A chain of length \(l\) lies in the smooth horizontal arm of an \(\Gamma\)-shaped tube. The other arm of the tube hangs vertically downwards and the corner of the tube is rounded. A fraction, \(k\), of the chain hangs over the corner and dips into the vertical arm. Show that the chain will be free of the horizontal arm in time \[ \sqrt{\frac{l}{g}} \log\left(\frac{1+\sqrt{(1-k^2)}}{k}\right). \] If the chain breaks when the tension is \(\frac{4}{25}\) of its weight, show that it will break when \(\frac{3}{5}\) of the chain is past the corner.
Prove that a coplanar system of forces may be reduced to a force through an assigned point and a couple. Show also that, in general, the system is equivalent to a single force. A square lamina \(ABCD\) lies on a smooth horizontal table and is subject to a force \(F\) acting at \(A\) along \(DA\) produced, a force \(2F\) at \(B\) along \(AB\) produced, and a force \(2F\) at \(C\) towards \(B\). Show that if the lamina is freely hinged at \(D\) it will not move. Find the force on the hinge.
A uniform ladder of weight \(w\) rests with one end on the ground and with the other against a vertical wall, its angle of inclination being \(45^\circ\). The coefficient of friction at each end is \(\frac{1}{2}\). A man of weight \(4w\) begins to climb the ladder. Show that slipping will commence when he has covered three-eighths of the length of the ladder.
A closed rectangular box is made of thin uniform sheet, its base being a square of side \(a\) and its height \(\frac{3}{2}a\). The base is made of double thickness of sheet and the rest of the box is made of sheet of single thickness. The box stands on a perfectly rough inclined plane four of its edges being parallel to the line of greatest slope. A horizontal force equal to one-third of the weight of the box is applied along the perpendicular bisector of the highest edge so as to tend to topple the box down the slope. Show that if the force is just able to topple the box then the inclination of the plane to the horizontal is \(\tan^{-1}\frac{5}{8}\).
A plane uniform lamina is bounded by a semicircle of radius \(a\). Find its centre of gravity. A second plane uniform lamina, of different surface density, is bounded by a square of side \(2a\). A composite plane lamina is formed by joining the base of the semicircular lamina to a side of the square one. What is the ratio of the surface densities if the centre of gravity of the composite lamina is at the mid-point of the common edge?
One end of a uniform rod of weight \(w\) and length \(5l\) is freely hinged, while the other is attached by a light elastic string of unstretched length \(2l\sqrt{2}\) to a point at the same level as the hinge and distant \(7l\) from it. In equilibrium the length of the string is \(3l\sqrt{2}\). A weight \(W\) is now attached to the mid-point of the rod, and the length of the string in the new equilibrium position is \(4l\sqrt{2}\). Show that \(W=5w/3\).
A particle is projected under gravity with initial velocity \(v\) from a point \(O\) at a height \(h\) above a horizontal plane and strikes the plane at a horizontal distance \(d\) from \(O\). Find \(d_1\), the maximum value of \(d\), and, if \(d_2\) is the corresponding maximum distance when the point of projection is at a depth \(h\) below the plane, prove that \[ \frac{d_1^2}{d_2^2} = \frac{v^2+2gh}{v^2-2gh}. \]
A smooth sphere collides with a second smooth sphere with the same mass which is at rest; the coefficient of restitution is \(e\), and \(\theta\) is the angle just before impact between the direction of motion of the first sphere and the line of centres. Show that the change in the direction of motion of the first sphere is a maximum if \(2\tan^2\theta=1-e\), and explain this result if \(e=1\).
A bead of mass \(m\) moves under gravity on a smooth wire in the form of a parabola with its axis vertical and vertex uppermost. Prove that the pressure on the wire is \(mkg/ \rho\) when the bead is at a point where the radius of curvature of the wire is \(\rho\), and determine the constant \(k\) if \(4a\) is the latus rectum of the parabola and \(v\) is the velocity of the bead when it is at the highest point of the wire.
Two gravitating particles, of masses \(m_1, m_2\), are moving freely in a plane under their gravitational attraction which is of magnitude \(km_1m_2/r^2\) when the particles are at a distance \(r\) apart. If throughout the motion the particles are at a constant distance \(d\) apart, find the angular velocity of the straight line joining them.