Four particles each of mass \(m\) are attached to the corners \(A, B, C, D\) of a rhombus formed of a light string. The whole is laid on a smooth horizontal table with the strings taut and the angle \(BAD\) acute and equal to \(2\alpha\). A blow \(P\) is applied to the mass at \(A\) in the direction \(CA\). Prove that the kinetic energy produced by the blow is \(P^2(1+2\sin^2\alpha)/8m\), also find the velocities of the particles at \(B\) and \(D\) just before they strike each other.
Two particles of equal mass are attached to the ends of a light rod. The rod can turn freely about a point \(O\) at distances \(l\) and \(l'\) respectively from the ends. Prove that, if the rod rotate about a vertical axis through \(O\) with angular velocity \(\omega\), it is inclined to the vertical at an angle \[ \cos^{-1}\left\{\frac{(l\sim l')g}{\omega^2(l^2+l'^2)}\right\}^{\frac{1}{2}}. \]
Prove that a force \(P\) can be replaced by forces \(X, Y, Z\) along the sides \(BC, CA, AB\) of a triangle in its plane in one way only. \par If \(P\) acts at right angles to \(BC\) at its middle point inwards, shew that \[ \frac{X}{b^2-c^2} = \frac{Y}{ab} = \frac{Z}{-ac} = \frac{P}{4\Delta}, \] where \(a,b,c\) are the sides and \(\Delta\) the area of the triangle.
Shew that couples of equal and opposite moment in one plane are in equilibrium. \par A heavy bar is suspended by two equal vertical strings of equal length. It is twisted round a vertical axis through a given angle \(\theta\) so that its centre rises vertically. Find the couple needed to preserve equilibrium.
State the principle of virtual work and explain how by its use unknown forces and stresses are eliminated from statical problems. \par \(ABCD\) is a rhombus of freely jointed rods lying flat on a smooth table and \(P, Q\) are the middle points of \(AB, AD\). Prove that if the system is held in equilibrium by tight strings joining \(P\) to \(Q\) and \(A\) to \(C\), the tensions in these strings are in the ratio of \(2BD\) to \(AC\).
A fire engine raises \(n\) gallons of water per minute from a reservoir and discharges it at a height \(h\) feet above the surface of the reservoir through a pipe whose cross-section is \(A\) square inches; find the horse power consumed, assuming a gallon of water to weigh 10 lbs.
One end of a light string is fixed, and the string, hanging vertically in a loop in which a ring of mass \(m\) moves, passes over a smooth fixed pulley and has a mass \(M\) tied to the other end. Shew that in the motion the tension of the string is \(\frac{3Mmg}{4M+m}\).
A particle is describing a circle uniformly; determine the radial force acting on it. \par Two particles are connected by a fine string passing through a smooth ring and describe horizontal circles in the same periodic time; shew that the particles are at the same vertical depth below the ring, and find the ratio in which the string is divided by the ring.
A body is projected from a given point with velocity \(V\), so as to pass through another point at a horizontal distance \(a\) from the point of projection and at a height \(b\) above it. Find an equation to determine the necessary angles of elevation. \par A shot has a range \(c\) on a horizontal plane when the angle of elevation is \(\alpha\) and just reaches the base of a vertical target of height \(2a\), where \(a=c\tan\alpha\). Shew that with the same initial velocity, and with elevation \(\theta+\alpha\), it will strike the target at a distance \(a\sin^2\theta\sec^2(\alpha+\theta)\) below the centre.
If \(y=a+x\log\frac{y}{b}\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) when \(x\) is zero. \par Shew that, if \(x\) is so small that the value of \(x^3\) and of higher powers may be neglected, then \[ y = a+x\log\frac{a}{b} + \frac{x^2}{a}\log\frac{a}{b}. \]