Prove that the moment of inertia of a uniform circular disc of mass \(M\) and radius \(a\) about the line \(l\) through its centre and perpendicular to its plane is \(\frac{1}{2}Ma^2\). The disc rotates about \(l\) in a medium that resists the motion. The resisting force on an element of area \(dS\) of either face is \(k v dS\), where \(k\) is a constant and \(v\) is the speed of the element. Show the angular velocity of the disc is reduced to half its initial value in the time \[ \frac{M\log 2}{2\pi ka^2}. \]
A heavy circular cylindrical axle of weight \(W\) and radius \(a\) rests in a V-shaped bearing, the two sides of the V being at equal angles \(\alpha\) with the vertical. If the coefficient of friction between the axle and the bearing is \(\mu\), where \(\mu < \cot\alpha\), show that the couple required to turn the axle is \(W a \mu \operatorname{cosec}\alpha / \sqrt{1+\mu^2}\). If a weight \(w\) hangs vertically by a string wrapped round the axle, find the greatest value of \(w\) that will not turn the axle.
A uniform solid elliptic cylinder is in stable equilibrium resting on a perfectly rough horizontal table. Show that no amount of loading along its highest generator will render it unstable if the eccentricity of its cross-section exceeds a certain value, to be found.
A uniform heavy wire of length \(l\) is tightly stretched between two points at distance \(a\) apart and on the same level. Show that, if \(c\) is the parameter of the catenary in which it hangs, then \[ \frac{l}{a} = 1 + \frac{1}{24}\left(\frac{a}{c}\right)^2, \] approximately. Telegraph wire is carried in spans of 300 ft. between successive supports, and the horizontal component of tension in it is equal to the weight of 4500 ft. of wire. Find by how much per mile the length of wire used exceeds the actual distance owing to the sag of the wire.
An engine and train of combined weight \(W\) tons can attain a limiting speed of \(V\) ft. per sec. on a level track. If the engine is assumed to exert a constant force equal to \(P\) tons under all conditions, and if the resistance to the motion at speed \(v\) ft. per sec. is taken as \(a+bv^2\) tons, where \(a\) and \(b\) are constants, show that the horse-power developed by the engine when the train is climbing steadily an incline of angle \(\alpha\) is \[ 4\cdot073\; PV \left\{1 - \frac{W\sin\alpha}{P-a}\right\}^{\frac{1}{2}}. \]
The barrel of a gun is locked in position so that if the gun were standing on a horizontal plane the elevation would be \(\alpha\). The gun is in fact placed transversely to the lines of greatest slope on an inclined plane of angle \(\beta\) with the horizontal. (The axles of the gun-carriage are parallel to the lines of greatest slope.) If \(V\) is the speed of projection, show that the time of flight is \(\dfrac{2V}{g}\sin\alpha\sec\beta\). Find also the ratio \(R/R_0\) of the actual range \(R\) on the plane to the range \(R_0\) that the gun would have if the plane were horizontal.
Three equal imperfectly elastic spheres lie on a smooth horizontal table and their centres are collinear. One of the outer spheres is then projected directly towards the central one. Show that there will be only three collisions if the coefficient of restitution \(e\) exceeds \(0 \cdot 172\) approximately. Find the percentage loss of kinetic energy if \(e=0 \cdot 2\).
A simple pendulum is making complete revolutions in a vertical plane in such a way that its greatest and least angular velocities are \(\omega_1\) and \(\omega_2\) respectively. Show that when the inclination of the pendulum to the downward vertical is \(\theta\) the angular velocity is \[ (\omega_1^2 \cos^2\tfrac{1}{2}\theta + \omega_2^2 \sin^2\tfrac{1}{2}\theta)^{\frac{1}{2}}. \] Show further that stationary values of the tension can never occur except at the highest and lowest positions, and find the corresponding formula for the tension at a general position in terms of its greatest and least values \(T_1\) and \(T_2\).
An inextensible thread is being unwound from a fixed circular reel of centre \(O\). The radius \(OC\) to the point of contact rotates with constant angular velocity \(\omega\), and the unwound portion \(CD\) is kept straight and in the plane of the reel. Show that the acceleration of the end \(D\) of the thread is always towards a certain point \(B\) at a fixed distance from \(C\) in \(OC\) produced, and is of magnitude \(\omega^2 DB\).
A block of wood of mass \(M\) is at rest but free to slide on a smooth horizontal table. A bullet of mass \(m\) is fired into the block with initial speed \(v\) and penetrates into it horizontally a distance \(a\), during which time the block simply slides in the same direction as that of the initial motion of the bullet. Calculate the force of resistance between the bullet and the block on the assumption that it remains constant during the penetration.