PYQs

1) Suppose the Lagrangian of a mechanical system is given by \(L=\dfrac{1}{2}m(a \dot x^2+2b\dot x\dot y+c\dot y^2 )-\dfrac{1}{2}k(a x^2+2bxy+c y^2 )\),

where \(a\), \(b\), \(c(>0)\), \(k(>0)\) are constant and \(b^2\neq ac\). Write down the Lagrangian Equations of motion and identify the system.

[2018, 20M]

2) Two uniform rods \(AB\) and \(AC\), each of mass \(m\) and length \(2 a\), are smoothly hinged together at \(A\) and move on a horizontal plane. At time \(t\), the mass centre of the rod is at the point \((\xi, \eta)\) referred to fixed perpendicular axes \(Ox\), \(O y\) in the plane, and the rods make angles \(\theta \pm \phi\) with \(O x\). Prove that the kinetic energy of the system is \(m\left[\xi^{2}+\eta^{2}+\left(\dfrac{1}{3}+\sin ^{2} \phi\right) a^{2} \theta^{2}+\left(\dfrac{1}{3}+\cos ^{2} \phi\right) a^{2} \phi^{2}\right]\). Also derive Lagrange’s equation of motion for the system if an external force with components \([X, Y]\) along the axes acts at \(A\).

[2017, 20M]

3) A hoop with radius \(r\) is rolling, without slipping, down an inclined plane of length \(l\) and with angle of inclination \(\phi\). Assign appropriate generalized coordinate to the system. Determine the constraints, if any. Write down the Lagrangian equation for the system. Hence or otherwise determine the velocity of the hoop, at the bottom of the inclined plane.

[2016, 15M]

4) Two equal rods \(AB\) and \(BC\) each of length \(l\) smoothly jointed at \(B\), are suspended from \(A\) and oscillate in a vertical plane through \(A\). Show that the periods of normal oscillation are \(\dfrac{2 \pi}{n}\), where \(n^{2}=\left(3 \pm \dfrac{6}{\sqrt{7}}\right) \dfrac{g}{l}\).

[2013, 15M]

5) Obtain the equations governing the motion of a spherical pendulum.

[2012, 12M]

6) A perfectly rough sphere of mass \(m\) and radius \(b\), rests on the lowest point of a fixed spherical cavity a radius \(a\). To the highest point of the movable sphere is attached a particle of mass \(m^{\prime}\) and the system is disturbed. Show that the oscillations are the same as of a simple pendulum of length \((a-b) \dfrac{4 m^{\prime}+\dfrac{7}{5} m}{m+m^{\prime}\left(2 - \dfrac{a}{b}\right)}\).

[2009, 30M]

7) A uniform rod of mass \(3m\) and length \(2l\) has its middle point fixed and a mass \(m\) is attached to one of its extremity. The rod, when in a horizontal position is set rotating about a vertical axis through its centre with an angular velocity \(\sqrt{\dfrac{28}{l}}\). Show that the heavy end of the rod will fall till the inclination of the rod to the vertical is \(\cos^{-1}(\sqrt{2}-1)\).

[2008, 30M]

8) Given points \(\mathrm{A}(0,0)\) and \(\mathrm{B}\left(x_{0}, y_{0}\right)\) not in the same vertical, it is required to find a curve in the \(x-y\) plane joining \(A\) to \(B\) so that a particle starting from rest will trom \(A\) to \(B\) along this curve without friction in the shortest possible time. If \(y=y(x)\) is the required curve find the function \(f(x, y, z)\) such that equation of motion can be written as \(\dfrac{d x}{d t}=f\left(x, y(x), y^{\prime}(x)\right)\).

[2006, 12M]

9) A particle of mass \(m\) moves under the influence of gravity on the inner surface of the paraboloid of revolution \(x^{2}+y^{2}=a z\) which is assumed frictionless. Obtain the equation of motion show that it will describe a horizontal circle in the plane \(z=h\), provided that it is given an angular velocity whose magnitude is \(\omega=\sqrt{\dfrac{2 g}{a}}\).

[2004, 12M]

10) Find the equation of motion for a particular of mass \(\mathrm{m}\) which is constrained to move on the surface of a cone of semi-vertical angle \(\alpha\) and which is subjected to a gravitational force.

[2001, 30M]