Problems

A small ring of mass \(m\) slides on a smooth wire in the form of the parabola \(y^2 = 4ax\), the \(x\)-axis being the downward vertical. It is connected to the focus by a light spring of natural length \(a\), and rests in equilibrium at the point \((a\lambda_0^2, 2a\lambda_0)\). Show that the modulus of the spring is \(mg\lambda_0^{-2}\). Show further that, if slightly displaced, the spring vibrates with period approximately \[ 2\pi\sqrt{\frac{a(\lambda_0^2+1)}{g}}. \]

Prove that, if \(ab' - a'b \neq 0\), the locus given by \[ x = at^2 + bt + c, \quad y = a't^2+b't+c', \] where \(t\) is a parameter, is a parabola. If the line joining the points with parameters \(t_1, t_2\) is parallel to \(y=mx\), prove that \[ t_1+t_2 = (b' - mb)/(ma-a'). \] Find the equation of the locus of middle points of chords of the parabola parallel to \(y=mx\).

Evaluate \[ \int_{-\pi}^{\pi} \cos m\theta \cos n\theta \,d\theta \] for all non-negative integral values of \(m\) and \(n\). If the function \[ f(\theta) = \sum_{r=0}^n a_r \cos r\theta, \] where \(n\) is a positive integer and the \(a_r\) are real constants, has the property that \(f(\theta) \ge 0\) for all real \(\theta\), prove by considering the integrals \[ \int_{-\pi}^{\pi} (1 \pm \cos\theta)f(\theta)\,d\theta, \] or otherwise, that \(-2a_0 \le a_1 \le 2a_0\).

The equations of motion of a particle of mass \(m\), moving under a force \((X,Y)\) in plane, are \[ m\frac{d^2x}{dt^2} = X, \quad m\frac{d^2y}{dt^2} = Y, \] referred to rectangular axes in the plane, with origin \(O\). Deduce that the rate of change of the moment about \(O\) of the momentum of \(m\) is equal to the moment about \(O\) of \((X,Y)\). Indicate any theorem of mechanics that is assumed in the course of the proof. Two particles of masses \(M,m\) are joined by a light inextensible string; \(M\) lies on a smooth horizontal table, and the string passes through a small hole \(O\) in the table so that \(m\) hangs below the table. Initially \(M\) is at a distance \(r_0\) from \(O\) and is moving horizontally at right angles to \(OM\) with velocity \(V_0\). Obtain an equation of the form \((dr/dt)^2 = f(r, r_0, V_0)\) for the distance \(r\) of \(M\) from \(O\) in the subsequent motion. Deduce from this equation that if \(M\) again moves at right angles to \(OM\), \(r\) must then equal one of the two values \(r_0\) and \(r_1\), where \[ r_1 = \rho\{1 + (1+2r_0/\rho)^{1/2}\} \quad \text{and} \quad \rho = MV_0^2/4mg. \] By considering the sign of \(f(r,r_0,V_0)\) show that \(r\) must lie between \(r_0\) and \(r_1\).

Prove that, if \(G\) is the centre of gravity of a uniform plane lamina of mass \(M\), \(P\) is any point of the lamina, \(I_G\) is the moment of inertia about the line through \(G\) perpendicular to the plane of the lamina, and \(I_P\) is the moment of inertia about the parallel line through \(P\), then \[ I_P = I_G + M(PG)^2. \] A uniform heavy rod \(AB\) of length \(2a\) is freely suspended from a fixed point \(O\) by two light rods \(OA, OB\), each of length \(2a\). The system is released from rest with \(AB\) vertical. Find the greatest velocity of \(A\) in the subsequent motion.

Three fixed points \(A, B, C\) are taken on a conic. Prove that there are infinitely many triangles \(PQR\), self-conjugate with regard to the conic, such that \(P, Q, R\) lie on \(BC, CA, AB\) respectively. Prove further that \(AP, BQ, CR\) meet in a point and find the locus of this point.

By transforming to polar coordinates, or otherwise, find the area of the loop of the curve \[ x^3 + y^3 = 3axy. \]

Find the locus of points \(P\) in the plane of a triangle \(ABC\) such that three forces through \(P\) whose lines of action pass through \(A, B, C\) and whose magnitudes are proportional to \(BC, CA\) and \(AB\) respectively can be in equilibrium.

A uniform circular hoop hangs in contact with a smooth vertical wall over a thin nail, which is perpendicular to the wall. A horizontal force \(P\) parallel to the wall acts on the hoop at the other end of the diameter through the nail. Show that equilibrium is possible for all values of \(P\) provided that the coefficient of friction between the nail and the hoop is not less than \[ \frac{1}{2\sqrt{2}}. \]

A uniform rod \(AB\) of length \(2b\) rests on the rim and inner surface of a smooth hollow hemispherical bowl of internal radius \(a\), which is fixed with the plane of its bounding circle horizontal. The end \(A\) of the rod is inside the bowl and a point \(C\) of the rod is in contact with the rim of the bowl. Find the reactions at \(A\) and \(C\) and show that \(b\) must lie between \(a\sqrt{2}\) and \(2a\).