2000

1) Prove that a central force motion is a motion in a plane and the areal velocity of a particle is constant.

[12M]

2) A telephonic wire weighing 0.04 lb per foot has a horizontal span of 150 feel and sag of 1.5 feet. Find the length of the wire and also find maximum tension.

[12M]

3) Assuming that the earth attracts points inside it with a force which varies as the distance from its centre, show that, if a straight frictionless airless tunnel be made from one point of the earth’s surface to any point, a train would traverse the tunnel in slightly less than three quarters of an hour. Assume the earth to be a homogeneous sphere of radius \(6400 \mathrm{km}\).

[12M]

4) A small bead is projected with any velocity along the smooth circular wire under the action of a force varying inversely as the fifth power of the distance from a entre of force situated on the circumference. Prove that the pressure on the wire is constant.

[10M]

1999

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

[10M]

2) A particle moves with a central acceleration \(\mu(r^3-c^4r)\), being projected fro an apse at distance \(c\) with a velocity \(\sqrt{\dfrac{2\mu}{3}}c^3\).Determine its path.

[10M]

1998

1) Two particles of masses \(m_1\) and \(m_2\) moving in coplanar parabolas round the sun collide at right angles and coalesce when their commom distance from the sun is \(R\). Show that the subsequent path of the combined particles is an ellipse of major axis \(\dfrac{(m_1+m_2)^2R}{2m_1m_2}\).

[10M]

1997

1) A shell bursts on contact with the ground and pieces from it fly in all directions with all velocities upto 80 units. Show that a man 100 units away is in danger for a time of \(\dfrac{5}{2} \sqrt{2}\) units if \(g\) is assumed to be of 32 unils.

[10M]

2) A particle moves under a force \(m \mu\left(3 a u^{4}-2\left(a^{2}-b^{2}\right) u^{5}\right\}, a>b\) and is projected from an apse at a distance a+b with velocity \(\sqrt{\dfrac{\mu}{a+b}} .\) Find the orbit.

[10M]

3) A particle is projected along the inner side of a smooth circle of radius a, the velocity at the lowest point being u. Show that if \(2 \mathrm{ag}<\mathrm{u}^{2}<5 \mathrm{ag}\), the particle will leave the circle before arriving at the highest point. What is the nature of the path after the particle leaves the circle?

[10M]

1996

1) One end of a light elastic string of natural length a and modulus \(2\mathrm{mg}\) is attached to a fixed point O and the other to a particle of mass \(\mathrm{m}\). The particle is allowed to fall from the position of rest at 0 . Find the greatest extension of the string and show that the particle will reach 0 again after a time \(\left(\pi+2-\tan ^{-1} 2\right) \sqrt{\dfrac{2 a}{g}}\)

[15M]

2) A stone is thrown at an angle \(\alpha\) with the horizon from a point in an inclined plane whose inclination to the horizon is \(\beta\), the trajectory lying in the vertical plane containing the line of greatest slope. Show that if \(\theta\) be the elevation of that point bf the path which is most distant from the inclined plane, then \(2 \tan \theta=\tan \alpha+\tan \beta\)

[15M]

3) A particle moves under gravity on a vertical circle, sliding down the convex side of smooth circular arc. If its initial velocity is that due to a fall to the starting point from a height h above the centre; show that it will fly off the circle when at a height \(\dfrac{2 h}{3}\) above the centre.

[15M]