\lambda OP, \mu OQ\). Shew that the magnitude and direction of their resultant are given by the form \((\lambda+\mu)OR\) along \(OR\), \(R\) being a certain point of \(PQ\), and extend the result to the cases of (a) two parallel forces, (b) several concurrent forces. Forces of given magnitudes and directions act in a plane at the points \(A_1, A_2, \dots, A_n\) respectively. Shew that if the system of points \((A_1, \dots, A_n)\) is rotated rigidly about any axis perpendicular to its plane, the forces remaining unaltered in magnitude and direction, the resultant always passes through a point fixed relatively to the system \((A_1, \dots, A_n)\).
Four rough uniform spheres equal in every respect are placed with three of them resting on a horizontal plane with their centres at a distance apart equal to three times their radius, and with the fourth resting symmetrically on top in contact with the other three. Find the limitations on the coefficients of friction between the spheres and between the spheres and plane, necessary for equilibrium.
A uniform inextensible rough string hangs over a fixed circular cylinder of radius \(a\) and horizontal axis, in a vertical plane perpendicular to this axis. \(\mu\) is the coefficient of friction between string and cylinder. If one free end of the string is at one point of the circular cross section where the tangent is vertical, shew that the greatest length of string which can hang vertically on the other side without causing slipping is \[ \frac{2\mu(1+e^{\mu\pi})}{\mu^2+1}a. \]
A light elastic string of modulus \(\lambda\) and natural length \(l\) has its ends attached to two fixed points \(AB\) distance \(l\) apart and on the same horizontal level. A fixed vertical circular wire of diameter \(l\csc\theta\) passes through \(A\) and \(B\) and has its centre below \(AB\). The string passes through a smooth ring of mass \(m\) which slides without friction on the wire. Shew that there is an oblique position of equilibrium if \(\frac{mg}{\lambda} < \cot\frac{\theta}{2} - \cos\frac{\theta}{2}\).
Particles projected with given speed \(u\) from a point \(O\) in all directions in a vertical plane containing \(O\) fall on the horizontal plane through \(O\). Find the maximum range subject to the condition that the particles pass beneath a barrier of height \(h\) above, and at horizontal distance \(d\) from \(O\), distinguishing between the different cases.
A particle falls under gravity from rest through a distance \(h\) on to a smooth fixed plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of restitution is \(e\). Determine the distance between the point of first impact and the point at which bouncing ceases and find the velocity of the particle at the latter point.
Two equal particles each of mass \(m\) are connected by a light smooth inextensible string which passes through a smooth ring fixed to the circumference of a fixed smooth horizontal circular wire of radius \(a\). One particle moves freely on the wire and the other hangs vertically below the ring. The system is slightly displaced from rest when the first particle is at the opposite end of the diameter through the ring. If \(\theta (<\pi)\) is the angle turned through by the radius to the bead, find an expression in terms of \(\theta\) for the tension in the string and the reaction on the wire, and shew that the velocity of the upper bead is \[ \sqrt{4ag \left(\frac{1-\cos\frac{\theta}{2}}{1+\sin^2\frac{\theta}{2}}\right)}. \]
A particle moves under gravity in a uniform medium which offers a resistance per unit mass equal to \(\kappa\) times the speed. Shew that there is a certain direction in which the velocity component remains unaltered and find the magnitude of this component in terms of \(g, \kappa\), and the initial conditions.
A train of total mass 800 tons with an engine of constant tractive force 20 tons weight and subject to a steady resistance of 15 tons weight runs from \(A\) to \(B\), a distance of six miles. The gradient is first downwards 1 in 1000 and then upwards 1 in 500, \(A\) and \(B\) being on the same horizontal level. Find the least time from rest to rest for the journey, and also the time taken for least expenditure of mechanical energy by the engine. It may be assumed that braking resistance up to twice the ordinary resistance can be applied at will, and that \(g = 32\) ft./sec.\(^2\).
A cylindrical body of any section can turn freely about a fixed horizontal axis which is parallel to its generators. While at rest under gravity the body is subjected to a certain horizontal blow perpendicular to the axis and at a point beneath such that in the subsequent motion the kinetic energy of the body is proportional to the depth of its centre of gravity below its highest possible position. With the usual notation \((M, h, \kappa^2)\) find the time taken from rest until the centre of gravity is on the same horizontal level as the axis. If in addition to these circumstances it is further observed that there is no impulsive reaction at the axis of suspension at impact, determine the magnitude of the blow given.