Concepts: centre of mass of a system of particles

System of Particles and Rotational Motion22 min readLesson 2 of 8

2.1 Why a system can be replaced by one point A cricket bat thrown in the air has every part moving differently, yet its motion looks like a translation of one point plus a rotation about it. That point is the centre of mass. For a system of particles the centre of mass is the mass weighted average of the positions of all the particles, so a heavy particle pulls the point closer to itself and a particle on the axis of symmetry contributes equally from both sides. The usefulness of the centre of mass comes from a theorem: the external force on a system equals the total mass times the acceleration of the centre of mass, exactly as if the entire mass were concentrated there. Internal forces, however complicated, cancel in pairs and never change the motion of the centre of mass. This is why the internal explosion of a shell does not move its centre of mass, and the debris continues along the original trajectory. For a two particle system on the x-axis, if the masses are m1 at x1 and m2 at x2, the centre of mass sits at the weighted average. When the masses are equal the point is at the midpoint; when one mass is much larger the point lies almost at that mass. The same logic extends to any number of particles and to three dimensions.

X_{cm} = \frac{m _{1} x _{1} + m _{2} x _{2}}{m _{1} + m _{2}}, R_{cm} = \frac{\sum m _{i} r _{i}}{\sum m _{i}}

Centre of mass of two particles and of a system

2.2 Continuous bodies and symmetry For a continuous body the sum becomes an integral, with each small mass element weighted by its position. For a uniform rod the integral gives the midpoint; for a uniform triangular lamina it lies at the centroid, one third of the height above the base; for a semicircular ring it lies on the axis of symmetry at a distance 2R over pi from the centre. These standard results are worth remembering because they are used as building blocks for composite bodies. Symmetry arguments save the integral. If a body has a plane of symmetry, its centre of mass lies in that plane. If it has two planes of symmetry, the centre of mass lies on their line of intersection. A uniform ring has its centre of mass at the centre, where there is no material at all, which shows that the centre of mass need not be inside the body. A bangle and a hollow sphere are the usual examples of this point. For a composite body made of simple parts, treat each part as a particle at its own centre of mass, and take the weighted average. For a plate with a hole, treat the hole as a part of negative mass. This trick gives the centre of mass of the remaining plate without any new integral.

2.3 Worked examples Two particles of masses 1 kg and 3 kg are placed at x equal to 0 and 4 m. The centre of mass is at 1 times 0 plus 3 times 4, divided by 4, which is 3 m, so it lies three quarters of the way toward the heavier mass. Reversing the masses moves the point to 1 m, showing that the heavier body always wins. A uniform rod of length 1 m has a 2 kg mass attached at one end and a 3 kg mass at the other. Treating the rod as 5 kg at its centre and adding the two masses gives the centre of mass at (1 times 0 plus 2.5 times 0.5 plus 3 times 1) divided by 6, which is about 0.79 m from the end with the 1 kg mass. The calculation is only a weighted average, but writing the table of masses and positions prevents mistakes. A semicircular wire of radius R has its centre of mass at 2R over pi from the centre along the axis of symmetry. A semicircular plate has it at 4R over 3 pi. The wire and the plate differ because the mass of the plate is spread through its area while the wire's mass is on its arc, which is a useful reminder that the mass distribution matters more than the outline.

X_{cm} = \frac{1}{M} \int x d m, y_{cm} = \frac{2 R}{π} for a semicircular ring

Centre of mass of a continuous body and the semicircular ring result

2.4 Motion of the centre of mass Treat the centre of mass as a particle carrying the whole mass and pushed only by the total external force. When that total force is zero, the point stays at rest or moves uniformly, whatever the internal parts do. Two skaters who push each other apart move in opposite directions, and their centre of mass stays where it was. A projectile exploding in mid air is the standard illustration. The fragments fly apart, but the centre of mass of all the fragments continues on the original parabolic path until a fragment hits the ground. The explosion changes the internal kinetic energy and the individual momenta, leaving the motion of the centre of mass untouched. Dividing the total momentum by the total mass gives the velocity of the centre of mass. This relation is why the conservation of momentum and the motion of the centre of mass are the same statement. In an isolated system the total momentum is constant, so the centre of mass moves uniformly, and the internal motion can be studied in the frame of the centre of mass where the total momentum is zero.

Common mistake

The centre of mass is not the same as the geometric centre. For a non-uniform body it is pulled toward the denser part, and for a body with a hole it moves away from the hole. Always use the mass weighting rather than the outline.

in hinglish

Centre of mass woh point hai jahan poora system ki mass concentrated maani ja sakti hai, aur woh mass weighted average hota hai. Do particles ke liye X = (m1x1 + m2x2)/(m1 + m2), isliye bhaari mass ke paas point aata hai. Continuous body ke liye integral lagta hai, aur symmetry se bahut kaam asaan ho jaata hai: plane of symmetry ho to CM usi plane mein hota hai, aur ring ka CM centre par hota hai jahan material hi nahi hota. Composite body ke liye har part ko uske CM par rakho aur weighted average lo; hole wale plate mein hole ko negative mass maan lo. External force total mass guna CM ka acceleration deta hai, isliye internal forces CM ki motion nahi badalte. Projectile phatne par bhi CM wahi parabola follow karta hai. Numerical mein table banao: mass, position, product, phir divide.