Systems Of Particles And Rotational Motion

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Patterns: Location Center of mass
Description: Point where we can assume mass of bodies are concentrated. For system of point masses it is calculated by . In case of uniformly distributed mass, we first check symmetry, if object is symetrical then CM will be on symmetrical line and to calculated other coordinates, we will assume object is made of number of similar types of small mass. Now we take small mass at distance r and will use the formula of . This can be integrated for entire mass of the object.
If an object is point of symmetry, centre of mass will be at symmetrical point but When some portion of this object is removed or some additional masses added on it then we use superposition principle by assuming cavity as an independent negative mass density system.
Question may ask- Location Center of mass of system of particles, rigid body having uniform and non uniform mass distribution, Rigid body with cavity or extra mass added into it.
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Patterns: Kinematics of centre of mass of system of particles
Description: Center mass of a system of particles follows laws of motion so centre of mass will move or not it depends on external force. But individual particles can move by mutual intersection. As displacement, velocity and acceleration are directions sensitive so, during the calculation of these physical quantities of C.M, direction of motion of individual particle will be taken into consideration. For displacement use . similar formula we can use for velocity and acceleration while replacing r with and for respective quantity.
Question may ask- Displacement of centre of mass, Velocity of CM, Acceleration of center of mass.
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Patterns: Moment of inertia
Description:  is inertia of the rotating body. For point mass, it is calculated as . It depends on axis of rotation and distribution of mass around the axis of rotation. Hence, when you compress a cylinder to disc or rectangular sheet to a line, both have same M.I.
During calculation of M.I of a rigid body, we will assume, body is made of similar types of large number of small masses, take any small mass at a distance r and M.I. of small mass while integrating this we will get value of I for entire body. In many cases, bodies move about different axis in such case we use Parallel and perpendicular axis theorems. In few cases, two standard objects can be jointed by welding or hinge joint. These types of joint also affect M.I. if the orientation of far object is changing with respect to given axis of rotation (like shaped object with one end is hinge with given axis) then total M.I will be due to combined effect and it will be equal to the sum of M.I. of individual bodies about given axis of rotation.
Question may ask: When point object moves, uniformly distributed mass moves, when two bodies are linked and bodies may or may not rotate about link.
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Patterns: Direction of friction in rolling motion
Description: When you throw the disc with rotation on a rough horizontal surface, if then torque will help to increase angular speed where it will reduce C.M. velocity hence direction will be opposite of direction velocity and if then torque will help to decrease angular speed where it will increase C.M. velocity hence direction of friction will be same of velocity direction. but when there will no friction act on the disc. But when you through the disc on the inclined plane in pure rolling motion then direction of friction will always be 'up the inclined plane' respective of motion of disc. And condition for pure rolling on inclined place where k- radius of gyration. Direction of friction also changes with height (from centre) of force applied on the rotating object. If direction of friction will be opposite of force applied and if , direction of friction will be same as direction of force applied but when there will no friction act on the object. K and R is radius of gyration and radius of the object.
Question may ask: Rotating disc reaches pure rolling from impure rolling, Body is in pure rolling, External Force applied at height from centre of rotating body, When object moves on the inclined plane
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Patterns: Velocity and acceleration of any arbitrary point on moving body
Description: In pure rotational motion, the velocity of points on the body depends on their distance from centre. in pure rolling motion, velocity of particles depends on both velocity of C.M. and its distance for centre and it is equal to or . Also when the object is rolling on moveble platform then where q is a point on the rolling object and is velocity of moving platform. Also acceleration of a point on rolling object depends on linear acceleration, tangential acceleration and normal acceleration. and it is equal to If platform is also acceleration then its acceleration will also add in into the above formula.
Question may ask: When body in pure translation, when body in pure rotation, When body is in a pure rolling motion, when body is in impure rolling, when the rotating body is on moving platform.
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Patterns: Concepts of angular acceleration and linear acceleration used together
Description: In many cases, we have seen that objects fall under gravity with the help of string which is just wrapped on it or string is passed over the pulley. in such types of question, we describe translational and rotational motion separately. During translational motion analysis, we keep all forces acting on object at the centre of mass and write equation for acceleration while during rotational motion analysis keep all force at a given location and write the equation for angular acceleration. Because , we will solve these equations for value of and .
Question may ask: when string wrapped on cylinder, When bodies are attached with string and string passes over a rough pulley.
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