A given set of coplanar forces reduces to a single resultant, and is such that the total moment about a point \(O\) is \(G\), while the sums of the components parallel to two perpendicular lines \(Ox, Oy\) are \(X\) and \(Y\) respectively. Find the equation of the line of action of the resultant. Forces of 1, 4, 2, and 6 lb. wt. act along the sides \(OB, BC, CD, DO\) respectively of a square \(OBCD\) with side of length \(a\). Find the magnitude of their resultant, and obtain the equation of the line of action referred to \(OB\) and \(OD\) as coordinate axes.
Two equal uniform smooth cylinders of radius \(r\) are placed inside a fixed hollow cylinder of internal radius \(R\) and a third equal cylinder is placed symmetrically on the first two, all the generators being parallel. Show that the system cannot remain in equilibrium unless \(R<6\cdot3\,r\), approximately.
Derive, with the usual notation, the equation \(y=c \cosh{x/c}\) for the catenary, and obtain also formulae for \(s\) and \(\psi\) in terms of the inclination \(\psi\) of the tangent to the horizontal. Show that the weight of the chain between the lowest point and any other point of a uniform catenary is the geometric mean between the sum and difference of the tensions at the two points.
A uniform rod of mass \(M\) is placed horizontally on a rough inclined plane of angle \(\alpha\), such that \(\tan\alpha < \mu\), the coefficient of friction. A string fastened to one end is pulled downwards in the direction of a line of greatest slope. If the pull is gradually increased until the rod just begins to move, find the point about which the rod will begin to turn. Show that the tension in the string will then be equal to \[ \lambda(\sqrt{2}-\lambda)\mu Mg\cos\alpha, \] where \(\lambda = \sqrt{(1+\tan\alpha/\mu)}\).
In starting a train the pull of the engine is at first a constant force \(P\), and after the speed attains a certain value \(u\) the engine works at a constant rate \(R=Pu\). Prove that the time \(t\) required for the engine to attain a speed \(v\), greater than \(u\), is given by \[ t = \frac{1}{2}\frac{M}{R}(v^2+u^2), \] where \(M\) is the total mass of the engine and train. Find also the distance travelled at this time in terms of \(v\). Calculate the time and distance taken in attaining from rest a speed of 45 m.p.h. if the total mass is 300 tons, the initial pull of the engine is 12 tons, and its horse-power is 420.
The end \(P\) of a rod \(PQ\) of length \(b\) describes a circle of centre \(O\) and radius \(a\), such that \(a < b\), with constant angular velocity \(\omega\). The other end \(Q\) slides along a fixed straight line through \(O\). Show that the acceleration of \(Q\) is proportional to the velocity of the point \(R\) in which the line \(PQ\) meets the line through \(O\) perpendicular to \(OQ\). Find the position of the instantaneous centre of the rod, and show that the equation of its locus in space is \[ (r-a)^2=b^2+(b^2-a^2)\tan^2\theta \] referred to polar coordinates \((r, \theta)\) with \(O\) as pole and \(OQ\) as initial line.
A shell of mass \(m_1+m_2\) is fired with a velocity whose horizontal and vertical components are \(u\) and \(v\). At the highest point of the path the shell explodes into two fragments of masses \(m_1\) and \(m_2\). The explosion produces additional kinetic energy \(E\) and occurs in such a way that the fragments separate in a horizontal direction. Find the distance apart of the points where the two fragments strike the ground. Show that for a given total mass \(M\) of the shell this distance has least value \(\frac{2v}{g}\sqrt{\frac{2E}{M}}\).
Obtain expressions for the radial and transverse components of acceleration of a point moving in a plane and referred to polar coordinates \((r, \theta)\). A particle of mass \(m_1\) is in motion on a smooth table and is attached by a light inextensible string, which passes freely through a small hole in the table, to a mass \(m_2\) that hangs vertically. Show that when the distance of \(m_1\) from the hole is \(r\), the tension in the string is \[ \frac{m_1 m_2}{m_1+m_2}\left(g+\frac{h^2}{r^3}\right), \] where \(g\) is the acceleration of gravity and \(h\) is a certain constant.
A uniform solid circular cylinder of mass \(M\) and radius \(a\) rolls with its axis horizontal up a plane of inclination \(\alpha\) and coefficient of friction \(\mu>\tan\alpha\), and does so under the action of a constant couple \(Q\). Show that if slipping is not to occur the couple must lie between certain limits, and obtain the values of these limits.
Explain what is meant by the ``equivalent simple pendulum'' for a rigid body free to rotate round a horizontal axis, and derive a formula for the length of the equivalent simple pendulum. Two equal uniform rods \(AB, BC\) each of length \(a\) are rigidly joined at \(B\) with the angle \(ABC\) a right angle. The system oscillates in a vertical plane about a smooth horizontal axis at \(A\). Find the length of the equivalent simple pendulum.