A plane system of forces in equilibrium acts on a rigid body formed of two rods \(AB, BC\) rigidly joined together at an angle at \(B\). Shew by examples that the reaction at \(B\) may consist (i) of equal and opposite forces in a line passing through \(B\), or (ii) of equal and opposite couples, or (iii) of equal and opposite forces in a line not passing through \(B\). A number of rigid rods \(AB, BC, \dots MN\) are rigidly joined together at \(B, C, \dots M\), forming an unclosed polygon, not necessarily plane. \(A\) and \(N\) are joined by a string at tension \(T\). Determine the reaction at each of the joints \(B, C, \dots M\).
Explain the term ``coefficient of friction.'' A uniform circular cylinder and a uniform square prism lie in contact on a horizontal plane (i.e.\ with axes horizontal). The weights are equal, and the diameter of the cylinder is equal to a side of the square. All the surfaces are equally rough, the coefficient of friction for each pair being \(\mu\). The plane is slowly tilted about a line parallel to the axes of the cylinder and prism, so that the cylinder lies above the prism. Shew that equilibrium will be broken by the cylinder slipping at its contact with the prism or with the plane according as \(\mu < 1\) or \(\mu > 1\), and find the inclination of the plane at which equilibrium is broken in each case.
Shew that for a lamina moving in a plane there is in general an instantaneous centre of zero velocity. Shew further that for two laminas moving in the same plane there is in general a relative instantaneous centre having the same velocity for both, and that it lies in the join of the instantaneous centres of the two laminas. A link \(C_1C_2\) is hinged to points \(C_1, C_2\) on rods \(A_1B_1, A_2B_2\) respectively. \(A_1, A_2\) move on a straight line \(Ox\), and \(B_1, B_2\) on a perpendicular straight line \(Oy\). Find the relative instantaneous centre of the rods.
Two masses \(M, m\) (\(M>m\)) are connected by a light inelastic string passing over a smooth peg. Find the acceleration. The system is released from rest when \(M\) is at a height \(h\) above an inelastic horizontal plane. Discuss the subsequent motion, and shew that, if the time to the first impact is \(t_0\), then motion ceases permanently at a time \(3t_0\) after the start.
A point moves in a straight line, its acceleration being always directed towards a fixed origin in the line and equal to \(\mu\) times its distance from the origin. Shew that the motion is periodic, and find the period. A heavy particle is attached to one point of a uniform light elastic string. The ends of the string are attached to two points in a vertical line. Shew that the period of a vertical oscillation in which the string remains taut is \(2\pi \sqrt{mh/2\lambda}\), where \(\lambda\) is the coefficient of elasticity of the string, and \(h\) is the harmonic mean of the unstretched lengths of the two parts of the string.
Shew that all the points in a vertical plane which can be reached by a projectile thrown from a given origin in the plane with given velocity lie within or on a parabola, and that this parabola is touched by all the trajectories. Prove that the time to reach a point on the enveloping parabola at distance \(r\) from the origin is \(\sqrt{2r/g}\).
A sphere of mass \(4m\) in motion collides with a sphere of mass \(m\) at rest. Assuming the spheres to be smooth and perfectly elastic, shew that the direction of motion of the more massive sphere cannot be deflected by the collision through an angle greater than \(14^\circ 29'\).
Find the radial and transverse components of acceleration of a point moving in a circle. A smooth solid circular cylinder of radius \(a\) is fixed in contact along its lowest generator with a horizontal plane. A particle slides on the cylinder in a plane perpendicular to the generators. If the particle leaves the cylinder at an angular distance \(\theta\) from the highest generator, prove that it meets the plane at a distance \(x\) from the line of contact given by \[ \frac{x}{a} = \sin^3\theta + \cos\theta (1+\cos\theta) \sqrt{\cos\theta(2-\cos\theta)}. \]
Two particles \(A, B\), each of mass \(m\), are attached to the ends of a light rod of length \(a\). The rod is horizontal and instantaneously at rest, when \(A\) receives an upward vertical impulse \(mv\). Prove that, in the subsequent motion, the vertical component of \(B\)'s velocity will always be downwards if \(v^2/2ga\) is less than the least positive root of the equation \[ x \sin(x + \sqrt{x^2-1}) = 1. \]
Three similar triangles \(PBA, AQB, BAR\) are described on the same side of \(AB\), the similarity being indicated by the order of the letters as stated. Prove that the triangles \(PAQ, RBQ\) are similar; hence, or otherwise, prove that the triangle \(PQR\) is similar to each of the given triangles.