Show that the centre of mass of a uniform thin hemispherical bowl of radius \(a\) is at a distance \(\frac{3}{8}a\) from the plane of the rim. The bowl is placed with its rough outer curved surface on a horizontal table. A smooth rod whose mass is half that of the bowl rests with one end on the smooth inner surface of the bowl and a point of the rod in contact with the rim. In a position of equilibrium the rod and the plane of the rim each make an angle \(\theta\) with the horizontal. Show that \(\theta = \frac{1}{4}\pi\).
An elastic string, of natural length \(l\) and modulus of elasticity \(mg/k\), has one end fixed at the point \(O\). To the other end is attached a particle of mass \(m\). The particle is dropped from \(O\). Find the distance through which it falls before it first comes to rest instantaneously, and show that the time taken for this to happen is \[\left[\sqrt{2}+\sqrt{k}\left\{\frac{1}{2}\pi + \cos^{-1}\left(\sqrt{\frac{2}{2+k}}\right)\right\}\right]\sqrt{\frac{l}{g}}.\]
A particle of mass \(m\) moves in a planar orbit under a central force of magnitude \(mf(r)\) directed towards the origin of plane polar coordinates \(r, \theta\). Show that (a) the radius vector to the particle sweeps out area at a constant rate \(\frac{1}{2}h\), where \(mh\) is the angular momentum about the origin; (b) if \(u = 1/r\), the equation of the orbit may be put in the form \[\frac{d^2u}{d\theta^2} + u = \frac{f(1/u)}{h^2u^2}.\] If the orbit of the particle is an ellipse \(l = r(1 + e \cos\theta)\), show that the semi-major axis, \(a\), of the ellipse equals \(l/(1-e^2)\). Find the force and show that the period of time is \(2\pi a^{\frac{3}{2}}l^{\frac{1}{2}}/h\). [The area of the ellipse is \(\pi ab\) where \(b = a (1-e^2)^{\frac{1}{2}}\).]
Equal particles lie at rest at equal intervals along a straight line on a smooth level table. The particle at one end of the line is struck towards its neighbour, hitting it after a time \(t_1\). The coefficient of restitution \(e\) is only slightly less than unity. Find the time that elapses until the \(n\)th particle begins to move, and show that the `collision wave' propagates at a velocity which ultimately is inversely proportional to the time elapsed. Sketch the trajectories of the particles in the (distance, time) plane, and use your sketch to indicate how a second collision wave of slower speed will propagate in the same direction some time later (no detailed calculation required).
The Earth is to be treated as a uniform sphere of density \(\rho\) and radius \(R\), with no atmosphere. Gravitational acceleration is given by \(GM(r)/r^2\), where \(r\) is the distance from the Earth's centre and \(M(r)\) is the mass within the sphere of radius \(r\). (i) A satellite orbits the Earth in a circular orbit just above the surface. Find the time taken for an orbit, in terms of \(G, \rho\) and any other relevant quantities. (ii) A tunnel is drilled diametrically through the Earth's centre, from one side to the other. If a particle is released from rest at one end, find the time it takes to fall the length of the tunnel. (iii) A particle falls to the Earth's surface starting from rest at distance \(2R\) from the centre. Find the time taken to reach the Earth.
A librarian picks up a row of identical books from a shelf, by pressing the outer two books between her hands sufficiently firmly that friction keeps the books in place while she raises the whole row. The covers of the books are all in vertical planes. The coefficients of limiting friction between pairs of books, and between her hands and the books, are all the same. Show that friction is nearest to limiting where her hands touch the books. If the maximum force she can exert with each hand is independent of the direction in which she applies it and equals the weight of \(N\) books, show that the maximum number of books which she can lift by this method is the largest integer less than \(2\mu N/(1+\mu^2)\), where \(\mu\) is the coefficient of limiting friction. How important for this calculation is the condition that the books be identical?