PYQs

1) A particle is free to move on a smooth vertical circular wire of radius \(a\). At time \(t=0\) it is projected along the circle from its lowest point \(A\) with velocity just sufficient to carry it to the highest point \(B\). Find the time \(T\) at which the reaction between the particle and the wire is zero.

[2017, 17M]

2) A particle of mass \(\mathrm{m}\), hanging vertically from a fixed point by a light inextensible cord of length \(l\), is struck by a horizontal blow which imparts to it a velocity 2\(\sqrt{g l}\). Find the velocity and height of the particle from the level of its initial position when the cord becomes slack.

[2014, 15M]

3) A particle of mass 2.5 \(\mathrm{kg}\) hangs at the end of a string, 0.9 \(\mathrm{m}\) long, the other end of which is attached to a fixed point. The particle is projected horizontally with a velocity 8 \(\mathrm{m} / \mathrm{sec}\). Find the velocity of the particle and tension in the string when the string is:
i) horizontal and
ii) vertically upward.

[2013, 20M]

4) A particle slides down the arc of a smooth cycloid whose axis is vertical and vertex lowest. Prove that the time occupied in falling down the first half of the vertical height is equal to the the time of falling down the second half.

[2010, 20M]

5) A particle is projected with velocity \(V\) from the cusp of a smooth inverted cycloid down the arc. Show that the time of reaching the vertex is 2\(\sqrt{\dfrac{a}{g}} \cot ^{\prime}\left(\dfrac{V}{2 \sqrt{a g}}\right)\), where \(\alpha\) is the radius of the generating circle.

[2009, 10M]

6) A smooth parabolic tube is placed with vertex downwards in a vertical plane. A particle slides down the tube from rest under the influence of gravity. Prove that any position, the reaction of the tube is equal to \(2 w\left(\dfrac{h+a}{\rho}\right)\), where \('w'\) is the weight of the particle, \('p'\) the radius of curvature of the tube, \('4a'\) its latus rectum and \('h'\) the initial vertical height of the particle above the vertex of the tube.

[2008, 12M]

7) A particle attached to a fixed peg \(O\) by a string of length \(l\), is lifted up with string horizontal and then let go. Prove that when the string makes an angle \(\theta\) with the horizontal, the resultant acceleration is \(g \sqrt{\left(1+3 \sin ^{2} \theta\right)}\).

[2007, 15M]

8) A particle is free to move on a smooth vertical circular wire of radius \(a\). It is projected horizontally from the lowest point with velocity 2\(\sqrt{g a}\). Show that the reaction between the particle and the wire is zero after a time \(\sqrt{\dfrac{a}{g}} \log (\sqrt{5}+\sqrt{6})\).

[2006, 12M]

9) A particle is projected along the inner side of a smooth vertical circle of radius \(a\) so that its velocity at the lowest point is \(u\). Show that if \(2 a g< u^{2} < 5 a g\), the particle will leave the circle before arriving at the highest point and will describe a parabola whose latus rectum is \(\dfrac{2\left(u^{2}-2 g a\right)^{3}}{27 g^{3} a^{2}}\).

[2005, 15M]

10) Two particles connected by a fine string are constrained to move in a fine cycloidal tube in a vertical plane. The axis of the cycloid is vertical with vertex upwards. Prove that the tension in the string is constant throughout the motion.

[2005, 15M]

11) If a particle slides down a smooth cycloid, starting from a point whose actual distance from the vertex is \(b\), prove that its speed at any time tis \(2 x b / T \sin (2 x \top / T)\), where \(T\) is the time of complete oscillation of the particle.

[2003, 15M]

12) A heavy particle of mass \(m\) slides on a smooth arc of a cycloid in a medium whose resistance is \(\mathrm{mv}^{2} / 2 \mathrm{c}, \mathrm{v}\) being the velocity of the particle and \(c\) being the distance of the starting point from the vertex. If the axis vertical and vertex upwards, find the velocity of the particle at the cusp.

[2002, 15M]