Physics Mechanics - How to Solve: Centre of Mass (COM) – Complete Guide for IIT JEE

How to Solve: Centre of Mass (COM) – Complete Guide for IIT JEE

(For Continuous Bodies, Velocity of COM, Collisions)

? Introduction

Mastering Centre of Mass (COM) unlocks 5-10 marks in IIT JEE—directly in Mechanics (20-25% of the paper). Whether it’s a colliding asteroid, a rotating rod, or a rocket ejecting fuel, COM is the hidden key. Miss it, and you lose marks on every collision, rotation, and system of particles question.

? WHAT YOU NEED TO KNOW FIRST

1. Basic COM of discrete particles – You must know how to find COM for 2-3 point masses.
2. Integration basics – You should be comfortable with definite integrals (for continuous bodies).
3. Conservation of momentum – Essential for collisions and velocity of COM problems.

(If you’re shaky on any of these, pause and revise them first.)

? KEY TERMS & FORMULAS

1. Centre of Mass (COM) for Continuous Bodies

Formula: [ \vec{R}_{COM} = \frac{1}{M} \int \vec{r} \, dm ] - (\vec{R}_{COM}) → Position vector of COM (m) - (M) → Total mass of the body (kg) - (\vec{r}) → Position vector of a small mass element (dm) (m) - (dm) → Infinitesimal mass element (kg)

MEMORISE THIS – You’ll use this for rods, discs, spheres, and irregular shapes.

Special Cases (Given on Exam Sheet, but memorise for speed): | Shape | COM Position | Formula | |-----------------|--------------------------------------|--------------------------------------| | Uniform Rod | Midpoint | ( \frac{L}{2} ) | | Semicircular Rod | ( \frac{2R}{\pi} ) from centre | ( \frac{2R}{\pi} ) (along symmetry axis) | | Triangular Lamina | Centroid (intersection of medians) | ( \frac{h}{3} ) from base | | Hemisphere | ( \frac{3R}{8} ) from base | ( \frac{3R}{8} ) (along symmetry axis) |

2. Velocity of COM

Formula: [ \vec{V}_{COM} = \frac{1}{M} \sum m_i \vec{v}_i \quad \text{(Discrete)} \quad \text{or} \quad \frac{1}{M} \int \vec{v} \, dm \quad \text{(Continuous)} ] - (\vec{V}_{COM}) → Velocity of COM (m/s) - (m_i, \vec{v}_i) → Mass and velocity of (i^{th}) particle - (M) → Total mass of the system

MEMORISE THIS – Used in collisions, explosions, and variable mass systems.

Key Property: - If no external force acts, (\vec{V}_{COM}) remains constant (Conservation of Momentum).

3. COM in Collisions

Formula (Conservation of Momentum): [ M \vec{V}_{COM} = \text{Constant} \quad \text{(Before = After collision)} ] - Elastic Collision: Both momentum & kinetic energy conserved. - Inelastic Collision: Only momentum conserved (KE lost).

MEMORISE THIS – Always check if external forces are zero before applying.

? STEP-BY-STEP METHOD

? How to Find COM of a Continuous Body (Step-by-Step)

1. Choose a coordinate system – Pick an origin (usually one end of the body).
2. Express (dm) in terms of (dx) (or (dA, dV)) –
3. For 1D (rod): (dm = \lambda \, dx) (where (\lambda = \frac{M}{L}))
4. For 2D (lamina): (dm = \sigma \, dA) (where (\sigma = \frac{M}{A}))
5. For 3D (solid): (dm = \rho \, dV) (where (\rho = \frac{M}{V}))
6. Write the COM integral – [ R_{COM} = \frac{1}{M} \int x \, dm ]
7. Substitute (dm) and limits –
8. For a rod of length (L): (R_{COM} = \frac{1}{M} \int_0^L x \lambda \, dx)
9. Solve the integral – Simplify and compute.
10. Check symmetry – If the body is symmetric, COM lies on the symmetry axis.

? How to Find Velocity of COM (Step-by-Step)

1. Identify all masses and their velocities – List (m_i) and (\vec{v}_i).
2. Write the COM velocity formula – [ \vec{V}_{COM} = \frac{1}{M} \sum m_i \vec{v}_i ]
3. Plug in values – Substitute masses and velocities.
4. Simplify – Combine terms.
5. Check for external forces – If no external force, (\vec{V}_{COM}) is constant.

? How to Solve Collision Problems Using COM (Step-by-Step)

1. Draw before & after diagrams – Label masses and velocities.
2. Check if external forces are zero – If yes, momentum is conserved.
3. Write conservation of momentum – [ m_1 \vec{v}_1 + m_2 \vec{v}_2 = m_1 \vec{v}_1' + m_2 \vec{v}_2' ]
4. If elastic, add KE conservation – [ \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2 ]
5. Solve for unknowns – Use algebra to find final velocities.
6. Find (\vec{V}_{COM}) – If needed, compute before and after collision.

✏️ WORKED EXAMPLES

? Example 1 – Basic: COM of a Non-Uniform Rod

Problem: A rod of length (L = 2 \, \text{m}) has linear mass density (\lambda = kx) (where (k = 1 \, \text{kg/m}^2)). Find its COM.

Solution (Step-by-Step):
1. Choose origin at (x=0).
2. Express (dm): [ dm = \lambda \, dx = kx \, dx ]
3. Total mass (M): [ M = \int_0^L dm = \int_0^2 kx \, dx = k \left[ \frac{x^2}{2} \right]0^2 = 2k = 2 \, \text{kg} ]
4. COM integral: [ R \int_0^2 x^2 \, dx ]
5. } = \frac{1}{M} \int_0^L x \, dm = \frac{1}{2} \int_0^2 x (kx) \, dx = \frac{k}{2Solve: [ R_{COM} = \frac{1}{2} \left[ \frac{x^3}{3} \right]_0^2 = \frac{1}{2} \times \frac{8}{3} = \frac{4}{3} \, \text{m} ]

What we did and why: - We used integration because the mass distribution was non-uniform. - Key step: Correctly expressing (dm = \lambda \, dx) and setting up the integral.

? Example 2 – Medium: Velocity of COM in a Collision

Problem: Two blocks, (m_1 = 2 \, \text{kg}) (moving at (4 \, \text{m/s})) and (m_2 = 3 \, \text{kg}) (at rest), collide inelastically. Find: (a) Final velocity of the combined mass. (b) Velocity of COM before and after collision.

Solution (Step-by-Step):
1. Before collision: - (v_1 = 4 \, \text{m/s}), (v_2 = 0) - (\vec{V}{COM} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} = \frac{2 \times 4 + 3 \times 0}{5} = \frac{8}{5} = 1.6 \, \text{m/s})
2. After collision (inelastic): - Combined mass = (5 \, \text{kg}) - By momentum conservation: [ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \implies 8 = 5 v_f \implies v_f = 1.6 \, \text{m/s} ] - (\vec{V}) remains 1.6 m/s (no external force).

What we did and why: - Inelastic collision → Only momentum conserved, not KE. - COM velocity remains constant because no external force acts.

? Example 3 – Exam-Style: COM of a Semicircular Wire

Problem (IIT JEE 2018-Style): A uniform semicircular wire of radius (R) and mass (M) lies in the (xy)-plane with its diameter along the (x)-axis. Find the (y)-coordinate of its COM.

Solution (Step-by-Step):
1. Choose origin at centre.
2. Parametrize the wire: - Angle (\theta) from (0) to (\pi). - Arc length (ds = R \, d\theta).
3. Mass element (dm): [ dm = \lambda \, ds = \left( \frac{M}{\pi R} \right) R \, d\theta = \frac{M}{\pi} d\theta ]
4. (y)-coordinate of (dm): [ y = R \sin \theta ]
5. COM integral: [ y_{COM} = \frac{1}{M} \int y \, dm = \frac{1}{M} \int_0^\pi (R \sin \theta) \left( \frac{M}{\pi} d\theta \right) = \frac{R}{\pi} \int_0^\pi \sin \theta \, d\theta ]
6. Solve: [ y_{COM} = \frac{R}{\pi} \left[ -\cos \theta \right]_0^\pi = \frac{R}{\pi} (-\cos \pi + \cos 0) = \frac{R}{\pi} (1 + 1) = \frac{2R}{\pi} ]

What we did and why: - Symmetry → COM lies on the (y)-axis ((x_{COM} = 0)). - Key step: Correctly setting up the polar integral for a semicircle.

❌ COMMON MISTAKES

? EXAM TRAPS