A family of ellipses, all having eccentricity \(e\), have for their major axes parallel chords of a fixed circle. Show that their envelope is an ellipse of eccentricity \[ \sqrt{(1-e^2)/(2-e^2)}. \]
A flat strip of wood, of mass \(M\), lies on a smooth horizontal table; a particle, of mass \(m\), rests on the strip, the upper surface of which is rough, and the coefficient of (dynamical) friction between the strip and the particle is \(\mu\). If a velocity \(U\) is suddenly given to the particle so that it moves along the strip (which may be assumed not to rotate), find for how long a time the particle slips on the strip; show also that at the instant when slipping ceases the distance through which the strip has moved is \[ m M U^2 / 2\mu g (m+M)^2. \] Find how far the particle slips along the strip, and verify that the loss of kinetic energy is equal to the work done against the frictional force.
A particle is projected with velocity \(V\) under gravity from a point \(O\) of a plane inclined at an angle \(\alpha\) to the horizontal. The direction of \(V\) makes an angle \(\beta\) with the upward direction of a line of greatest slope of the plane and lies in a vertical plane through that line. Show that (i) the time taken for the particle to attain the maximum distance from the plane is one-half the time elapsing before it strikes the plane, (ii) the particle strikes the plane normally if \(2\tan\alpha = \cot\beta\). The particle is projected as before with an assigned velocity \(V\) so as to strike the plane at a point \(P\) above \(O\) and distant \(a\) from it; show that if \(P\) is within range there are two possible values of \(\beta\), given by \[ \sin(\alpha+2\beta) = \sin\alpha + (ag/V^2)\cos^2\alpha. \]
Water, of density \(\rho\) lb./ft.\(^3\), is pumped from a well and delivered at a height \(h\) ft. above the level in the well in a jet of cross-section \(A\) sq. in. with velocity \(v\) ft./sec. Find the horse-power at which the pump is working. If the water strikes a vertical wall normally and falls to the ground without recoil, find in lb. wt. the force exerted on the wall.
Establish the equivalence of the two definitions of simple harmonic motion in a straight line (i) as the projection of uniform circular motion, (ii) as motion under an acceleration proportional to the distance \(x\) from a fixed point \(O\) in the line and directed towards \(O\). In such a motion the speed is 16 cm./sec. when \(x=6\) cm., and equals 12 cm./sec. when \(x=8\) cm. Find the amplitude and period of the motion. In simple harmonic motion, show that the energy, averaged with respect to time, is one-half kinetic and one-half potential. If in the above example the mass of the moving particle is 50 gm., find the total energy, stating in what units this energy is measured.
A particle of mass \(m\) is projected with velocity \(v_0\) along a smooth horizontal table and the motion is opposed by a resistance \(mkv^3\), where \(v\) is the velocity of the particle after it has travelled a distance \(x\) in a time \(t\). Obtain the relations \[ \frac{2t}{x} = \frac{1}{v_0} + \frac{1}{v}; \quad t = \frac{x}{v_0} + \frac{1}{2}kx^2. \]
A flywheel is mounted on an axle, of radius 3 in., so as to be capable of rotating in smooth bearings with its axis horizontal; the moment of inertia of the flywheel about its axis is 112\(\frac{1}{2}\) lb. ft.\(^2\) A light rope is coiled round the axle and carries at its free end a mass of 200 lb. The system is released from rest with the 200 lb. mass hanging with the free portion of the string vertical. Find the acceleration of this mass in the ensuing motion; find also the tension of the rope in lb. wt. After the mass has descended a distance of 10 ft. the rope slips from the axle. Find, in lb. wt., the magnitude of the force that must then be applied tangentially to the rim of the flywheel in order to bring the flywheel to rest in 3 sec., the radius of the flywheel being 18 in. [Take \(g\) to be 32 ft./sec.\(^2\)]
Define the mass-centre of \(n\) coplanar point-masses \(m_i\) (\(i=1,2,\dots,n\)), situated at points \((x_i, y_i)\), and prove that it is a unique point. If the coordinates \(x_r, y_r\) of one of the masses \(m_r\) are changed to \((x_r+\xi_r, y_r+\eta_r)\), show that the coordinates of the mass-centre are changed by \((m_r\xi_r/M, m_r\eta_r/M)\), where \(M\) is the total mass. Generalise this to cover the case in which any number of the point-masses are moved in their plane. A circular disc of uniform thin paper, of radius \(a\), is cut along a radius and one of the quadrants is folded over so as to lie in the plane of the remaining three quadrants. Find the distance of the mass-centre of the folded paper from the centre of the circle. [It may be assumed that the mass-centre of a sector of the circle, of angle \(2\beta\), is at a distance \((2a \sin\beta)/(3\beta)\) from the centre of the circle.]
A light horizontal rod \(AB\) bears a load \(W\) at its middle point and is freely hinged to a vertical wall at \(B\). It is supported by a wedge, of angle \(\beta\), at \(A\); the lower face of the wedge is in contact with the smooth horizontal ground. The rod is perpendicular to the wall and lies in a vertical plane through a line of greatest slope of the inclined face of the wedge, which is rough. A horizontal force \(P\), perpendicular to and towards the wall, is applied to the wedge as shown in the figure. Find, in terms of \(P, W\) and \(\beta\), the normal reaction and the frictional force exerted on the rod at \(A\). Show that, if \(\cot\beta<\mu\), where \(\mu\) is the coefficient of friction, no force \(P\), however great, will push the wedge towards the wall. If, however, \(\cot\beta>\mu\), show that the mechanical advantage of the system (regarded as a device for lifting \(W\)) is \(2\cot(\lambda+\beta)\), where \(\lambda=\tan^{-1}\mu\). \centerline{\includegraphics[width=0.6\textwidth]{1950_12_13_Mechanics_Q8_Fig.png}}
A plane framework \(AEBCD\) consists of seven light smoothly jointed rods such that the rods \(AE, EB\) and \(DC\) are horizontal, and the rods \(AD, DE, EC, CB\) are inclined at an angle \(\alpha\) to the vertical. The framework is hung by vertical chains attached at \(A\) and \(B\), and a load \(P\) is suspended from \(D\). Calculate from a force diagram, or otherwise, the tensions and thrusts in the rods. Write down the corresponding system of tensions and thrusts in the rods when a load \(\lambda P\) hangs from \(C\) and no load is borne at \(D\). By superposing these two systems, find the tension or thrust in \(ED\) when the framework carries loads \(P\) and \(\lambda P\) at \(D\) and \(C\) respectively. State quite briefly why it is justifiable to superpose these systems of forces.