06%20-%20Dynamics%20%20of%20%20a%20%20System%20%20of%20%20Particles

06%20-%20Dynamics%20%20of%20%20a%20%20System%20%20of%20%20Particles

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6 - DYNAMICS OF A SYSTEM OF PARTICLES Page 1 6.1 Centre of mass The centre of mass of a system of n particles having masses m 1 , m 2 , … , m n and position vectors 1 r , 2 r , … , n r respectively is defined as cm r = n m ... 2 m 1 m n r n m ... 2 r 2 m r m 1 1 + + + + + + = M n r n m ... 2 r 2 m r m 1 1 + + + , where M = m 1 + m 2 + … + m n = the total mass of the system. M cm r = + + + n r n m ... 2 r 2 m r m 1 1 Assuming that the mass remains constant, the time derivative of the above equation gives dt r d M cm = dt r d m 1 1 + dt r d m 2 2 + … + dt r d m n n M cm v = m 1 1 v + m 2 2 v + … + m n n v = 1 p + 2 p + … + n p = P Here P , which is the vector sum of the linear momenta of the particles of the system, is called the total linear momentum of the system and is the product of the total mass and velocity of the centre of mass of the system. Differentiating the above equation w.r.t. time, we get dt v d M cm = dt p d 1 + dt p d 2 + … + dt p d n = dt P d = 1 F + 2 F + … + n F = F = m 1 1 a + m 2 2 a + … + m n n a = M cm a Here, 1 F 2 F , …, n F are the forces acting on the particles of the system producing the accelerations 1 a , 2 a , … , n a respectively and F is the resultant force. dt
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06%20-%20Dynamics%20%20of%20%20a%20%20System%20%20of%20%20Particles

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