1- A rod is lying on the x-axis with one end at x = 0 and other at x = L. Some impulsive force is applied on the rod and as result it's center begins to move with constant velocity u towards positive y-direction and the rod also acquired a constant angular velocity of magnitude w directed in negative z direction. What is the speed of a point (i) at x = L / 3. (ii) at x = 2 L / 3. After the rod rotates by 90 degree angle, what is locational coordinates of the center and what is the speed of a point which is at distance L / 6 from the center ?
2-A On a rough inclined plane, there are two points marked as P and Q. From P first a uniformly dense ring is released from P. This rolls without slip and reaches at the point Q after time T , with speed of center as v and angular momentum L about axis through the center and normal to its plane. Now the experiment is repeated with a uniformly dense cylinder of same mass as the ring. The value of the corresponding quantities at Q are now T2 , v2 and L2 respectively.Find the values of the ratios T2 / T , v2/ v. and L2 / L.
3- There are two spherical uniformly dense planets of masses m and 4m, radii r and 2 r respectively. Their centers are separated by the distance 6 r.The combined gravitational potential due to the two planets is found to be maximum at certain point P between the planets.What is the gravitational potential at the point P? If some one wishes to throw a particle to infinity, from a point situated near to these two planets, then will it be correct to say that point P is the best choice from where a particle should be thrown to infinity ?
4-A satellite of mass m is rotating around the earth in a circular orbit of radius. 2 Re.The gravitational potential energy of the satellite is - G Me m /2Re and kinetic energy is G Me m /4Re if the potential energy is assumed zero at the infinity. If the potential energy on the earth surface is taken as zero, then what will be kinetic and gravitational potential energy of the same satellite ?
5-A uniformly dense rod is falling freely under gravity with it's length horizontal. Suddenly the rod is caught at one end which remains now hinged.During this event some impulsive force was applied on the rod so the momentum of the rod must have changed due to this event but as the torque of this impulsive force about that particular end must be zero ( due to the obvious regions), so the angular momentum about this end should have remained constant during this event. Using this fact obtain the angular speed of the rod just after the event, in the terms of length and velocity of the rod before the event. Also obtain the fraction of the kinetic energy lost during this event.
6- It is known that gravitational field of earth at distance r from center of the earth is g r /Re with symbols having usual meaning. How much work against gravity will be needed to move a unit mass from center of the earth to it's surface ?
