Problems

A particle of constant mass \(m\) moves on a straight line under a force which is a function of position on the line. If \(v\) denotes the velocity of the particle when its potential energy is \(\Omega\), shew that \[ \frac{1}{2}mv^2 + \Omega \] remains constant throughout the motion. \par If the mass \(m\) of the particle is not constant, but is a function of \(v\), shew that \[ \int mv dv - \int m \dot{v} dv + \Omega \] remains constant. \par An electron, whose mass is \[ \frac{m_0}{\sqrt{(1-v^2/c^2)}}, \] moves from rest under a constant force. Shew that the increase of mass of the electron is proportional to the distance travelled.

Two particles, whose masses are \(m_1\) and \(m_2\), move on a straight line. Prove that the kinetic energy of the system is \[ \frac{1}{2}M\dot{\xi}^2 + \frac{1}{2}\mu\dot{x}^2, \] where \(M=m_1+m_2, \mu=m_1m_2/M, \xi\) is the distance of the centre of gravity from a fixed origin on the line, and \(x\) the distance between the particles. \par If there are no forces other than the mutual attraction between the particles, \(F\), shew that \[ \mu\ddot{x} + F = 0. \] Shew further that, if \(F=\gamma m_1 m_2 / x^2\), and the particles start from rest at distance \(c\) apart, they will collide after a time \(\pi\sqrt{(c^3/8\gamma M)}\).

Shew that the tangential and normal components of acceleration of a point moving on a given curve are \(\ddot{s}\) and \(\dot{s}^2/\rho\). \par A smooth wire is in the form of an arc of the cycloid \(s=4a\sin\psi\) between consecutive cusps, and is fixed with its axis vertical and its ends upwards. A bead slides on the wire. If the bead is let go from rest at a point of the wire, determine its position at any subsequent time. \par If the bead is let go from rest at a cusp, shew that the angular velocity of the direction of motion is constant during each half-period, and that the complete hodograph consists of two circles.

Explain and establish the principle of conservation of linear momentum. \par The base of a solid hemisphere of mass \(M\) and radius \(a\) rests on a horizontal plane. A particle of mass \(m\) is placed on the highest point of the hemisphere and is slightly disturbed. Assuming the surfaces to be smooth, shew that, so long as the particle remains in contact with the hemisphere, the path of the particle in space is an arc of an ellipse. \par Shew also that, if \(\theta\) denotes the angle which the radius through the particle makes with the vertical at time \(t\), then \[ \left(\frac{d\theta}{dt}\right)^2 = \frac{2g(1-\cos\theta)}{a(1-k\cos^2\theta)}, \] where \(k=m/(M+m)\).

Form the equation whose roots are \(\omega^{-1}p+\omega q, p+q, \omega p+\omega^{-1}q\), where \(\omega^3=1\) (\(\omega\ne 1\)). Solve the equation \[ x^3+6x^2-12x+32=0, \] by reducing it to the form of equation so obtained.

Find the sum to \(n\) terms of the recurring series \[ 1+2x+3x^2+9x^3+\dots, \] for which the scale of relation is \[ u_n - u_{n-1}x + 10u_{n-2}x^2 - 8u_{n-3}x^3. \]

\(ABCD\) is a quadrilateral inscribed in a conic \(S\), and circumscribed to a conic \(\Sigma\). \(AD, BC\) meet in \(Y\); \(AB, CD\) in \(Z\); \(AC, BD\) in \(X\). Taking \(XYZ\) as triangle of reference, shew that the equations of \(S\) and \(\Sigma\) can be written \begin{align*} S &\equiv x^2+y^2+z^2=0, \\ \Sigma &\equiv ax^2+by^2+cz^2+2fyz=0, \end{align*} where \[ a(b+c) = bc - f^2. \]

Interpret the equations \[ S+\lambda S'=0, \quad S+\lambda L^2=0, \quad S+\lambda LT=0, \] where \(S\) and \(S'\) are conics, \(L\) a straight line, and \(T\) a tangent to the conic \(S\). \par A circle meets a conic in four points \(A, B, C\) and \(D\). Shew that there are two parabolas and one rectangular hyperbola through these four points, and that the tangents at any one point \(A, B, C\) or \(D\) to the circle, hyperbola and parabolas form a harmonic pencil.

Find the maximum and minimum values of the function \[ u=x^3+y^3+z^3, \] where \(x,y\) and \(z\) are connected by the relations \begin{align*} x+y+z &= a, \\ x^2+y^2+z^2 &= a^2. \end{align*}

State and prove the formula for integration by parts, and shew that \[ \int_0^1 x^n(1-x)^m dx = \frac{m!n!}{(m+n+1)!}. \]