Problems

Two weights \(P\) and \(Q\) are resting, one on each of two equally rough inclined planes, and are connected by a light smooth string passing over the horizontal line of intersection of the planes. The two portions of the string lie in lines of greatest slope of the planes. If \(Q\) is on the point of motion downwards, prove that the greatest weight that can be added to \(P\) without disturbing equilibrium is \[ \frac{P \sin 2\lambda \sin(\alpha+\beta)}{\sin(\alpha-\lambda)\sin(\beta-\lambda)}, \] where \(\alpha, \beta\) are the inclinations of the planes, and \(\lambda\) is the angle of friction.

A railway wagon of mass 8 tons, travelling at 8 feet per second, collides with a similar stationary wagon. Each of the four buffer springs in contact exerts a force of 2 tons when the buffers are fully extended, and requires an additional force of 1 ton for each inch of compression. The motion of a buffer in either direction is also resisted by a constant frictional force of \(\frac{1}{2}\) ton. Prove that each buffer is compressed 3 inches, and that the speeds of the wagons when contact ceases are \((4\pm 2\sqrt{3})\) feet per second.

A small smooth heavy ring is free to slide on a fixed parabolic wire whose axis is vertical and vertex downwards. The ring is projected from the vertex \(A\) with velocity \(\sqrt{2gh}\), and after passing the extremity \(B\) of the arc proceeds to describe an equal parabola freely. If \(c\) is the vertical height of \(B\) above \(A\), prove that the latus rectum is \(4(h-2c)\).

A particle of mass \(2M\) on a smooth horizontal table is connected by a light inextensible string passing through a small smooth hole in the table to a particle of mass \(M\) hanging freely. The upper particle, at a certain moment, is moving at right angles to the string with velocity \(u\), at a distance \(a\) from the hole. When next moving at right angles to the string, its velocity is \(v\), at a distance \(b\) from the hole. Express \(a\) and \(b\) in terms of \(u\) and \(v\).

For a certain rowing eight, the resistance to motion is \(\frac{1}{8}v^2\) lb., where \(v\) is the speed in feet per second. The total mass is 2000 lb., and it may be assumed that a constant propulsive force of 300 lb. is maintained for 1\(\frac{1}{4}\) seconds, after which for \(\frac{3}{4}\) second the propulsive force is zero. If the speed at the beginning of a stroke is 12 feet per second, prove that at the end of the working part of the stroke the speed is 15 feet per second, and find the speed \(\frac{3}{4}\) second later. [\(e^{0.56}=1.75\)] \item[(i)] If \(y=x^{n-1}\log x\), prove that \[ \frac{d^n y}{dx^n} = \frac{(n-1)!}{x}. \] \item[(ii)] Prove that the limit of \[ \frac{\cos^2 \pi x}{e^{2x}-2e^x}, \] as \(x\) approaches the value \(\frac{1}{2}\), is \(\dfrac{\pi^2}{2e}\).

Prove that \[ \frac{1+2x-x^2+2\sqrt{x-x^3}}{1+x^2} \] is a maximum or minimum when \(x = -1\pm\sqrt{2}\).

Evaluate

1. [(i)] \(\int \left(\frac{1}{x}+\frac{1}{x^2}\right)\log x dx\);
2. [(ii)] \(\int \tan x(1+\sec x)dx\);
3. [(iii)] \(\int \frac{1}{(1+x)\sqrt{1+2x}}dx\).

A uniform rod of length \(2a\) and weight \(W\) is supported by a string of length \(2l\), whose ends are fastened to the ends of the rod and which passes over a smooth peg. A weight \(2W\) is attached to the rod at a distance \(c\) from its middle point. Shew that the lengths of the string on the two sides of the peg are \(l(3a-2c)/3a\) and \(l(3a+2c)/3a\).

Prove that, if three forces are in equilibrium, they must lie in a plane, and must either meet in a point or be parallel. Four light rods freely jointed at their extremities form a quadrilateral. Each rod is acted on at its middle point in a direction perpendicular to its length by a force acting outwards whose magnitude is proportional to the length of the rod. Shew that in equilibrium the quadrilateral can be inscribed in a circle, and that the reactions at the corners are all equal and act along the tangents to the circle.

State the laws of friction and find the least force that will keep a weight \(W\) at rest on a rough inclined plane, where \(\lambda\) the angle of friction is \(< \alpha\) the inclination of the plane to the horizontal. Two uniform heavy rods, each of length \(2a\), are freely jointed to each other. They are placed symmetrically in a vertical plane across a rough cylinder of radius \(r\) which is fixed with its axis horizontal. Shew that the least angle each rod can make with the horizontal in equilibrium is given by \[ a\cos^2\alpha \cos(\alpha+\lambda) = r\sin\alpha\cos\lambda, \] and find an equation satisfied by the greatest angle each rod can make with the horizontal in equilibrium.