Problems

A uniform thin chain, 20 feet long and weighing 10 lb., rests in a small space on the ground. One end of it is given a constant vertical acceleration of 5 feet per second per second by a force applied to that end. Determine the work which has been done by this force when the whole chain is just clear of the ground.

If the normals at four points of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) are concurrent, and if two of the points lie on the line \[ lx + my = 1, \] prove that the other two lie on the line \[ \frac{x}{a^2l} + \frac{y}{b^2m} + 1 = 0. \] Hence, or otherwise, shew that if the feet of two of the normals from a point \(P\) to the ellipse are coincident, then the locus of the middle point of the chord joining the feet of the other two normals is \[ \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right)^2 \left(\frac{a^6}{x^2} + \frac{b^6}{y^2}\right) = (a^2-b^2)^2. \]

Find the solution of the equation \[ (x+2)^2 y'' - 2(x+2) y' + 2y = 0 \] which represents a curve touching the parabola \(y = x^2 + 1\) at the point \((1, 2)\).

A block of mass \(M\) with a plane base is free to slide on a smooth horizontal plane. The block contains a spherical cavity, of radius \(a\), whose surface is smooth. A particle of mass \(m\) slides on the surface of the cavity starting from rest at the level of the centre. Shew that when the radius to the particle makes an angle \(\psi\) with the vertical, the velocity of the particle relative to the block is \[ \{2ga \cos\psi (M+m)/(M+m\sin^2\psi)\}^{\frac{1}{2}}, \] assuming the motion of every point to be parallel to the same vertical plane.

The polar coordinates at time \(t\) of a particle moving in a plane are \(r\) and \(\theta\). Shew that its velocity \(v\) is given by \[ v^2 = \left(\frac{dr}{dt}\right)^2 + \left(r\frac{d\theta}{dt}\right)^2. \] A small ring of mass \(m\) is free to slide along a smooth parabolic wire in a horizontal plane and it is attached to one end of an inextensible string. In the same horizontal plane and in the neighbourhood of the focus there is a table, and the string is threaded through a smooth hole in the table at the focus. At the other end of the string hangs a bead of mass \(M\). The ring is held at one end of the latus rectum (which is of length \(4a\)) and then released. Shew that it will not come to rest again until it reaches the other end of the latus rectum. Also shew that its angular velocity about the focus is greatest as it crosses the axis of the parabola, and find this maximum value.

Shew that in general two triangles can be inscribed in the hyperbola \(xy=k^2\) with sides parallel to the lines \[ y+lx=0, \quad y+mx=0, \quad y+nx=0. \] State under what conditions the triangles will be real. Shew that if \(l, m, n\) vary, the area of either triangle is proportional to \[ \frac{(m-n)(n-l)(l-m)}{lmn}. \]

Find the coordinates of the node of the curve \[ (x+y+1)y + (x+y+1)^2 + y^3 = 0, \] and the area of the loop at the node.

Two particles of masses \(m\) and \(m'\) are joined by a light inextensible string of length \(a+b\) and rest on a smooth horizontal plane at points \(A, B\) at distances \(a, b\) from a smooth vertical peg \(O\) round which the string passes so that initially the two portions \(OA, OB\) are at right angles. Shew that if the first particle is projected with velocity \(u\) parallel to \(OB\), its distance \(r\) from \(O\) at time \(t\) is given by \(\dot{r}^2 = a^2 + \frac{m}{m+m'}u^2t^2\) if the string is still in contact with the peg.

A flat disc, with its plane horizontal, is spinning in frictionless bearings at an angular velocity \(\omega_1\) about a vertical axis through its centre, its moment of inertia about that axis being \(I\). A uniform ring of mass \(m\) and radius \(R\), with its plane horizontal and its centre on the axis of the disc, is lowered on to the latter while spinning in its own plane about its centre with an angular velocity \(\omega_2\) in the opposite direction to \(\omega_1\). If the coefficient of friction between the ring and the disc be \(\mu\), derive an expression for the time during which relative slipping will continue.

Referred to rectangular axes, the equations of a curve are given in the parametric form \[ x = at + bt^2, \] \[ y = ct + dt^2, \] where \(a, b, c, d\) are constants such that \(ad-bc\) is not zero. Shew that the curve is a parabola and that the chord joining the points whose parameters are \(t_1\) and \(t_2\) is given by the equation \[ \begin{vmatrix} x & y & t_1 t_2 \\ a & c & t_1+t_2 \\ b & d & -1 \end{vmatrix} = 0. \] Further, if the tangents at these two points are at right angles, shew that the chord passes always through the point \((x_0, y_0)\), where \[ \frac{cx_0 - ay_0}{c^2+a^2} = \frac{dx_0-by_0}{2(ab+cd)} = \frac{bc-ad}{4(b^2+d^2)}. \]