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Physics Smart Booklet 49 7.System of Particles and Rotational Motion Physics Smart Booklet Theory + NCERT MCQs + Topic Wise Practice MCQs + NEET PYQs
Physics Smart Booklet 50
Physics Smart Booklet 51 System of Particles and Rotational Dynamics A group of particles that can be identified and distinguished from other particles or groups is called a system. The motion of the system as a whole can be analysed by applying the laws of mechanics. This is achieved by using the concept of ‘centre of mass’. Thus, identifying the centre of mass is quite important in the study of dynamics of a system. Rigid body: It is a body whose shape and size do not change during its state of rest or motion. A classic example of a physical system is that of a rigid body, in which the relative distance between any two particles remains unaltered during its motion. Study of rigid body motion involves physical parameters like moment of inertia, torque, angular velocity, angular momentum, translational energy and rotational energy. Centre of mass of a system of two particles Centre of mass is the point at which the entire mass of a body is supposed to be concentrated. If we have discrete system of particles as shown in the figure, then centre of mass is defined as 1 1 2 2 3 3 cm 1 2 3 m R m R m R . . . R m m m . . . + + + = + + + cm i i 1 R m R M =  The coordinates of centre of mass 1 1 2 2 cm i i 1 2 3 m x m x . . . 1 X m x m m m M + + = = + +  1 1 2 2 3 3 cm i i 1 2 3 m y m y m y . . . 1 Y m y m m m . . . M + + + = = + + +  1 1 2 2 cm i i 1 2 m z m z . . . 1 Z m z m m . . . M + + = = + +  For a system having continuous distribution of the mass, the coordinates of cm are cm cm dm cm 1 1 1 X xdm, Y y , Z zdm M M M = = =    • Position of centre of mass is independent of coordinate system choosen. • Centre of mass depends on the shape of the body and distribution of mass. • Centre of mass coincides with geometric centre bodies where mass is homogeneous. • Centre of mass remains unchanged in rotatory motion while in translatory motion position changes. • If small position of mass m2 is removed from a larger position of mass m1. Then centre of mass of the remaining part is 1 1 2 2 cm 1 2 m x m x x m m − = − Motion of centre of mass If a system of particles of masses m1, m2, . . move with velocities v1, v2, v3 . . . respectively • Then velocity of centre of mass is given by 1 1 2 2 n n 1 i CM 1 2 n m v m v . . . m v m v V m m . . . m M + + + = = + + +  (M → total mass of the body)
Physics Smart Booklet 52 • Momentum of the centre of mass is Mv m v m v . . . m v cm 1 1 2 2 n n = + + + + P P P . . . P CM 1 2 n = + + + If V 0, P 0 CM CM = = i.e., in the frame of reference of CM, the momentum of a system is zero. • Acceleration of CM is given by 1 1 2 2 n n i i CM 1 2 n m a m a . . . m a m a a m m . . . m M + + = = + +  Or Ma m a m a . . . m a CM 1 1 2 2 n n = + + F F F . . . F ext 1 2 n = + + + This is equation of motion of centre of mass. If F 0 a 0 v ext CM CM =  =  = constant. If F 0, ext = no external force acts on a system, then the velocity of its CM remains constant.  velocity of CM is not affected by internal forces. • If F 0 a 0 V ext CM CM =  =  = constant, then p = constant. This leads to conservation of linear momentum. • If a system of 2 particles of mass m1 and m2 separated by a distance x initially at rest, moving towards each other under the action of attractive force then the 2 particles collide at their centre of mass. Here F12 = − F21 or m1a1 = m2a2  1 2 2 1 a m a m = Since initial momentum = 0, centre of mass is at rest. 1 2 CM 1 1 2 2 2 1 v m V 0, m v m v or v m = = =  Ratio of distances covered by particles before collision is 1 2 2 1 x m x m = Rotational motion: A rigid body undergoes rotational motion when each of its particles travel in a circle centered on a straight line, called the axis of rotation Rotational variables: The rotational variables are the angular equivalents of the linear quantities position, displacement, velocity and acceleration Angular position (): It is the position of a fixed line perpendicular to the axis of rotation, fixed in the body, relative to a fixed axis. It is also called the angular coordinate. r s  = where s → arc length described by a point on the reference line relative to the fixed axis. r → radius of the arc. Its SI unit is the radian (rad) without any dimensions. Angular displacement: It is the difference in the angular coordinates of rotating body at times t1 and t2 = t1 + t  = 2 − 1  Reference line Rotation axis Z x y A rigid body rotating anticlockwise in the xy- phase about z axis the rotation axis Lightly shaded