Three rigid plates \(A, B, C\) are moving in any manner in one plane; prove that the instantaneous centres of the motion of \(A\) with reference to \(B\), and of \(B\) with reference to \(C\), and of \(C\) with reference to \(A\) are in one straight line. Find the instantaneous centre of the motion of the crank of an engine with reference to the piston.
Shew that if the cubic equation derived by clearing of fractions the equation \[\frac{a}{x+a} + \frac{b}{x+b} = \frac{c}{x+c} + \frac{d}{x+d}\] has a pair of equal roots, then either one of the numbers \(a, b\) is equal to one of the numbers \(c, d\) or \[\frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d}.\]
Determine the stresses in the given frame under the loads as shewn. [A diagram of a roof truss is provided. It consists of an isosceles triangle with base horizontal, apex angle \(60^\circ\). The two equal sides are bisected, and these bisection points are connected by a horizontal member. The bisection points are also connected to the opposite base vertices. A load of \(10\text{cwt\) is applied at the apex. A load of \(5\text{cwt}\) is applied at the left bisection point. A load of \(10\text{cwt}\) is applied at the right bisection point. The truss is supported at the two base vertices. The left support is on rollers on a surface inclined at \(30^\circ\) to the horizontal, with the support force normal to this surface. The right support is a fixed hinge.]} Distinguish between the members in compression and in tension.
Having given that \[ \frac{x^2-yz}{a} = \frac{y^2-zx}{b} = \frac{z^2-xy}{c}, \] prove that \[ \frac{a^2-bc}{x} = \frac{b^2-ca}{y} = \frac{c^2-ab}{z}; \] and shew that the product of one of the first three expressions and one of the last three is \[ (a+b+c)(x+y+z). \]
The cartesian coordinates of a point on a curve are given functions of a parameter: determine the equations of the tangent and normal at any point of the curve. The coordinates of any point on a three cusped hypocycloid may be written as \[ x=a(2\cos\theta - \cos 2\theta), \quad y=a(2\sin\theta+\sin 2\theta). \] Prove that any tangent to the curve cuts the curve again in two points at the constant distance \(4a\) apart and that the tangents at these points intersect at right angles on the circle through the vertices of the curve.
A carriage is moving in a straight line with velocity \(v\) and acceleration \(f\); find the magnitude and direction of the acceleration of any given point on the rim of one of the wheels of radius \(a\).
Find the approximate increment in the radius of the circumscribed circle of a triangle \(ABC\) when the side \(a\) receives a small increment \(x\), the other sides remaining unaltered; and deduce that, if the three sides receive small increments \(x, y, z\), then the increment in the radius is approximately \[\frac{1}{2}\cot A \cot B \cot C (x \sec A + y \sec B + z \sec C).\]
The weight on a suspension bridge is so arranged that the total load carried by the chains including their own weight is uniformly distributed across the span of the bridge. Shew that the curve of the chain is a parabola. The span of the bridge is 100 feet and the sag at the middle of the chain is 10 feet; if the total load on each chain is 25 tons, find the greatest tension in each chain and the tension at the lowest point.
Shew graphically or otherwise that the equation \(10^{x-1} = 2x\) has only two real roots and by means of tables find each of these correct to three places of decimals.
Discuss the conditions of equilibrium of a system of given coplanar forces. Prove that in all cases a system of coplanar forces can be replaced by two forces, one of which acts through a given point and the other along a given straight line.