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
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.
Explain how the principle of virtual work may be used to determine the unknown reactions of a system in equilibrium. A regular octahedron formed of twelve equal uniform rods of weight \(w\) freely jointed is suspended from one corner. Prove that the thrust in each horizontal rod is \(3w/\sqrt{2}\).
An aeroplane has a speed \(u\) in still air. A wind is blowing with velocity \(w (
State Newton's Laws of Motion. A smooth wedge of mass \(M\) and angle \(\alpha\) is free to move on a smooth horizontal plane in a direction perpendicular to its edge. A particle of mass \(m\) is projected directly up the face of the wedge with velocity \(V\). Prove that it returns to the point on the wedge from which it was projected after a time \[ 2V(M+m\sin^2\alpha)/\{(m+M)g\sin\alpha\}. \] Also find the pressure between the particle and the wedge at any time.