A particle, moving under gravity, is resisted by a frictional force which acts in the opposite direction to the velocity and is a function of the speed \((g)\) only. Shew from the equations of horizontal and vertical motion that the horizontal velocity continually diminishes, and that \[ \frac{d}{dx}\left(\frac{v}{u}\right) = -\frac{1}{2}\frac{d}{dy}\left(\frac{v^2}{u^2}\right) = -\frac{g}{u^2}, \] where \(x\) is measured horizontally and \(y\) vertically upwards, and \((u,v)\) are the components of velocity in these directions. Deduce that
Shew that a plane system of forces acting on a rigid body is equivalent either to a single force or to a couple. Forces of magnitudes 1, 1, 2, 2 act on a rigid body in the sides \(AB, BC, CD, DA\) of a square \(ABCD\). Shew that the system is equivalent to a single force, and find its magnitude and line of action.
Three uniform heavy rods \(AB, BC, CA\) of lengths 3, 4, 5 feet, and weights \(3W, 4W, 5W\), are freely jointed together at their ends to form a triangular framework. The framework is suspended by a string attached to a point \(H\) of \(AC\), and \(AC\) is horizontal. Prove that the length \(AH\) is 12/5 feet. Find the horizontal and vertical components of the reaction at \(A\) on the rod \(AB\).
Four uniform rods \(AB, BC, CD, DE\), each of length \(2a\) and weight \(w\), are freely hinged together at \(B, C\) and \(D\), and the chain hangs in equilibrium from two supports at the same level to which the ends \(A\) and \(E\) are freely attached. If the rods \(AB, BC\) make angles \(\theta, \phi\) with the horizontal, prove that \(\tan\theta = 3\tan\phi\). If the horizontal component of the force exerted by a support is \(3w/2\), prove that the distance \(AE\) is about \(6.6a\).
A bead of mass \(m\) slides on a smooth circular hoop which is fixed in a vertical plane, and the bead is attached to a particle of mass \(M\) by a light inelastic string passing through a small smooth ring which is fixed at the highest point of the hoop. Shew that the potential energy of the system is \[ C-mga\left(x - \frac{M^2}{m^2} \frac{1}{x}\right), \] where \(a\) is the radius of the circle, and \(ax\) is the distance of the bead from the ring. Shew that, if \(M \le 2m\), there is a position of unstable equilibrium in which \(x=M/m\). Find the positions of equilibrium, and discuss their stability, if \(M>2m\).
Find the form of a uniform flexible inelastic string which hangs at rest under the action of gravity with its ends attached to fixed supports. The ends of a uniform string of length \(2na\) are held at the same level at a distance \(2a\) apart. Prove that the depth of the lowest point of the string below the level of the supports is \[ na - \frac{1-e^{-\theta}}{\theta}a, \] where \(\theta\) is defined by the equation \[ \sinh\theta = n\theta. \]
A particle moves with constant acceleration on a straight line. Shew that the velocity at the middle of any time-interval is equal (i) to the mean velocity in the interval, (ii) to the average of the velocities at the beginning and end of the interval. A particle moving on a straight line travels distances \(AB, BC, CD\) of lengths 44, 51, 40 feet in three successive intervals of 2, 3, 4 seconds. Shew that these observations are consistent with the hypothesis that the particle has constant retardation, and, on this hypothesis, find the distance from \(D\) to the point \(E\) where the particle comes to rest, and find the time taken in the motion from \(D\) to \(E\).
In starting an engine of mass \(m\) the pull on the rails is at first constant and equal to \(R/u\), and after the velocity attains a value \(u\) the engine works at a constant rate \(R\); throughout the motion there is a frictional resistance which is proportional to the square of the velocity, and the greatest steady velocity at which the engine can travel is \(w (>u)\). Prove that the distance in which the engine, starting from rest, attains a velocity \(V\) between \(u\) and \(w\) is \[ \frac{mw^3}{6R} \log \frac{w^3-u^3}{(w^3-V^3)^2}. \quad (\text{Note: original has }(w^3-u^3)(w^3-V^3)^2 \text{ in denominator, which seems unlikely}) \] Let's check the OCR: `log (w^3-u^3)(w^3-V^3)^2`. Let's assume this is correct and transcribe as seen. \[ \frac{mw^3}{6R} \log \frac{w^6}{(w^3-u^3)(w^3-V^3)^2}. \quad (\text{The numerator is unclear in scan, but seems to be } w^6) \] Re-examining: A simpler form might be intended. The OCR text is `mw^3/6R log (w^6) / (w^3-u^3)(w^3-V^3)^2`. Let's assume the \(w^6\) is correct.
Two smooth perfectly elastic spheres, one of mass \(M\) and the other of smaller mass \(m\), are initially at rest. The sphere of mass \(M\) is projected so that it collides with the other. Shew that the direction of motion of \(M\) cannot be deflected by the collision through an angle greater than \(\sin^{-1}(m/M)\).
A particle is projected in a given vertical plane from a point \(O\) of the plane with a velocity \(\sqrt{(2gh)}\). Shew that the points of the plane which are accessible by projection from \(O\) with the given velocity lie on or beneath the parabola having \(O\) as focus and a horizontal line at height \(h\) above \(O\) as tangent at the vertex. Shew that the time taken to reach a point on this parabola at distance \(r\) from \(O\) is \(\sqrt{(2r/g)}\).