Three equal smooth pencils are tied together by a string and laid on a smooth table. Find the tension of the string when there is a pressure \(P\) between the pencils in contact with the table.
Two equal uniform ladders are jointed at one end and stand with the other ends on a rough horizontal plane. A man whose weight is equal to that of one of the ladders ascends one of them. Prove that the other will slip first. If it begins to slip when he has ascended a distance \(x\), prove that the coefficient of friction is \((a+x)\tan\alpha/(2a+x)\), \(a\) being the length of each ladder, and \(\alpha\) the angle each makes with the vertical.
State the principle of Virtual Work. Four equal uniform rods of weight \(W\) are freely jointed so as to form a square \(ABCD\) which is suspended from \(A\) and is prevented from collapsing by an inextensible string joining the middle points of \(AB\) and \(BC\). Prove that the tension of the string is \(4W\) and find the magnitude and direction of the reaction at \(B\).
Find the centroids of an arc and a sector of a circle. Shew that the centroid of a segment of a circle divides the line of symmetry into parts in the ratio of 2:3, if the breadth of the segment is small compared with the radius.
State Newton's Second Law of Motion and shew how it leads to the equation \(P=mf\). A pulley of mass \(m\) is connected with a mass \(4m\) by a string which hangs over a fixed smooth pulley; a string with masses \(m\) and \(2m\) at its extremities is hung over the pulley. If the system is free to move, find the acceleration of each of the masses.
Find the velocity of the centre of inertia of two particles whose masses and velocities are given. Shew that the kinetic energy of two masses is equal to that of the sum of the masses moving with the velocity of the centre of inertia together with that of each mass moving with its velocity relative to the centre of inertia.
Find the range of a projectile on an inclined plane through the point of projection. Two particles are projected with velocities \(u, u'\) and elevations \(\theta, \theta'\) from the same point at the same time and in the same vertical plane. Shew that the difference of the times of their passing through the other point common to their paths is \[ 2uu' \sin(\theta \sim \theta') / g(u\cos\theta+u'\cos\theta'). \]
State the principles by which we are enabled to calculate the changes in velocity produced by the impact of two smooth elastic spheres. A smooth elastic sphere falls vertically with velocity \(u\) on a smooth wedge which lies on a smooth table. Calculate the velocity of the wedge after the impact, the wedge and the table being supposed inelastic.
An elastic string of natural length \(a\) has one end fixed and a weight attached to the other. When it hangs vertically it is stretched a length \(c\), and when it revolves as a conical pendulum making \(n\) revolutions per second it is stretched a length \(z\). Prove that \(gz = 4\pi^2 n^2 c(a+z)\).
Find the conditions that \(ax^2+bx+c\) may be positive for all real values of \(x\). Shew that for real values of \(x\) the fraction \((2x^2-5x+2)/(x^2-4x+3)\) assumes all values from \(-\infty\) to \(+\infty\). Draw a graph of the function for all values of \(x\) from \(-\infty\) to \(+\infty\).