Physics · System Of Particles And Rotational Motion · NEET
Because the mass is spread evenly along the rod, for every small mass element dm at a distance +x from the centre, there is an equal mass element at -x. When you add up all the position*mass terms, these pairs cancel. The only point where the total cancels to zero is the geometric centre. NCERT calls this reflection symmetry. So a uniform rod of length L has its COM at L/2.
The centre of mass does not have to lie on the body. For a uniform ring, every mass element on the rim has a matching element on the exactly opposite side. Their position vectors cancel in pairs, so the balance point is the empty geometric centre. The same is true for a hollow spherical shell: its COM is at the centre, where there is no mass.
Yes, but only when the body is uniform (mass spread evenly). For a homogeneous body the centre of mass coincides with the centroid, the purely geometric centre of the shape. If the density is not uniform, the COM shifts toward the heavier side and no longer sits at the centroid.
Break the shape into simple uniform pieces (squares, rectangles, discs). Put each piece's mass at its own geometric centre, then treat these as point masses and use X = (m1*x1 + m2*x2 + ...)/(m1 + m2 + ...) and the same for Y. This is the NCERT L-shape lamina method. You never integrate; you just combine known centres.
Cutting a shape in half destroys the symmetry, so the COM is no longer at the old centre. It shifts onto the axis of symmetry but away from the flat edge. For a semicircular ring the COM is at 2R/pi from the centre; for a half disc it is at 4R/(3*pi). These need integration, so NEET usually gives them as remembered results.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is at the midpoint of the rod. For a uniform rod of length L, the COM lies at L/2 from either end, on the rod's axis.
Both have their centre of mass exactly at their geometric centre. For the ring this centre is an empty point with no material, which is allowed because the COM need not lie on the body.
For both a uniform solid sphere and a uniform hollow spherical shell, the centre of mass is at the geometric centre of the sphere.
No. For a ring, a hollow shell, or an L-shaped lamina, the centre of mass can lie in a region with no material. It is only a mathematical balance point.
When the body is non-uniform (density varies) or its symmetry is broken, for example after cutting a piece away or bending the shape. Then you must use the composite-body formula or integration.
