The document discusses concepts related to rotational mechanics including: 1) Centripetal acceleration and the maximum velocity at which a small sphere can stay in contact with a hemispherical surface. 2) Conditions for stable, unstable, and neutral rotational equilibrium. 3) Calculating moments of inertia for various objects including rods, plates, rings, and cylinders. 4) Using parallel axis and perpendicular axis theorems to calculate moments of inertia.

1. Rotational Mechanics What is common to the motion of a ceiling fan, a bicycle wheel, a mixed blade or a compact disc? Prepared By Anand Arora copyrights © youmarks.com

2. About Centripetal Acceleration A small glass sphere is at an angle of θ =60 0 with the vertical on a smooth hemispherical surface of radius r o . What should be the maximum velocity of projection v in vertical plane so that it stays in contact of the spherical surface v copyrights © youmarks.com

3. Concepts A person leans in opposite direction when he/she lifts a heavy load in one hand In case of stable equilibrium , the center of mass goes higher on being slightly displaced. For unstable equilibrium, it goes lower for neutral equilibrium , it stays at same height Cyclist bend at an angle for Rotational Equilibria, Whats that angle? Pseudo Torque and the case when it becomes equal to zero. copyrights © youmarks.com

4. Concepts Cont…. A uniform rod is kept vertically on a horizontal smooth surface at some point O. If it is rotated slightly and released it falls down on the horizontal surface. The lower end will fall_______________ A mass m is moving with a constant velocity parallel to the axis. Its angular momentum w.r.t the origin is _________ copyrights © youmarks.com

5. Concepts Cont…. A closed cylindrical tube containing some water (not filling the entire tube) lies in a horizontal plane. If the tube is rotated about a perpendicular bisector, the moment of inertia of water about its axis ______ When tall buildings are constructed on earth duration of day night slightly increases. Explanation!!!! copyrights © youmarks.com

6. Strings A horizontal homogeneous cylinder of mass M and radius r is pivoted about its axis of symmetry. A string is wrapped several times around the cylinder and tied to a body of mass m resting on a support positioned so that a string has no slack. If the body is released find its acceleration and tension in the string. copyrights © youmarks.com

7. Moment of Inertia Rod about its center ML 2 /12. Rectangular Plate of side a and b is M(a 2 +b 2 )/12 Circular Ring about its center MR 2 Circular Plate about its center MR 2 /2 Hollow cylinder MR 2 Solid cylinder MR 2 /2 Hollow Sphere 2/3 MR 2 Solid Sphere 2/5 MR 2 Use Parallel Axis and Perpendicular Axis Theorams copyrights © youmarks.com

8. Friction Case Show that a cylinder will slip of an inclined plane at an angle of incination θ if the coefficient of static friction between plane and cylinder is less than 1/3 (tan θ ). copyrights © youmarks.com

9. Problem Constraints A rod of length l is contrained to move in a vertical plane containing its length between two smooth perpendicular planes. When it is released from the vertical position by pulling its bottom gently, find its acceleration and angular velocity as a function of θ made by it with horizontal copyrights © youmarks.com

10. Angular Momentum Projecile- Angular Momentum about the point of projection at the highest point of trajecory and at each point as a function of time? Conservation Law and Torque –L relation A wheel is rotating at an angular speed of Ώ about its axis which is kept vertical. An identical wheel initially at rest is dropped into the same axle and the two wheels start rotating with a common speed. Find that common speed? copyrights © youmarks.com

11. Pulley Problem (How??) Two masses M and m are conneted by a light string going over a light pulley of radius r. The pulley is free to rotate about its axis which is kept horizontal. The moment of inertia of the pulley about the axis is I. The system is released from rest. Find the angular momentum of the system when the mass M has descended through a height h. The string does not slip over the pulley copyrights © youmarks.com

12. A Good Problem A uniform cylinder of radius R is spinned about its axis to the angular velocity α and then placed in a corner. The coefficient of friction between the cylinder and both corner walls is equal to k. How many turns will the cylinder accomplish before it stops. copyrights © youmarks.com

13. Note The problems in this lecture or the lectures to come are for discussions. If you face any (ANY) problem, please post it to our discuss problems / discussion forum section. copyrights © youmarks.com