Three equal uniform rods of length \(l\) and weight \(w\) are smoothly jointed together to form a triangle \(ABC\). This triangle is hung up by the point \(A\); and by two strings each of length \(l/\sqrt{2}\) a weight \(W\) is attached to \(B, C\). The system hangs under gravity. Prove that the thrust along the rod \(BC\) is \[ \frac{1}{\sqrt{3}}\{w+W(1+\sqrt{3})\}. \]
A uniform beam \(AB\) lies horizontally on two rough parallel rails at points \(A\) and \(C\). Prove that the least horizontal force applied at \(B\) in a direction perpendicular to \(AB\), which is able to move the beam, is the smaller of the two forces \(\mu W \frac{b-a}{2a-b}\) and \(\frac{1}{2}\mu W\), where \(AB\) is \(2a\), \(AC\) is \(b\), \(W\) is the weight of the beam, and \(\mu\) is the coefficient of friction at each point of contact.
Inelastic particles are projected horizontally from different points of a vertical tower with velocities due to the height of the tower vertically above the points of projection, and impinge on a rough horizontal plane. Assuming that the coefficients of impulsive and of dynamical friction are each unity, prove that all the particles come to rest on the circumference of a circle, and find its radius.
An elastic string has its ends attached to two points in a horizontal plane, the distance between the points being equal to the natural length of the string. A particle of weight \(w\) is attached to the middle point of the string and is let go from rest from the position in which the string is unstretched. Prove that in the subsequent motion the greatest length of the string will be double its natural length if the modulus of elasticity is \(w\sqrt{3}\).
\(P(a,b,c)\) is a point of the surface \(F(x,y,z)=0\). \(F\) and as many of its partial derivatives as may be required are continuous at \(P\). Also one at least of the first order derivatives of \(F\) at \(P\) is different from zero. If \(Q(\xi, \eta, \zeta)\) is any point in space near to \(P\), show that as \(Q\) tends to \(P\) the order of smallness of the least distance \(\delta\) from \(Q\) to the surface is determined by that of \(F(\xi, \eta, \zeta)\). Hence determine the conditions for contact of order \(n\) at \((a,b,c)\) between \(F(x,y,z)=0\) and the curve \[ x=\theta(t), \quad y=\phi(t), \quad z=\psi(t), \] it being given that the latter passes through \((a,b,c)\) for \(t=t_0\). What is, in general, the highest order of contact that can be obtained between a straight line and a surface?
For the surface \[ x=f(u,v), \quad y=\phi(u,v), \quad z=\psi(u,v), \] define the ``elements'' of length and area, and obtain the connexion between them. A nearly spherical ellipsoid has the equations \[ x=a\sin\theta\cos\phi, \quad y=(a+\epsilon)\sin\theta\sin\phi, \quad z=(a+\omega)\cos\theta, \] where \(\epsilon\) and \(\omega\) are small compared with \(a\). Find the first order terms in the element of area and hence show that the area of the ellipsoid is approximately \[ 4\pi a^2 + \frac{4}{3}\pi a(\epsilon+\omega). \]
Examine the nature (as regards convergence etc.) of the following series, distinguishing the various cases that may arise:
Show how a definite integral may be defined as the common bound of two aggregates of approximative sums. Deduce from your definition that, if \(f(x)\) is integrable and \(H
Defining the Legendre Polynomial of degree \(n\) (positive integral) by the equation \[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^n, \] show that
Define the Weierstrassian Elliptic Function \(\wp(u)\) as the sum of a double series and verify that it is doubly periodic. Prove that, if \(u+v+w=0\), then \[ \begin{vmatrix} 1 & \wp(u) & \wp'(u) \\ 1 & \wp(v) & \wp'(v) \\ 1 & \wp(w) & \wp'(w) \end{vmatrix} = 0. \]