An acute-angled isosceles triangular prism stands on a rough horizontal plane, and one of its side faces is subjected to increasing uniform normal pressure. Shew that equilibrium will be broken by sliding or tumbling as the angle of friction is less or greater than the vertical angle of the prism, supposed less than \(60^\circ\).
Solve the equations \begin{align*} x + y + z &= 5 \\ x^2 + y^2 + z^2 &= 13\frac{1}{2} \\ x^3 + y^3 + z^3 &= 44. \end{align*}
Obtain the tangential equation of the conic given by the general equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Define conjugate points and conjugate lines with regard to a conic, and obtain the condition (1) that two points be conjugate, (2) that two lines be conjugate. Through two given points \(P\) and \(Q\) pairs of conjugate lines are drawn; prove that the locus of their point of intersection is a conic through \(P\) and \(Q\). Discuss the case when \(P\) and \(Q\) are conjugate points.
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 \[\frac{1}{2}\cot A \cot B \cot C (x \sec A + y \sec B + z \sec C).\]