Problems

In starting an engine of mass \(m\) the pull on the rails is at first constant and equal to \(R/u\), and after the velocity attains a value \(u\) the engine works at a constant rate \(R\); throughout the motion there is a frictional resistance which is proportional to the square of the velocity, and the greatest steady velocity at which the engine can travel is \(w (>u)\). Prove that the distance in which the engine, starting from rest, attains a velocity \(V\) between \(u\) and \(w\) is \[ \frac{mw^3}{6R} \log \frac{w^3-u^3}{(w^3-V^3)^2}. \quad (\text{Note: original has }(w^3-u^3)(w^3-V^3)^2 \text{ in denominator, which seems unlikely}) \] Let's check the OCR: `log (w^3-u^3)(w^3-V^3)^2`. Let's assume this is correct and transcribe as seen. \[ \frac{mw^3}{6R} \log \frac{w^6}{(w^3-u^3)(w^3-V^3)^2}. \quad (\text{The numerator is unclear in scan, but seems to be } w^6) \] Re-examining: A simpler form might be intended. The OCR text is `mw^3/6R log (w^6) / (w^3-u^3)(w^3-V^3)^2`. Let's assume the \(w^6\) is correct.

Two smooth perfectly elastic spheres, one of mass \(M\) and the other of smaller mass \(m\), are initially at rest. The sphere of mass \(M\) is projected so that it collides with the other. Shew that the direction of motion of \(M\) cannot be deflected by the collision through an angle greater than \(\sin^{-1}(m/M)\).

A particle is projected in a given vertical plane from a point \(O\) of the plane with a velocity \(\sqrt{(2gh)}\). Shew that the points of the plane which are accessible by projection from \(O\) with the given velocity lie on or beneath the parabola having \(O\) as focus and a horizontal line at height \(h\) above \(O\) as tangent at the vertex. Shew that the time taken to reach a point on this parabola at distance \(r\) from \(O\) is \(\sqrt{(2r/g)}\).

A particle moves in a plane under a force directed towards an origin \(O\); using polar coordinates with \(O\) as origin to describe the position of the particle at time \(t\), prove that \(r^2\dot{\theta}\) is constant. A particle of mass \(m\) moves on a smooth horizontal table, and is attached to a point \(O\) of the table by a light elastic string of natural length \(a\) and modulus \(\lambda\). Initially the particle rests at a point \(P\) at distance \(a\) from \(O\). It is projected with velocity \(\sqrt{\left(\frac{4\lambda a}{3m}\right)}\) in a direction at right angles to \(OP\). Shew that the greatest elongation of the string in the subsequent motion is \(a\).

(i) Define an involution pencil and prove that the pairs of tangents from a fixed point to conics touching four fixed lines form an involution pencil. \par (ii) A fixed conic meets corresponding rays of an involution pencil, whose vertex O is not on the conic, in P, P' and Q, Q'. Prove that the lines PQ, P'Q', PQ', P'Q touch a fixed conic which also touches the double lines of the pencil.

Deduce from the triangle of forces that the resultant of two parallel forces is, in general, a third force whose moment about any point is the sum of the moments of the two forces about the point. \par A uniform straight rod of length \(3a\) is hinged freely at one end to a wall, and is supported in a horizontal position by a knife-edge distant \(2a\) from the wall. Find where the bending moment is greatest.

Through a given point inside a parallelogram construct a straight line which shall divide the area of the parallelogram into two parts as unequal as possible.

Define a determinant, and prove from your definition that if two rows or two columns of a determinant are equal the determinant vanishes. \par Find the factors of \[ \begin{vmatrix} a & b & c \\ b+c & c+a & a+b \\ a^3 - abc & b^3 - abc & c^3 - abc \end{vmatrix}. \]

Prove that through a given point there are two conics confocal with a given ellipse, one being an ellipse and the other a hyperbola, and that they intersect at right angles. \par Find the locus of the pole of a given straight line with respect to a system of confocal conics. \par Prove that the tangents to a conic from a given point T make equal angles with the tangent at T to either of the confocal conics through T.

The crane ABCD is built up from freely hinged light rods, and is hinged to the horizontal ground at A and B. A weight W is suspended by a light chain passing over a small frictionless pulley at C to B. Find, by means of a force diagram or otherwise, the tensions or thrusts in the rods. \par [A diagram is shown with points A and B on a horizontal line. The structure ABCD is above this line. Angles are given as \(\angle DAB = 45^\circ\), \(\angle ADC = 45^\circ\), \(\angle CBA = 30^\circ\), and \(\angle BCD=30^\circ\). A weight W hangs from C.]