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IIT-JEE Download Sample Question Papers

IIT Physics Sample Paper 1

Rotational Kinematics & Dynamics

1. Auniform thin rod AB of mass 3m and length 2a is free to rotate in a horizontal planeabout a smooth fixed vertical axis which passes through the midpoint O of the rod.Two small smooth rings each of mass m are free to slide on the rod. At time t=0 therings are on the opposite side of O and are at a distance a/2 from O. The rod is thengiven an initial angular velocity 2 g/a, the rings being initially atrest relative to therod.

1. Show that when the rings are about to slip off the rod its angular velocity is g/a.

2. Find the speed of either ringat this instant.3. If the point O is at a height a above the horizontal plane show that the distance

between the points where the ring strikes the plane is 2a (6 + 2 3).2. Two particles m 1 and m 2 are connected by a light elastic string of natural length l o

and are placed on a smooth horizontal table with separation between m 1 and m 2 as lo . m 2 is projected with a velocity v o along the table at right angles to the string.Show that the max length l of the particle is given by: l 2 (l-l o)=m 1 m 2 v o 2 (l+l o)/k(m 1 +m 2 ) where k is force const of string

3. Two equal particles are attached to the end of alight rodand a 3 rd equal particle is connected by an inextensible string to a point on the rod atdistance a and b from its ends. The latter particle is projected at right angles to the rodwith a velocity v o . Show that when the string becomes taut the velocity of the

particle is changed to v 1 =v o (a 2 +b 2 )/2(a 2 +ab+b 2 ) and that the rod begins toturn with angular velocity (a-b/a 2 +b 2 )v 1 .

4. Arod of length 2a and mass M is in motion in a horizontal plane with speed u at rightangle to itslength when it collides with a small elastic sphere of equal mass whose

center lies in the same plane. If the sphere is free to move, Prove that angular velocityacquired by the rod cannot exceed (1+e)u 6/4a.

5. Acube rests on a rough plane of inclination q (< p /4) with 2 of its upper and two ofitslower edges horizontal. A rope is attached to the midpoint of its upper most edgeand is pulled parallel to the greatest slope of the plane. Show that it will be impossibletodrag the cube without upsetting it if the coefficient of friction exceeds (1-tan q )/2

6. Auniform rod of length 2a,mass m is set in motion by a suddenblow J acting at A and inclined at q to rod as shown. Find velocity of A of rodimmediately after the impact and Prove that energy communicated is J 2 (1+3sin 2 q

)/2m.

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7. Asmall block of mass m isplaced inside a hollow cone rotating about avertical axis with angular velocity w as in the figure. The semivertical angle of thecone is q and the coefficient of friction between the cone and the block is m . If the

block is to remain at a constant height h above the apex of the cone, what are themaximum and minimum values of w ?

8. The shown arrangement is ina horizontal plane. The particle with mass m is restricted to move in an elliptical orbitwith shown dimension. Theparticle has velocity v at the shown position towardsright. Length of rod is 2l. The coefficient of restitution between the particle and therod is 1/3. The moment of inertia of rod is I, about O.

1. Find the locus of mid-pointof spring of force constant K 3 .2. Find the frequency of the rod.3. What would be the maximum angular displacement of the oscillating system.

(Assume only one collision)

9. The system shown is initially at rest.Find the acceleration of thebobbins A and mass m. Assume that A can roll without

slipping along the incline. Given that the MI about instantaneous axis of rotation of

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bobbins A,B,C is I A , I B ,I C . Assume their masses to be m A , m B , m c andinternal and external radii to be r and R.

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