A frame consists of five light rods \(AB, BC, CA, CD, DA\) freely jointed together. \(A\) is a fixed hinge, \(AB\) is horizontal, \(AD\) is vertical, and \(C, D\) are above \(AB\). \(AB=24\) in., \(AC=BC=20\) in., \(CD=15\) in., and \(AD=25\) in. A load \(W\) is suspended from \(B\), and the frame is kept in equilibrium by a horizontal force acting at \(D\). Determine the stresses in the rods, stating which are in tension and which in compression.
A wheel of radius \(R\) is fixed to an axle of radius \(r\), and the system can turn freely about a fixed horizontal axis. A flexible cord has one end wound round the axle and after passing under a light pulley of diameter \(R+r\), from which a mass \(M\) is suspended, passes over the wheel and is then attached to a mass \(m\). Prove that, assuming that the cord does not slip on the wheel and that the wheel and axle have no inertia, the acceleration of \(m\) \[ = 2g(2m-M)/\{4m+(1-r/R)M\}, \] and find the tension of the string.
A bullet of mass \(m\) moving horizontally with velocity \(v\) penetrates a distance \(c\) into a block of wood. Prove that, if the bullet moving with the same velocity strikes a block of similar material of thickness \(t\) that is free to move horizontally, the bullet will pass through the block if \[ t < cM/(M+m), \] where \(M\) is the mass of the block. Shew further that, if \(t\) is small compared with \(c\), the velocity of the bullet on emerging from the block is approximately \(v(1-t/2c)\).
A mass \(m\) is suspended from a spring causing an extension \(a\). If a mass \(M\) is added to \(m\) find the periodic time of the ensuing motion, and the amplitude of the oscillation.
Pencils and ranges of conics, and their relation to the theory of confocal conics.
Curvature of surfaces.
Continued fractions.
The Riemann integral.
Newtonian potentials.
Series of variable terms, in particular power series.