A bead of mass \(A\) can slide freely on a horizontal wire and is attached to a mass \(B\) by a light inextensible string of length \(l\). When \(A\) is at rest, the mass \(B\) is let fall from rest at a point on the wire distant \(l\) from \(A\). Determine the velocities of \(A\) and \(B\) when the line joining the masses is vertical.
The point of suspension of a simple pendulum initially at rest is made to move in a horizontal straight line with constant acceleration \(g/2\). If the bob (weight \(W\)) is initially at rest and vertically below the point of suspension, prove that throughout the motion the inclination of the string to the vertical is always less than \(54^\circ\), and find the tension in the string when the inclination is \(30^\circ\).
Find \[ \text{(i) } \int \frac{dx}{x^2\sqrt{x^2+1}}, \quad \text{(ii) } \int_0^\infty \frac{xdx}{(1+x^2)^2}; \] and shew that, if \(a\) and \(b\) are positive, \[ \int_0^\pi \frac{\sin^2x\,dx}{a^2 - 2ab \cos x + b^2} = \frac{\pi}{2a^2} \quad \text{or} \quad \frac{\pi}{2b^2}, \] according as \(a\) is greater or less than \(b\).
Tangents \(q_1, q_2, q_3\), are drawn at three points \(P_1, P_2, P_3\) on the parabola \(y^2 = 4ax\), and \(Q_1, Q_2, Q_3\) are the vertices of the triangle formed by \(q_1, q_2, q_3\) (\(Q_1\) being opposite to \(q_1\), etc.). Through \(Q_1\) are drawn lines parallel to \(q_1\) and to \(P_2 P_3\); and similarly for the other vertices. Prove that the six lines thus obtained all touch the parabola \[ (y - 2as_1)^2 + 8a(x-as_2) = 0, \] where \[ s_1 = t_1+t_2+t_3, \quad s_2 = t_2 t_3 + t_3 t_1 + t_1 t_2, \] \(t_1, t_2, t_3\) being the parameters of \(P_1, P_2, P_3\) in the parametric representation \(x=at^2, y=2at\).
Explain how to reduce the solution of a dynamical problem to that of a statical problem. A uniform rod of length \(l\) attached at one end to a fixed point by a smooth universal joint rotates freely under gravity as a conical pendulum. If \(\omega\) is the angular velocity of the vertical plane through the rod, and \(\alpha\) the constant inclination of the rod to the vertical, prove that \[ \omega^2 = \frac{3g}{2l}\sec\alpha. \]
A light rod 4 ft. long is free to rotate about one end which is fixed and carries a massive particle at the other end. The rod is kept in equilibrium in a horizontal position by a light inextensible string which joins the middle point of the rod to a fixed point 3 ft. vertically above the particle. Prove that small oscillations of the system are of period 1.92 sec. approximately.
Draw the curves
Interpret the equation \(S - \lambda uu' = 0\), where \(\lambda\) is a constant and \begin{align*} S &= ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \\ u &= lx+my+nz=0, \quad u' = l'x+m'y+n'z=0, \end{align*} are the equations, in a system of homogeneous coordinates in a plane, of a conic and two straight lines. In a plane are given a triangle \(ABC\), a conic \(S\) not passing through a vertex of \(ABC\), and a point \(O\) not on a side of \(ABC\). A variable line through \(O\) meets \(S\) in \(P\) and \(Q\), and the conic through \(A, B, C, P, Q\) cuts \(S\) again in \(X\) and \(Y\). Prove that \(XY\) touches a fixed conic \(\Gamma\). Shew also that, for given \(ABC\) and \(O\), two different conics \(S\) and \(S'\) will give the same conic \(\Gamma\) provided that a conic can be drawn through \(A, B, C\) and the four points of intersection of \(S\) and \(S'\).
A particle of mass \(m\) is describing an orbit in a plane under a force \(\mu m r\) towards a fixed point at a distance \(r\). Taking this point as origin of coordinates, shew that, if when the particle is at a point \((a,b)\) it has a velocity with components \(u,v\) parallel to the axes, the orbit will be given by \[ \mu(bx-ay)^2 + (vx-uy)^2 = (av-bu)^2. \]
A particle of mass \(m\) resting on the highest point of a fixed sphere of radius \(a\) and coefficient of friction \(\frac{1}{4}\) is slightly disturbed and slides down the sphere in a vertical plane. Prove that when the radius to the particle makes an angle \(\theta\) with the vertical, the angular velocity, \(\dot{\theta}\), of the radius is given by the equation \[ \frac{d}{d\theta}(\dot{\theta}^2 e^{-\theta}) = \frac{g}{a}(2\sin\theta - \cos\theta)e^{-\theta}, \] and shew that the normal reaction on the sphere at this instant is \[ \frac{mg}{2}[3(\cos\theta+\sin\theta)-e^\theta]. \]