A uniform rod of mass \(m\) lying on a horizontal table is hit at its midpoint by a particle, also of mass \(m\), sliding along the table with velocity \(v\) perpendicular to the rod. Immediately after this impact one end of the rod hits an inelastic stop. If \(e\) is the coefficient of restitution between the particle and the rod, find the condition that the particle should immediately hit the rod again. If \(e\) is very small, will the final angular velocity of the rod be (i) practically the same as if the particle had coalesced with the rod, (ii) practically the same as if the rod had initially been free to turn about one fixed end?
Forces \(P, Q, R\) and \(S\) act in the sense indicated along the sides \(AB, BC, CD, DA\) of a square \(ABCD\) whose sides are of length \(l\). Reduce this system to
A square table of weight \(W\) has side \(2a\) and height \(b\). The top is uniform and it has four equal legs at its corners. It stands on a rough horizontal floor of coefficient of friction \(\mu\). Given that \[ \frac{b}{a} < \frac{1-\mu^2}{\mu}, \] find the magnitude and elevation of the least force that will make the table slide on the floor parallel to a side. Show also that this force may be applied at any point of the table without toppling it.
A uniform heavy beam of length \(2l\) and weight \(2W\) rests on two supports at distance \(\frac{1}{4}l\) from either end. Weights of magnitude \(W\) are suspended from each end and an upward force \(2kW\) (\(k<2\)) is applied to the mid-point of the beam. Derive expressions for the shearing force and the bending moment at a general point of the beam and prove that the numerically greatest bending moment occurs at the supports if \(k<\frac{5}{8}\) and at the mid-point if \(k>\frac{5}{8}\).
A light inextensible string is wound a number of times round a horizontal circular cylinder. The two ends of the string hang vertically downwards supporting weights \(w\) and \(32w\) respectively. The coefficient of friction between the string and the cylinder is \(\dfrac{1}{\pi}\log_e 2\). Derive the equation for the variation of the tension along the string when the string is on the point of slipping, and find the least number of turns of the string round the cylinder for the system to be in equilibrium.
A sphere of radius \(a\) is intersected by a plane at a distance \(\frac{1}{2}a\) from its centre. A solid of revolution consists of the larger of the two parts into which the sphere is so divided together with the right circular cylinder of radius \(\frac{\sqrt{3}}{2}a\) and height \(\frac{1}{2}a\) whose base coincides with the plane section of the sphere. The density of the whole solid is uniform. Find the mass centre of the solid and the moment of inertia about the axis of symmetry.
A wooden body of mass \(5m\) is projected at an angle to the vertical from a point of a horizontal plane. When it is at the highest point of its trajectory it is hit by a bullet of mass \(m\) flying vertically upwards. The bullet becomes embedded in the body which then falls on the plane at the same point as if the bullet had not hit it. Show that if the bullet was fired from a point of the plane its muzzle velocity must have been \((2\cdot21)^{\frac{1}{2}}\) times the vertical component of the velocity of projection of the wooden body.
A small ring of mass \(m\) can slide on a fixed smooth wire which is in the form of a single arc of the cycloid, \(x=a(\theta-\sin\theta)\), \(y=-a(1-\cos\theta)\), from \(\theta=0\) to \(\theta=2\pi\), the positive \(y\)-axis being vertically upwards. The ring is released from rest at the point \(x=0, y=0\). Prove that its a vertical velocity is greatest when it has fallen through a vertical distance equal to \(a\). Calculate the time taken by the ring in falling to the lowest point of the wire.
The power output of a car at speed \(v\) is \[ W \frac{v^3 w^2}{(v^2+w^2)^2}, \] where \(W\) is the weight of the car and \(w\) is 30 m.p.h., so that, if the weight of the car is \(1\frac{1}{4}\) tons, its power at 30 m.p.h. is 56 h.p. The car climbs a hill inclined at an angle \(\alpha\) to the horizontal. Show that if \(\sin\alpha > \frac{3}{32}\) its speed will decrease on the hill in all circumstances. Find the inclination of the steepest hill on which the car can maintain a speed of 15 m.p.h.
A pendulum consists of a thin straight uniform rod of mass \(M\) and length \(2l\) swinging about a certain point \(P\) of itself. It is found that when a particle of mass \(m\) is attached to the lowest point of the rod the period \(T\) of the pendulum does not alter, whatever the value of \(m\). Find the position of \(P\). The rod with the particle attached is now allowed to swing about a point of itself a small distance \(\xi\) above \(P\). Show that the period \(T'\) will change by the amount \(\dfrac{3\xi}{4l}\dfrac{2m-M}{4m+M}T\) approximately.