The footway of a suspension bridge is horizontal, and is suspended by vertical rods attached at equal intervals along it. The upper ends of the rods are attached to a light cable. The tensions in all the vertical rods are equal. Show that the points of attachment to the cable lie on a parabola.
Show that the centre of gravity of a uniform semicircular rod is at a distance from the centre equal to \(2/\pi\) times the radius. A circular disc, whose mass per unit area is \(\sigma r\), where \(\sigma\) is a constant and \(r\) the distance from the centre, is divided into two by a diameter. Find the centre of gravity of either half.
Prove that the gain of the kinetic energy of a particle in any interval is equal to the work done on it by the applied forces in that interval. A particle is projected directly up a plane inclined at an angle \(\alpha\) to the horizon, with initial velocity \(u\) given by \[ u^2=2gh(\sin\alpha + \mu \cos\alpha), \] where \(\mu\) is the coefficient of friction. Show that it traverses a distance \(h\), and that it stays at the highest position if \(\tan\alpha < \mu\). If \(\tan\alpha > \mu\), find its velocity when it returns to the starting point, and explain the difference between the initial and final kinetic energies.
Two flywheels, whose radii of gyration are in the ratio of their radii, are free to revolve in the same plane, a belt passing round both. Initially one, of mass \(m_1\) and radius \(a_1\), is rotating with angular velocity \(\Omega\), and the other, of mass \(m_2\) and radius \(a_2\), is at rest. Suddenly the belt is tightened, so that there is no more slipping at either wheel. Show that the second wheel begins to revolve with angular velocity \[ \frac{m_1 a_1}{(m_1+m_2)a_2} \Omega. \]
A 20 h.p. motor lorry, weighing 5 tons, including load, moves up a hill with a slope of 1 in 20. The frictional resistance is equivalent to 13 lbs. weight per ton, and may be supposed independent of the velocity. Find the maximum steady rate at which the lorry can move up the slope, and the acceleration capable of being developed when it is moving at 6 miles per hour.
A particle moves in a straight line under a force to a fixed point in the line proportional to its distance from the point. Find its motion. A spring, whose natural length is \(l_0\), is free to vibrate horizontally, one end being fixed. The force required to shorten the spring by an amount \(x\) is \(Ex\). The mass of the spring is \(M\), and its centre of gravity may be supposed to be always at its middle point. The spring is compressed by an amount \(x_0\), a mass \(m\) is placed at the end, and that end is then released. Find the velocity of the particle when it leaves the spring.
Show that any possible motion of a system of particles still satisfies the equations of motion if the same uniform velocity is compounded with the velocity of every particle and the forces between the particles remain unaltered. A pendulum consisting of a light rod of length \(l\) and a heavy bob hangs freely. The point of support is suddenly made to move horizontally with uniform velocity \(v\). Show that the pendulum will describe a complete revolution if \(v > 2\sqrt{(gl)}\).
If the feet of the perpendiculars from a point \(P\) on the sides of a triangle \(ABC\) are collinear, shew that \(P\) lies on the circumcircle of the triangle. If \(P\) and \(Q\) are extremities of a diameter of the circumcircle shew that the pedal lines of \(P\) and \(Q\) are perpendicular.
Shew that the locus of the mid-points of a system of parallel chords of a parabola is a straight line parallel to the axis. \(P, P'\) are two points on a parabola. Ordinates \(PV, P'V'\) are drawn to the diameters through \(P', P\) respectively. Shew that \(PP'V'V\) is a parallelogram.
State and prove the harmonic properties of a quadrilateral. \(P\) is a variable point upon a conic which circumscribes the triangle \(ABC\). \(AP, BC\) meet in \(Q\); \(AB, PC\) in \(R\). Shew that \(QR\) always passes through a fixed point.