Insights to rotational motion
- 1. Insights into Circular and Rotational Mechanics with Problem Solving Bought to you by Youmarks Preparation Prepared by, Anand Arora Part-I : Revolving chair problem in a restaurant Copyrights (C) youmarks.com
- 2. The Terms and concepts of Physics that you should know before… Torque Angular Momentum Angular Rotational Analogue of distance, velocity and acceleration Moment of Inertia Theoram of perpendicular and parallel axis Conservation of Angular Momentum Kinetic Energy of Rotational Motion Combined Rotational and Translational Motion Copyrights (C) youmarks.com
- 3. Few things that should always be considered A rotational mechanics problem is no different than a translational mechanics problem with just few understanding between the two and basic dynamics been worked out clearly by the student. The steps that you should follow while approaching a rotational mechanics problem are: Understand the problem carefully. ( Language of the problem) Draw the Free Body Diagram or Motion Picture of the given situation either on the paper or visualize it if problem is simpler. It would be a rigid body under consideration and you have to clearly mark the motion on the respective parts of the system under consideration. Also, point out in the side and XY axis under your reference. Understand and clearly indicate the motion that is described in the problem in the picture or the FBD under consideration. Draw all the forces acting on the individual bodies and associate the appropriate direction to it. Work out for the sources of rotational motion. You can check this by visualising the forces which can give the torque effect to the body. This means check out whether there is any force acting on a body at a distance from a fixed point in the body which stays at rest w.r.t the body or in some other condition check out the forces which can give a circular motion impact to the body. Wrte the equations of motion both linear and rotaional . Check for the relations between the linear motion and rotational motion. Work out Moment of Inertia carefully under the axis of rotation in consideration. These are broadly some of the guidelines which can help you solve much of the problems . However these need not to be taken as fixed guidelines religiously and further thought process should never be stopped withouth checking out logic for the problem Copyrights (C) youmarks.com
- 4. Dinner at the Revolving Restaurant? Most of you guys must have been to a restaurant where the floor is rotating around a center with the dinner tables been rotating along the floor in orbital paths and some times there are restaurants where tables are also rotating and the floor been revolving. ( Quite few of them name their tables as planets with planets rotating and revolving around the sun..) Now there is a very interesting rotational mechanics problem associated with it. Suppose the restaurant arena of revolution is of radius R where 8 tables are placed along the circumference of the arena at equal distance from each other. Assume that the masses of table chair setup are M each. Suppose the chairs are not joined to the floor permanently and only the friction force is holding them on the floor which is revolving at a constant angular velocity w. Calculate the frictional coefficient , u between each chair and the floor to have all the chairs not moving w.r.t the revolving floor.? Consider dimensions of chair to be negligible as compared to R. Copyrights (C) youmarks.com
- 5. Dinner at the Revolving Restaurant? We first drew subset (A) in the picture so as to have an idea of what we are talking about. Now since the friction is holding up the chairs against the floor. Considering the motion of each chair w.r.t the floor makes them stationary and thus we equated all forces along the X- Axis as showing in the picture subset (B). That gives an easy solution of friction coefficient to us. Conceptually understood as motion of each chair would be identical since the masses are equal of each chair. Copyrights (C) youmarks.com
- 6. Dinner at the Revolving Restaurant? Now suppose the chairs are of different masses say M, 2M, 3M and 4M with two sets of each masses of chair. Or imagine people are sitting on these chairs totaling these masses to them. Can you place the chairs in this problem so that they can remain stationary for a particular value of u between each chair and floor? If not , what could be the frictional values between each chair and floor for the entire chairs to remain stationary w.r.t the revolving restaurant.? Copyrights (C) youmarks.com
- 7. Dinner at the Revolving Restaurant? Since u, frictional coefficient is independent of M, one would obviously point out that masses can be placed at any pattern along the circumference. However, we should look into other aspects of the problem now…. Consider any diameter of the revolving restaurant. Since there are masses along each side of diameter, there is a frictional force acting on the revolving platform also. To let the platform remain in equilibrium in the horizontal direction, both the forces across the diameter must equate and cancel out . Since this is directly proportional to the masses of chairs. We should place masses in such a way that sum of masses along each diameter is constant as shown in subset ( C). Copyrights (C) youmarks.com
- 8. Dinner at the Revolving Restaurant? Now suppose it is also given that mass of platform is m and platform initially is at rest and starts rotating with the circular motion provided by a shaft which rotates the platform by rotating the central most point of the platform. What amount of work is to be done by the shaft to reach an angular velocity ,w for the platform and chairs of mass , M sticking to it due to friction. Copyrights (C) youmarks.com
- 9. Dinner at the Revolving Restaurant? This is just a small extension to the scenario above. As, we all know that work done on the system is change in the kinetic energy of the system. Hence subset (D) easily summarizes the results we needed. Also, one more thing there is no work done by the frictional force since relative motion is zero between the chairs and also considering whole chair + platform as system, friction is an internal force. Copyrights (C) youmarks.com
- 10. In the end…… Copyrights (C) youmarks.com
- 11. Thank you. We will continue serving you….. Copyrights (C) youmarks.com