Two weights \(P\) and \(Q\) rest on a rough double inclined plane, connected by a fine string passing over a small smooth pulley at the common vertex. The angle of friction \(\epsilon\) is the same for both planes, and \(Q\) is on the point of motion downwards. Prove that the greatest weight which can be added to \(P\) without disturbing the equilibrium is \[ \frac{P \sin 2\epsilon . \sin(\alpha+\beta)}{\sin(\alpha-\epsilon).\sin(\beta-\epsilon)}, \] where \(\alpha, \beta\) are the inclinations of the planes to the horizontal.
A uniform rod \(AB\) of weight \(W\) and length \(l\) rests on a horizontal table whose coefficient of friction is \(\mu\). A string is attached to \(B\) and is pulled gently in a horizontal direction perpendicular to the rod. As the tension is gradually increased, show that the rod begins to turn about \(C\), where \(AC\) is \(l(1-\frac{1}{\sqrt{2}})\), and that the tension of the string is then \(\mu W (\sqrt{2}-1)\).
A uniform rod \(PQ\), of length \(l\), rests with one end \(P\) on a smooth fixed elliptic arc whose major axis is horizontal, and the other end \(Q\) on a smooth fixed vertical plane at a horizontal distance \(h\) from the centre of the ellipse. If \(\theta\) is the inclination of the rod to the horizontal, \(2a\) and \(2b\) are the axes of the ellipse, and \(\alpha\) is the eccentric angle at \(P\), prove that \[ a\cos\alpha+h=l\cos\theta, \] and \[ a\tan\alpha=2b\tan\theta. \] What happens if \(h=0\), and also \(a=2b=l\)?
A train consists of an engine and tender, of mass \(M\) tons, and two coaches, each of mass \(m\) tons. At the start the buffers are in contact, and when the coupling chains are tight the buffers are \(a\) feet apart. The train starts with the engine exerting a constant tractive force \(F\) tons weight. Neglecting resistance, show that the second coach starts with velocity \(v\) feet per second, where \[ v^2 = 2ga \cdot \frac{F(2M+m)}{(M+2m)^2}. \]
\(A\) is a point on the ground, \(l\) feet distant from a vertical wall \(BC\), \(h\) feet high, so that \(AB\) is horizontal and equal to \(l\). It is required to throw a stone from \(A\) with a given velocity \(u\) feet per second to hit a stationary cat at \(C\). Neglecting air resistance, find an equation to give the two directions in which the stone can be thrown if the task is possible; and show that the least value of \(u\) that makes it possible is \[ \sqrt{g(h+\sqrt{h^2+l^2})}. \]
Two masses \(m_1\) and \(m_2\) are connected by a light spring and placed on a smooth horizontal table. When \(m_2\) is held fixed, \(m_1\) makes \(n\) complete vibrations per second. Show that if \(m_1\) is held fixed, \(m_2\) will make \(n\sqrt{m_1/m_2}\) vibrations per second, and if both are free, they will make \(n\sqrt{(m_1+m_2)/m_2}\) vibrations per second, the vibrations in all cases being in the line of the spring.
A motor car is running at a constant speed of 60 feet per second. It is found that the effective horse-power at the road wheels is 18. Find the resistance to motion. Assuming that the resistance varies as the square of the speed, and that the effective horse-power at the road wheels remains constant and equal to 18, prove that the distance required for the car to accelerate from 20 feet per second to 40 feet per second is \[ 750 \log_e \frac{26}{19} \text{ feet}. \] The car weighs 3300 lbs., and in both cases the road is level.
If \[ y=\sin(\log x), \] prove that \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+y=0. \] The work that must be done to propel a ship of displacement \(D\) for a distance \(s\) in time \(t\) is proportional to \(s^2 D^{2/3}/t^2\). Find approximately the percentage increase of work necessary when the distance is increased 1\%, the time is diminished 1\%, and the displacement of the ship is diminished 2\%.
A closed circular cylinder of height \(h\) is to be inscribed in a given sphere of radius \(R\). If the whole surface of the cylinder, including the base and lid, is to be a maximum, prove that \[ \frac{h^2}{R^2} = 2\left(1-\frac{1}{\sqrt{5}}\right). \]
Find the asymptotes of the curve \[ y^2 = \frac{a^3x}{a^2-x^2} \] and find the radius of curvature at the origin. Sketch the curve.