IFoS PYQs 3
2008
1) If \(t\) be the time taken and \(t^{\prime}\) the time it takes from \(\mathrm{P}\) to reach the horizontal plane that passes through the point of projection, show that height of \(\mathrm{P}\) above the horizontal plane is equal to \(\dfrac{1}{2} g t t^{\prime}, g\) being the acceleration due to gravity.
[10M]
2) For a particle executing simple harmonic motion, it is observed that its distances from the middle point of the path are \(x, y, z\) at three consecutive seconds. Show that the time of a complete oscillation is equal to
\[\dfrac{2 \pi}{\cos ^{-1}\left(\dfrac{x+z}{2 y}\right)}\][13M]
3) A bullet is fired with a velocity u into a plank. It is observed that after penetrating the plank, the velocity of the bullet reduces to \(4 / 5 \mathrm{u}\). It then strikes another similar plank having different thickness. If the velocity of the bullet becomes zero immediately after it passes through the second plank, calculate the ratio between the thicknesses of the two planks.
[13M]
2007
1) \(AB\), \(BC\) are two equal, similar rods freely hinged at \(B\) and lie in a straight line on a smooth table. The end \(A\) is struck by a blow perpendicular to \(AB\). Show that the resulting velocity of \(A\) is \(3 \dfrac{1}{2}\) times of \(B\).
[10M]
2) Two like rods \(A B\) and \(BC\), each of length \(2a\) are freely jointed at \(B\); \(A B\) can turn round the end \(A\), and \(C\) can move freely on a vertical straight line through \(A\) and they are then released Initially the rods are held in a horizontal line, \(\mathrm{C}\) being in coincidence with \(\mathrm{A}\) and they are then released Show that when the rods are inclined at an angle \(\theta\) to the horizontal, the angular velocity of either is
\[\sqrt{\left(\dfrac{3 g}{a}, \dfrac{\sin \theta}{1+3 \cos ^{2} \theta}\right)}\][13M]
2006
1) Find the work done in stretching an elastic string from length \(b\) to length \(c\), the unstretched length of the string being \(a\).
[10M]
2) A particle is projected from an apse at distance ‘c’ with velocity \(\sqrt{2 \mu / 3} \mathrm{c}^{3}\), If the force directed to the centre be \(\mu\left(r^{3}-c^{4} r\right),\) determine the equation of the orbit (\(\mu\) being a constant).
[14M]
3) A uniform beam rests tangentially upon a smooth curve in a vertical plane and one end of the beam rests against a smooth vertical wall, If the beam is in equilibrium in any position, find the equation to the curve.
[13M]
2005
1) A circular wire of radius a and density p attracts a particle according to
\[\gamma \dfrac{m_{1} m_{2}}{(\text { distance })^{2}}\]If the particle is placed on the axis of the wire at a distance b from the centre find its velocity when it is at a distance \(x\) if is placed at a small distance from the centre on the axis show that the time of a complete oscillation is
\[a\sqrt{\dfrac{2 \pi}{\gamma \rho}}\][10M]
2) If two particles are projected in the same vertical plane with velocities \(u\) and \(u'\) at angles \(\alpha\) and \(\alpha^{\prime}\) with the horizontal, show that the interval between their transits through the other point comnon to their paths is \(\dfrac{2 u u^{\prime} \sin \left(\alpha-\alpha^{\prime}\right)}{g u \cos \alpha+u^{\prime} \cos \alpha^{\prime}}\) assuming \(\alpha>\alpha'\).
[10M]
3) Two particles of masses \(\mathrm{m}\) and \(\mathrm{M}\) move under the force of their mutual attraction, if the orbit of \(\mathrm{m}\) relative to \(\mathrm{M}\) is a circle of radius \(a\) described with velocity \(v\) show that \(v=\left[\dfrac{G(M+m)}{a}\right]^{u / 2}\).
[10M]