1995

1) Two bodies, of masses \(\mathrm{M}\) and \(\mathrm{M'}\) are attached to the lower end of an elastic string whose upper end is fixed and hang at rest, \(\mathrm{M}^{\prime}\) falls off. Show that the distance of \(\mathrm{M}\) from the upper end of the string at time \(t\) is \(a+b+c \cos \left(\sqrt{\dfrac{g}{b}} t\right)\)

[20M]

2) A particle of mass \(\mathrm{m}\) moves under a central attractive force \(m \mu\left(\dfrac{5}{r^{3}}+\dfrac{8 c^{2}}{r^{3}}\right)\) and is projected from an apse at a distance c with velocity \(\dfrac{3 \sqrt{\mu}}{c}\), Prove that the orbit is \(r=c \cos \left(\dfrac{2 \theta}{3}\right)\) and that it will arrive at the origin after a time \(\dfrac{\pi}{8} \dfrac{c^{2}}{\sqrt{\mu}}\)

[20M]

3) If \(t\) be the time in which a projectile reaches a point \(\mathrm{P}\) in its path and \(t\) “ the time from \(\mathrm{P}\) till it reaches the horizontal plane through the point of projection, show that the height of \(\mathrm{P}\) above the horizontal plane is \(\dfrac{1}{2} g t t^{\prime}\) and the maximum height is \(\dfrac{1}{8} g\left(t+t^{\prime}\right)^{2}\).

[20M]

1994

1) Show that the length of an endless chain which will hang over a circular pulley of radius a so as to be in contact with two-thirds of the circumference of the pulley is \(a\left\{\dfrac{3}{\log (2+\sqrt{3})}+\dfrac{4 \pi}{3}\right\}\)

[10M]

2) A smooth rod passes through a smooth ring at the focus of an ellipse whose major axis is horizontal and rests with its lower end on the quadrant of the curve which is farthest removed from the focus. Find its position of equilibrium and show that its length must at least be \(\left(\dfrac{3 a}{4}+\dfrac{a}{4} \sqrt{1+8 e^{2}}\right)\) where \(2 a\) is the major axis and \(e ,\) the eccentricity

[10M]

3) If in a simple harmonic motion, the velocities at distances a, b, c from a fixed point on the straight line which is not the centre of force, be \(u\), \(v\), \(w\) respectively, show that the periodic time \(T\) is given by \(\dfrac{4 \pi^{2}}{T^{2}}(b-c)(c-a)(a-b)=\vert \begin{vmatrix}u^{2} & v^{2} & w^{2} \\ a & b & c \\ 1 & 1 & 1\end{vmatrix}\vert\).

[10M]

4) A gun is firing from the sea-level out to sea. It is mounted in a battery heters high up and fired at the same elevation \(\alpha\). Show that the range is increased by \(\dfrac{1}{2}\left[\left(1+\dfrac{2 g h}{u^{2} \sin ^{2} \alpha}\right)^{1 / 2}-1\right]\) of itself, u being the velocity of projectile.

[10M]

1993

1) A point executes simple harmonic motion such that in two of its positions, the velocities are u and \(v\) and the corresponding accelerations are \(\alpha\) and \(\beta\). Show that the distance between the positions is \(\dfrac{v^{2}-u^{2}}{a-\beta}\).

[10M]

2) A particle moves under a force

and is projected form an apse at a distance \(a+b\) with velocity \(\dfrac{\sqrt{\mu}}{a+b},\) show that its orbit is \(r=\) \(a+b \cos \theta\).

[10M]

1992

1) A particle is moving with central acceleration \(\mu\left(\mathrm{r}^{5}-\mathrm{c}^{4} \mathrm{r}\right)\) being projected from an apse at a distance \(c\) with a velocity \(\sqrt{\left(\dfrac{2 \mu}{3}\right)} \mathrm{c}^{3}\). Show that its path is the curve

[10M]

2) A particle is projected with a velocity whose horizontal and vertical components are respectively \(u\) and \(v\) from a given point in a medium whose resistance per unit mass is \(k\) times the speed. Obtain the equation of the path and prove that if \(\mathrm{k}\) is small, the horizontal range is approximately

[10M]

3) A particle slides down the arc of a smooth vertical circle of radius a being slightly displaced from rest at the highest point of the circle. Find the point where it will strike the horizontal plane through the lowest point of the circle.

[10M]