A light framework of three rods \(BC, CA, AB\), freely jointed together to form an equilateral triangle of side \(a\), is suspended by three strings \(OA, OB, OC\), each of length \(l\), from a point \(O\), and a weight \(W\) is suspended by three equal strings each of length \(l'\) connecting it to \(A, B\) and \(C\). Shew that the thrust in each of the rods is \[ \frac{Wa}{3\sqrt{3}} \{ (3l^2-a^2)^{-\frac{1}{2}} + (3l'^2-a^2)^{-\frac{1}{2}} \}. \]
The velocity of a point is varying in direction and in magnitude. Explain precisely what is meant by saying that the acceleration is the rate of change of its velocity. If the velocity of a point is compounded of two velocities, each variable in magnitude and in direction, shew that the acceleration of the point is compounded of the respective rates of change of those velocities. Deduce the specification of the acceleration of a point whose velocity is given as the resultant of two components of constant magnitudes \(u\) and \(v\) making an angle \(\alpha\) with each other and each rotating with angular velocity \(\omega\).
A mass \(M\) rests on a smooth table and is attached by two inelastic strings to masses \(m, m'\) (\(m' > m\)), which hang over smooth pulleys at opposite edges of the table. The mass \(m'\) falls a distance \(x\) from rest, and then comes into contact with the floor (supposed inelastic). Shew that \(m\) will continue to ascend through a distance \(y\) given by \[ \frac{y}{x} = \frac{(m'-m)(M+m)}{m(M+m+m')}. \] Shew further that when \(m'\) is jerked into motion again as \(m\) falls it will ascend a distance \(x(M+m)^2 / (M+m+m')^2\).
Two equal particles \(A, B\) are attached to the ends of a spring which is held by its ends vertically and unstretched, \(A\) being uppermost. \(B\) is released, and at the moment at which it first comes to rest \(A\) is also released. Describe fully the subsequent motion, and shew that \(B\) comes to rest again once.
Shew that the inclination (\(\theta\)) of a conical pendulum to the vertical is given by \[ \sec\theta = l\omega^2/g. \] One end of a string of length 10 ft. is attached to a fixed point and two masses attached to the middle point and the other end of the string respectively swing round with equal angular velocities in horizontal circles at depths 4 ft. and 7 ft. below the point of suspension, the two portions of the string being always in one plane with the vertical through the point of suspension. Shew that the ratio of the upper mass to the lower is \(49/15\).
The pressure of the steam in the cylinder of a steam engine (internal cross-section \(A\); length of stroke \(l\)) is \(p_0\) lbs. per unit area during the first half of the forward motion of the piston. During the latter half no more steam is admitted, and the pressure of the steam in the cylinder falls according to the law \(pv^\gamma = \text{const.}\), where \(v\) is its volume. Shew that \(W\), the work done during the whole forward motion of the piston, is \[ p_0Al \cdot \{\gamma - 2^{1-\gamma}\}/2(\gamma-1). \] If the piston drives a flywheel against a constant couple \(G\), shew that the angular velocities \(\omega_1, \omega_2\) of the wheel at the beginning and end of the half-revolution are connected by the relation \[ I(\omega_2^2 - \omega_1^2) = 2W - 2\pi G. \]
State the principal relations that exist between a plane figure consisting of straight lines and circles and its inverse. Two circles intersect in a point \(O\). Prove that the inverse of \(O\) with regard to any circle which touches the two given circles lies on one or other of two fixed orthogonal circles.
Shew how to construct, with ruler and compasses, the radical axis of two non-intersecting circles when one lies within the other. Prove that, if the polars of a point with regard to three circles, whose centres are collinear, are concurrent, then the circles are coaxal.
Shew that, if two pencils of four rays have the same cross ratio and one ray in common, then the intersections of their other corresponding rays are collinear. \(ABCD\) is a quadrilateral. The sides \(AB, AD\) are fixed in position, and the lines \(BC, CD, DB\) each pass through one of three collinear fixed points. Prove that the locus of \(C\) is a straight line.
Reciprocate with regard to a focus the theorem, ``If \(A, B\) are the tangents to two confocal conics at one of their points of intersection, then \(A\) is the locus of the poles of \(B\) with regard to the family of confocals.'' Prove either the theorem or its reciprocal.