Prove that, if three forces are in equilibrium, they must lie in a plane, and must either meet in a point or be parallel. Four light rods freely jointed at their extremities form a quadrilateral. Each rod is acted on at its middle point in a direction perpendicular to its length by a force acting outwards whose magnitude is proportional to the length of the rod. Shew that in equilibrium the quadrilateral can be inscribed in a circle, and that the reactions at the corners are all equal and act along the tangents to the circle.
State the laws of friction and find the least force that will keep a weight \(W\) at rest on a rough inclined plane, where \(\lambda\) the angle of friction is \(< \alpha\) the inclination of the plane to the horizontal. Two uniform heavy rods, each of length \(2a\), are freely jointed to each other. They are placed symmetrically in a vertical plane across a rough cylinder of radius \(r\) which is fixed with its axis horizontal. Shew that the least angle each rod can make with the horizontal in equilibrium is given by \[ a\cos^2\alpha \cos(\alpha+\lambda) = r\sin\alpha\cos\lambda, \] and find an equation satisfied by the greatest angle each rod can make with the horizontal in equilibrium.
Explain how the principle of virtual work may be used to determine the unknown reactions of a system in equilibrium. A regular octahedron formed of twelve equal uniform rods of weight \(w\) freely jointed is suspended from one corner. Prove that the thrust in each horizontal rod is \(3w/\sqrt{2}\).
An aeroplane has a speed \(u\) in still air. A wind is blowing with velocity \(w (
State Newton's Laws of Motion. A smooth wedge of mass \(M\) and angle \(\alpha\) is free to move on a smooth horizontal plane in a direction perpendicular to its edge. A particle of mass \(m\) is projected directly up the face of the wedge with velocity \(V\). Prove that it returns to the point on the wedge from which it was projected after a time \[ 2V(M+m\sin^2\alpha)/\{(m+M)g\sin\alpha\}. \] Also find the pressure between the particle and the wedge at any time.
Prove that, if the sum of the resolutes in a given direction of the external forces on any number of particles is zero, the sum of the momenta of the particles in that direction is constant. Two particles \(A\) of mass \(m\) and \(B\) of mass \(2m\) are connected by a light inextensible string of length \(a\) which is stretched at full length perpendicular to the edge of a smooth horizontal table. The particle \(B\) is drawn just over the edge of the table and is then released from rest in this position. Describe the nature of the subsequent motion and shew that after \(A\) leaves the table the centre of mass of the two particles describes a parabola of latus rectum \(8a/27\).
Prove that \(v^2/\rho\) is the acceleration along the normal inwards of a point moving with velocity \(v\) in a curve, where \(\rho\) is the radius of curvature at the point. A circular cylinder of radius \(a\) is placed in a fixed position with its axis horizontal on a smooth horizontal plane. A perfectly elastic particle is placed on the highest generator of the cylinder and being slightly displaced slides down the cylinder. Prove that the distance between consecutive points at which it strikes the horizontal plane is \(40a\sqrt{2}/27\).
Define simple harmonic motion and shew how to find the period of oscillation when the acceleration at any distance from the centre is given. A light elastic string of unstretched length \(l_0\) is suspended from a fixed point and has a mass \(m\) attached to its other end. In equilibrium its stretched length is \(l\). A blow \(B\) is applied vertically upwards to \(m\). Find \(B\) so that the string just becomes slack in the subsequent motion. If the blow applied had been twice this value of \(B\), shew that \(m\) would have risen to a height \(l-l_0+3B^2/2gm^2\) above its equilibrium position.
Shew that the value of a determinant is zero if two of its rows or two of its columns are identical. Deduce that \[ \begin{vmatrix} a_1+pb_1+qc_1, & b_1, & c_1 \\ a_2+pb_2+qc_2, & b_2, & c_2 \\ a_3+pb_3+qc_3, & b_3, & c_3 \end{vmatrix} \] is independent of \(p\) and \(q\). Use the equations \(x^3+px^2+qx+r=0, x^2+bx+c=0\) to express \[ (r+bc-pc)^2 - (cq-c^2-br)(bp-b^2-q+c) \] as a determinant of the fifth order.
If \(\dbinom{n}{r}\) denotes the number of combinations of \(n\) things taken \(r\) at a time, shew that \[ \binom{n}{r} = \frac{n}{r}\binom{n-1}{r-1} \text{ and derive an interpretation for } \binom{n}{0}. \] Prove that \[ \binom{m+n}{r} = \binom{m}{r}\binom{n}{0} + \binom{m}{r-1}\binom{n}{1} + \binom{m}{r-2}\binom{n}{2} + \dots + \binom{m}{1}\binom{n}{r-1} + \binom{m}{0}\binom{n}{r}. \] By means of the identity \(1-x^6 = (1-x^3)(1+x+x^2)\), or otherwise, prove that \[ \binom{6m}{n}\binom{6m}{n} + \binom{6m-1}{n}\binom{6m-2}{n} + \binom{6m-2}{n}\binom{6m-4}{n} + \dots + \binom{3m}{n}\binom{0}{n} \] \[ = \binom{n+6m-1}{6m-1}\binom{n}{0} - \binom{n+6m-4}{6m-3}\binom{n}{1} + \binom{n+6m-7}{6m-6}\binom{n}{2} - \dots + (-1)^m \binom{n+1}{0}\binom{n}{2m}. \]