\(ABCD\) is a rhombus formed of freely jointed light rods. \(AC\) is vertical, \(A\) being the higher end, and \(B, D\) are tied by strings of equal length to a fixed point in the line \(AC\). Weights \(W, W' (W>W')\) are suspended from \(A, C\). If the rhombus is constrained to remain in a vertical plane, prove that in the position of equilibrium the fixed point divides \(AC\) in the ratio \(W':W\).
An engine of 250 horse-power pulls a load of 150 tons up an incline of 1 in 75. Taking the road resistance to be 16 lbs. per ton, find the greatest speed attainable on the incline.
An elastic sphere strikes obliquely an equal sphere at rest. Find the angle through which the direction of motion is deflected, and prove that if the line of motion of the centre of the moving sphere before contact is tangential to the fixed sphere the angle of deflection is \(\tan^{-1}\frac{(1+e)\sqrt{3}}{5-3e}\), where \(e\) is the coefficient of restitution.
A particle is projected along the inner side of a smooth circle of radius \(a\), the velocity at the lowest point being \(u\). Shew that if \(u^2<5ga\) the particle will leave the circle before arriving at the highest point and will describe a parabola whose latus-rectum is \(2(u^2-2ga)^3/27g^2a^2\).
Find the resultant acceleration of a point which moves in any manner round a circle. \par The wheel axles of a motor car are 4 feet long and the height of the C.G. is 2 feet. Find the speed of the car if in going round a level track of 400 feet radius the inner wheels just leave the ground.
A particle moves with an acceleration towards a point equal to \(\mu \times\) distance from the point. Find the amplitude of the motion having given the velocity in any position, and find also the time of a complete oscillation. \par A weightless rod of length 3 feet, with equal heavy rings at the ends, one of which can slide on a smooth horizontal wire, is describing small oscillations in the vertical plane containing the wire. Shew that the period of oscillation is about 1.36 seconds.
Solve the equation \[ (x-1)(x+2)(x+3)(x+6)=160. \] Eliminate \(x,y,z\) from \[ x+y-z=a, \quad x^2+y^2-z^2=b^2, \quad x^3+y^3-z^3=c^3, \quad xyz=d^3. \]
Find the conditions that \(ax^2+2bx+c\) should be positive for all real values of \(x\). \par Prove that the function \(\frac{(x-b)(x-c)}{x-a}\) can take all values for real values of \(x\) if \(a\) lies between \(b\) and \(c\); but if this condition does not hold it can take all values except certain values which lie in an interval \(4\sqrt{(a-b)(a-c)}\).
If \[ (1+x)^n = c_0+c_1x+c_2x^2+\dots+c_nx^n, \] where \(n\) is a positive integer, find \(c_0^2+c_1^2+\dots+c_n^2\). \par If \begin{align*} s_0 &= c_0+c_3+c_6+\dots, \\ s_1 &= c_1+c_4+c_7+\dots, \\ s_2 &= c_2+c_5+c_8+\dots, \end{align*} prove that \[ s_0^2+s_1^2+s_2^2=1+s_1s_2+s_2s_0+s_0s_1. \]
Shew that every mixed periodic continued fraction, which has more than one non-periodic element, is a root of a quadratic equation with rational coefficients whose roots are both of the same sign. \par Find the value of the \(2n\)th convergent to the continued fraction \[ \frac{1}{2+}\frac{1}{4+}\frac{1}{2+}\frac{1}{4+}\dots. \]