A homogeneous solid block, made of material weighing 112 lb. per cubic foot, is in the shape of a rectangular parallelepiped with a square base of side 5 feet and a height of \(h\) feet. The block rests on level horizontal ground, the coefficient of friction between the block and the ground being 0.6, and it is subjected to a thrust of total magnitude \(10\sqrt{2}\) tons weight, uniformly distributed along an upper edge and acting at an angle of 45\(^\circ\) with the vertical, downwards and towards the block. Determine the range of permissible values of \(h\) for equilibrium, and state how, for values of \(h\) outside this range, equilibrium would be broken when the thrust is applied.
Any number \(n\) of coplanar forces having components \((X_r, Y_r)\) act at the points whose rectangular Cartesian coordinates are \((x_r, y_r)\) respectively (\(r=1, 2, \dots, n\)). If the forces are all turned through the same angle in the same sense, prove that whatever the value of the angle their resultant always passes through the point whose coordinates are \[ \frac{MP+NQ}{P^2+Q^2}, \quad \frac{MQ-NP}{P^2+Q^2}, \] where \begin{align*} P &= \Sigma X_r, \quad Q = \Sigma Y_r, \\ M &= \Sigma(x_r X_r + y_r Y_r), \quad N = \Sigma(x_r Y_r - y_r X_r). \end{align*}
The upper ends of three equal similar light springs obeying Hooke's law are fastened to smooth rings which slide on a fixed horizontal rod, and the lower ends of the springs are fastened to three points \(A, B, C\) of a light rigid rod; \(B\) is between \(A\) and \(C\), and the lengths \(AB, BC\) are not equal. A weight is attached to the rod at the point \(D\) between \(A\) and \(B\), and the system is allowed to hang freely. If in the equilibrium position the spring through \(C\) is to be compressed, prove that the distance of \(D\) from \(B\) must be greater than \[ \frac{AB \cdot AC}{AC+BC}. \]
State the principle of virtual work for the equilibrium of a system of bodies subject to frictionless constraints. \par Four uniform rods \(AB, BC, CD, DA\), each of weight \(W\) and length \(2a\), are freely jointed together to form a rhombus, which is suspended from \(A\). A point \(E\) in \(AB\) and a point \(F\) in \(CD\), at equal distances \(l\) (not equal to \(a\)) from \(B\) and \(D\) respectively, are joined by a light rod of length \(2c\) which keeps the rhombus in the form in which the angle \(BAC\) is equal to \(\alpha\). Determine the condition that the rod \(EF\) should be in tension and find the magnitude of the tension.
Obtain the equations \[ y = c \cosh \frac{x}{c}, \quad s = c \sinh \frac{x}{c} \] for the form of a uniform heavy string fixed at its ends and hanging freely. \par A uniform string of length \(2l\) hangs symmetrically across two smooth pegs which are fixed at a distance \(2a\) apart at the same level. Shew that the parameter \(c\) of the catenary in which the middle portion hangs is given by \(l=ce^{a/c}\). \par By considering the variation of the tension at the peg for symmetrical displacements, shew that the equilibrium is stable if \(l>ae\).
A body of mass \(M\) moves in a straight line under the action of a force which works at constant power \(P\), and against a resistance equal to \(P/C^2\) times the velocity, \(C\) being constant. If the body starts from rest, shew that after time \(t\) its velocity \(u\) is given by \[ u^2 = C^2 [1 - e^{-2Pt/(MC^2)}] \] and that the distance \(s\) traversed is equal to \[ \frac{MC^3}{2P} \log \frac{C+u}{C-u} - \frac{MC^2}{P} u. \]
A bead of mass \(m\) slides on a smooth straight wire inclined at an angle \(\alpha\) to the vertical, and is attached to one end of a long light inextensible string, which passes over a fixed smooth peg vertically above a point \(A\) of the wire at a height \(h\) and carries a mass \(M\) hanging freely at its other end. The vertical portion of the string passes the wire without interference. Shew that, if the bead is released from rest at \(A\), it will move upwards if \(M>m\) and downwards if \(M
State the principle of the conservation of linear momentum for the motion of any number of particles. \par Four particles of equal mass at four consecutive vertices \(A, B, C, D\) of a regular hexagon are on a smooth horizontal table and are connected by light inextensible strings \(AB, BC, CD\), which are taut. The particle at \(A\) is projected with velocity \(V\) parallel to and in the same sense as \(DC\). Find the velocity of the particle at \(D\) just after the string \(AB\) becomes taut again.
A particle moves in a straight line through a fixed point \(O\) so that, if \(x\) is its distance from \(O\) at time \(t\), its equation of motion is \[ \frac{d^2x}{dt^2} + n^2x = n^2l. \] Shew that its motion is a simple harmonic oscillation about the point \(x=l\). The equation of motion for a simple harmonic oscillation about \(O\) may be assumed to be \[ \frac{d^2x}{dt^2} + n^2x = 0. \] A spring of natural length 1 foot, negligible inertia and modulus 10 lb. weight has one end \(S\) fixed, and a mass of 2 lb. is fixed to the other end. The spring is parallel to a rough horizontal table on which the mass can move in a straight line through the fixed end of the spring, the coefficient of friction between the mass and the table being \(\frac{1}{4}\). If \(A\) and \(B\) are the positions of limiting equilibrium, \(SA\) being greater than \(SB\), find the lengths of \(SA\) and \(SB\). \par The spring is extended to a length 1.75 feet and the mass is released from rest. Shew that the subsequent motion of the mass is a simple harmonic half-oscillation about \(A\), followed by a half-oscillation about \(B\), and so on, and determine (i) the length of the spring when the mass finally comes to rest, and (ii) the total number of half-oscillations performed.
A particle of mass \(m\) is attached to one end of a light string, the other end of which is fastened to a ring of mass \(m\) which slides on a fixed rough horizontal rod. The system is released from rest with the string taut and along the rod. Shew that in order that the ring should not slide on the rod during the ensuing motion the coefficient of friction between the ring and the rod must be not less than \(\frac{3}{4}\).