LOM-07-PROBLEMS

PROBLEMS y 1. A particle of mass m, moving in a circular path of radius R with a constant 1 speed v2 is located at point (2R, 0) at time t=0and a man starts moving with a velocity v1 along the +ve y-axis from origin at time t = 0. Calculate the linear momentum of the particle w.r.t. the man as a function of time. (0, 0) 2. A hemispherical bowl of radius R=0.1m is rotating about its own axis (which is vertical) with an angular velocity  . A particle of mass 10-2kg on the frictionless inner surface of the bowl is also rotating with the same  . The particle is at a height h from the bottom of the bowl. Obtain the relation between h and  . What is the minimum value of  needed, in order to have a non-zero value of h? 3. In the figure, masses m1, m2 and M are 20 kg, 5 kg and 50 kg respectively. The co-efficient of friction between M and ground is P1 zero. The coefficient of friction between m1 and M and that be- tween m2 and ground is 0.3. The pulleys and the strings are mass- less. The string is perfectly horizontal between P1 and m1 and also between P2 and m2. The string is perfectly vertical between P1 and P . An external horizontal force F is applied to the mass M. Take g = 10 m/s2. (a) Draw a free body diagram of mass M, clearly showing all the forces. (b) Let the magnitude of the force of friction between m1 and M be f1 and that between m2 and ground be f2 . For a particular F it is found that f1 = 2f2. Find f1 and f2. Write equations of motion of all the masses. Find F, tension in the string and accelerations of the masses. ` 4. Two blocks of mass m1 = 10 kg and m2 = 5 kg connected to each other by a massless inextensible string of length 0.3 m are placed along a diameter of turn table. The coefficient of friction between the table and m1 is 0.5 while there is no friction between m2 and the table. The table is rotating with an angular velocity of 10 rad/sec about a vertical axis passing through its centre O. The masses are placed along the diameter of the table on either side of the centre O such that mass m1 is at a distance of 0.124 m from O. The masses are observed to be at rest with respect to an observer on the turn table. (i) Calculate the frictional force on m1. (ii) What should be the minimum angular speed of the turn table so that the masses will slip from this position. (iii) How should the masses be placed with the string remaining taut so that there is no frictional force acting on the mass m1.