UNIT \u2013 5 CENTRE OF MASS & ROTATIONAL MOTION<\/span><\/strong>\u00a0\u00a0<\/span><\/strong><\/p>\n In previous units, we have studded object having single particle of finite dimensions, but in this unit, we will study the objects having system of particles, so we have to define a rigid body & their centre of mass<\/span><\/p>\n 1.Rigid Body:-<\/span><\/strong><\/p>\n A body which has definite shape & definite size is called rigid body.<\/span><\/p>\n The distance between the different pairs of particles should not change by applying any force, however no anybody is perfectly rigid body.<\/span><\/p>\n 2. The concept of system, internal force and external force:-<\/span><\/strong><\/p>\n A collection of any number of particles which interact with each other is called a system, any object of finite size may be considered as a system.<\/span><\/p>\n The force exerted by the particles of a system on, one another is called internal forces.<\/span><\/p>\n The force exerted by any external agencies on a system is called external forces; the motion of the body is due to external force acting on the body.<\/span><\/p>\n 3. Types of motion:-\u00a0<\/span><\/strong><\/p>\n There may be of three types of motion of a body<\/span><\/p>\n If all the particles of a body are moving with the same velocity, than the motion of body is called of body is called translator motion or straight line motion.<\/span><\/p>\n If all the particles of a rigid body are rotating about a fixed axis in circular path perpendicular to the axis, then the motion of body is called pure rotational motion.<\/span><\/p>\n In pure rotational motion, the velocity of the different particle is different \u2013 different<\/span><\/p>\n However their angular velocity is same irrespective of their distance from the axis of rotation<\/span><\/p>\n 4. Concept of centre of mass:-<\/span><\/strong>imp<\/sup><\/span><\/strong><\/p>\n Centre of mass of a rigid body may be defined as a point at which whole mass of the body may be supposed to exist.\u00a0<\/span><\/p>\n All the external force exerted on the system, are applied only on the centre of mass of the body.<\/span><\/p>\n Centre of mass is only a mathematical concept.\u00a0<\/span><\/p>\n Centre of mass may or may not be inside a body, it may also be at outside for system of particles.<\/span><\/p>\n 5. Position of centre of mass of bodies of regular shape:-<\/span><\/strong><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Uniform hollow sphere \u2013<\/span>\u00a0\u00a0\u00a0\u00a0<\/span>centre of sphere<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Uniform solid sphere \u2013 centre of sphere\u00a0<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Uniform circular ring \u2013centre of ring<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Uniform circular disc \u2013centre of disc<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Uniform rod \u2013centre of rod\u00a0<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Cube, square, rectangle – point of intersection of diagonal\u00a0<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Triangle \u2013point of intersection of medians\u00a0<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Solid, hollow cylinder \u2013middle point of the axis of cylinder\u00a0<\/span><\/p>\n \uf0d8<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>Cone \u2013 on the axis of cone at a point distant 3h\/4 from the vertex o where h= OA is height of cone<\/span><\/p>\n 6. Centre of mass of a two particle system:-<\/span><\/strong>\u00a0imp<\/sup><\/span><\/strong><\/p>\n Suppose a system of two particles having mass\u00a0<\/span> Now from Newton\u2019s 2<\/span>nd<\/sup><\/span>\u00a0law of motion\u00a0<\/span><\/p>\n &\u00a0<\/span> Adding eq<\/span>n\u00a0<\/sup><\/span>\u00a0<\/span>1 and 2 we get<\/span>\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n So<\/span>\u00a0\u00a0<\/span> =\u00a0<\/span> Or<\/span>\u00a0\u00a0\u00a0<\/span> Or\u00a0<\/span> Or<\/span>\u00a0\u00a0\u00a0<\/span> Dividing & multiplying\u00a0<\/span> So\u00a0<\/span> As\u00a0<\/span> Comparing with<\/span>\u00a0\u00a0<\/span> We get position vector of centre of mass as<\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n Or<\/span>\u00a0\u00a0<\/span> Thus product of centre of mass with its position vector is equal to sum of product of the particles & their respective position vectors.<\/span><\/p>\n Centre of mass of two particle system lies at the line joining of the points.\u00a0<\/span><\/p>\n If centre of mass lies at origin\u00a0<\/span><\/p>\n Then\u00a0<\/span> If\u00a0<\/span> i.e. centre of mass always lies near to heavy body.<\/span><\/p>\n 7. Centre of mass of n-particle system:-<\/span><\/strong><\/p>\n Suppose a system of n particles having masses\u00a0<\/span> Now again suppose that\u00a0<\/span> Here\u00a0<\/span>\n
\n
\n
\n
\n
\u00a0&\u00a0<\/span>
\u00a0placed at A & B, having position vectors\u00a0<\/span>
\u00a0&<\/span>
. Now again suppose that\u00a0<\/span>
\u00a0&\u00a0<\/span>
\u00a0are the external & internal force on A &\u00a0<\/span>
\u00a0&\u00a0<\/span>
\u00a0are internal & internal force on B.<\/span><\/p>\n
\u00a0=\u00a0<\/span>
+<\/span>
\u00a0\u00a0<\/span>\u2026.1<\/span>\u00a0\u00a0\u00a0<\/span>(<\/span>
\u00a0is force on<\/span>\u00a0\u00a0<\/span>
\u00a0due to\u00a0<\/span>
)<\/span><\/p>\n
\u00a0=\u00a0<\/span>
+<\/span>
…2<\/span>\u00a0\u00a0\u00a0<\/span>(<\/span>
\u00a0is force on\u00a0<\/span>
\u00a0due to\u00a0<\/span>
)<\/span><\/p>\n
+<\/span>
+<\/span>
<\/p>\n
As from Newton\u2019s 3<\/span>rd<\/sup><\/span>\u00a0law of motion\u00a0<\/span><\/p>\n
=\u00a0<\/span>
\u00a0<\/span>
\u00a0<\/span>
<\/p>\n
\u00a0=\u00a0<\/span>
+<\/span>
<\/p>\n
\u00a0=\u00a0<\/span>
\u00a0\u00a0<\/span>(total external force\u00a0<\/span>
=\u00a0<\/span>
+<\/span>
)<\/span><\/p>\n
<\/p>\n
<\/p>\n
<\/p>\n
\u00a0L.H.S. we get<\/span><\/p>\n
\u00a0<\/span><\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n
=\u00a0<\/span>
<\/p>\n
\u00a0<\/span><\/p>\n
\u00a0<\/span><\/p>\n
<\/p>\n
<\/p>\n
<\/p>\n
\u00a0>\u00a0<\/span>
\u00a0\u00a0<\/span>then\u00a0<\/span>
<<\/span>
\u00a0<\/span><\/p>\n
,\u00a0<\/span>
,<\/span>
\u2026..<\/span>
. Whose position vector are\u00a0<\/span>
\u00a0having velocities\u00a0<\/span>
.\u00a0<\/span><\/p>\n
\u00a0are the external forces acting on the system & each particle experience internal forces due to rest (n-1) particles as\u00a0<\/span><\/p>\n
<\/p>\n
\u00a0is force on i<\/span>th<\/sup><\/span>\u00a0particle due to j<\/span>