Problems

One corner of a long rectangular strip of paper of breadth \(b\) is folded over so that it falls on the opposite edge, and so that the portion folded over is triangular. Shew that the minimum area of this portion is \(2\sqrt{3} b^2/9\).

Find the limits of \(\frac{x^3+y^3}{x-y}\) as \(x\) and \(y\) tend to zero

1. [(a)] along the curve \(y = x - x^2\);
2. [(b)] along the curve \(y = x - x^3\);
3. [(c)] along the curve \(y = x - x^4\).

For what positive value of \(\alpha\) does \(\frac{x^\alpha+y^\alpha}{x^2-y^2}\) tend to 1 as \(x\) and \(y\) tend to zero along the curve \(y = x-x^\alpha\)?

Three particles \(A, B, C\) each of the same mass rest on a smooth table at the corners of an equilateral triangle; \(AB\) and \(BC\) being tight inextensible strings. \(A\) is given a velocity \(v\) in the direction \(CB\). Shew that when the string \(AB\) again tightens \(C\) starts off with velocity \(\frac{v}{15}\).

Evaluate the integrals \[ \int_0^1 \sqrt{\frac{1+x}{1-x}} \,dx, \quad \int \frac{2x^2-2x-5}{2x^2-5x-3} \,dx, \quad \int_0^\pi \sin^5 x \,dx, \quad \int x \sin x \,dx. \]

Evaluate \(\int_0^2 \frac{dx}{(3-x)\sqrt{2x^2+4x+9}}\), the positive value of the root being taken. Indicate how you would proceed to evaluate the integral if \(3-x\) were replaced by \((3-x)^2\).

The ends of a bar of length \(l\) are fastened to studs which slide each in one of two communicating slots passing through \(O\) and forming a cross at right angles to each other. The centre of the bar is constrained to describe a circle round \(O\) with uniform speed. Shew that each extremity of the bar describes a simple harmonic motion, and that the velocity of a point on the bar distant \(a\) from one extremity is perpendicular to the line joining \(O\) to the point of the bar distant \(a\) from the other extremity.

A plane cuts off from a sphere a volume equal to \(\frac{7}{27}\) of the whole. Find the ratio in which the diameter perpendicular to the plane is divided by it.

A crane is built of light jointed bars as in the figure. Sketch the force diagram, showing which members are in tension or thrust, and find the reactions at the two points of support.

If any of the bars would buckle and break under a thrust \(P\), find the greatest weight the crane would lift.

Explain the principle of virtual work. A tripod of three equal light rods of length \(l\), loosely jointed together at the top, rests on a smooth table, their lower ends being held together by three equal horizontal strings of length \(a\), which join them in pairs. A weight \(W\) is hung from the top. Find the tensions of the strings.

Explain the construction of the funicular polygon, showing in particular what it becomes when the system of forces (a) is in equilibrium, (b) reduces to a couple. Forces of magnitudes 1, 2, 3, 4, 5 act along the successive sides of a regular pentagon, taken round it in the same direction. Sketch figures to show the magnitude, direction and line of action of the resultant force.