Centre of Mass Class 11: All Formulas & Formula Sheet with Free PDF Download (Physics, JEE & NEET)
Centre of Mass Formula Sheet — Competishun
Centre of Mass Class 11: All Formulas & Formula Sheet with Free PDF Download (Physics, JEE & NEET)
Centre of Mass is a fundamental concept in Class 11 Physics and a scoring topic in JEE and NEET. Every year, 2-3 questions from this chapter appear, testing your understanding of COM definition, calculation methods, momentum conservation, and collisions. A clear grasp of centre of mass also forms the foundation for rotational dynamics and rigid body motion.
This page gives you a complete Centre of Mass formula sheet for Class 11 physics, covering COM definition, calculation methods for discrete and continuous systems, standard results for common shapes, momentum, and collisions. Download the free PDF below and keep it handy for quick revision before your JEE Main, JEE Advanced, or NEET exam.
Download the Centre of Mass Formula Sheet PDF
Get all centre of mass formulas, including COM definition, calculation methods, standard results for common shapes, momentum, and collisions in one clean PDF, free. Perfect for JEE & NEET revision.
Download Free PDFWhat is Centre of Mass in Physics?
The centre of mass (COM) is a point where the entire mass of a system or body is assumed to be concentrated. It is the weighted average position of all particles in the system. For a system of particles, the COM is given by the formula:
| System Type | Formula |
|---|---|
| Discrete particles | R = (Σ mᵢ rᵢ) / Σ mᵢ |
| Continuous body | R = (∫ r dm) / ∫ dm |
| COM coordinates | xCM = Σ mᵢ xᵢ / Σ mᵢ, yCM = Σ mᵢ yᵢ / Σ mᵢ |
| The centre of mass can lie inside or outside the body. For symmetrical bodies, it lies at the geometric centre. | |
Methods to Find the Centre of Mass
There are several methods to find the centre of mass depending on the system.
- Method 1: For Discrete Particles — Use R = (Σ mᵢ rᵢ) / Σ mᵢ. This is the most direct method.
- Method 2: For Continuous Bodies — Use R = (∫ r dm) / ∫ dm. Integrate over the entire volume.
- Method 3: Using Symmetry — If the body has symmetry (sphere, cylinder, cube), the COM lies at the geometric centre.
- Method 4: Using Standard Results — Memorise the COM of common shapes (rod, triangle, hemisphere, etc.).
- Method 5: For Composite Bodies — Divide the body into simpler shapes, find their individual COM, then combine using the weighted average formula.
Standard Results — COM of Common Shapes
Memorising these standard results saves time in JEE and NEET problems. These are frequently asked in direct formula-based questions.
| Shape | Position of COM |
|---|---|
| Uniform rod | At midpoint (L/2 from one end) |
| Uniform triangular lamina | Centroid (one-third of median from the base) |
| Solid hemisphere | 3R/8 from the base |
| Hollow hemisphere | R/2 from the base |
| Solid cone | H/4 from the base |
| Hollow cone | H/3 from the base |
| Semicircular disc | 4R/(3π) from the centre |
| Semicircular ring | 2R/π from the centre |
| These results are for uniform density. For non-uniform density, the COM depends on the mass distribution. | |
Motion of the Centre of Mass
The centre of mass behaves like a single particle with the total mass of the system. This is a powerful concept for solving problems.
| Quantity | Formula |
|---|---|
| Velocity of COM | VCM = (Σ mᵢ vᵢ) / Σ mᵢ |
| Acceleration of COM | ACM = (Σ mᵢ aᵢ) / Σ mᵢ |
| Total momentum | P = M VCM |
| Newton's 2nd law for COM | Fext = M ACM |
| If no external force acts, the velocity of the COM remains constant (conservation of momentum). | |
Momentum & Collisions
Collisions are an important application of centre of mass and momentum conservation. Here are the key formulas.
| Concept | Formula |
|---|---|
| Linear momentum | P = mv |
| Conservation of momentum | m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ |
| Elastic collision (1D) | v₁ = (m₁−m₂)u₁/(m₁+m₂) + 2m₂u₂/(m₁+m₂) |
| Perfectly inelastic collision | v = (m₁u₁ + m₂u₂)/(m₁+m₂) |
| Coefficient of restitution | e = (v₂−v₁)/(u₁−u₂) |
| Kinetic energy (elastic) | ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂² |
| For elastic collisions, both momentum and kinetic energy are conserved. For inelastic collisions, only momentum is conserved. | |
All Centre of Mass Formulas at a Glance
Here is a quick reference table of all the key formulas from the Centre of Mass chapter.
| Formula | What It Means |
|---|---|
| R = Σ mᵢ rᵢ / Σ mᵢ | Centre of mass for discrete particles |
| R = ∫ r dm / ∫ dm | Centre of mass for continuous bodies |
| VCM = Σ mᵢ vᵢ / Σ mᵢ | Velocity of centre of mass |
| P = M VCM | Total momentum of a system |
| Fext = M ACM | Newton's second law for COM |
| m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | Conservation of momentum |
| e = (v₂−v₁)/(u₁−u₂) | Coefficient of restitution |
| Memorize these formulas and understand the conditions under which each applies (e.g., external forces, elastic vs. inelastic). | |
Worked Examples
| Problem | Solution |
|---|---|
| Two masses 2 kg and 3 kg are at (1,0) and (4,0). Find COM. | xCM = (2×1 + 3×4)/(2+3) = 14/5 = 2.8 m. |
| A uniform rod of length 2 m has its COM at? | For a uniform rod, COM is at midpoint = 1 m from either end. |
| Two masses 4 kg and 6 kg move at 3 m/s and 2 m/s in same direction. Find VCM. | VCM = (4×3 + 6×2)/(4+6) = 24/10 = 2.4 m/s. |
| Find the COM of a solid hemisphere of radius R. | COM = 3R/8 from the base. |
| Practise these types of problems to become comfortable with applying centre of mass formulas in different scenarios. | |
Common Mistakes to Avoid
- Confusing COM with centre of gravity: They coincide only in a uniform gravitational field. COM is purely about mass distribution.
- Forgetting the denominator in COM formula: Always divide by the total mass. This is a very common error.
- Using wrong standard results: Memorise the COM of common shapes carefully. There is no shortcut for these.
- Not checking if external forces are present: The velocity of COM changes only if an external force acts. Internal forces do not change it.
- Misapplying collision formulas: For elastic collisions, both momentum and kinetic energy are conserved. For inelastic, only momentum is conserved.
Why Centre of Mass Matters for JEE and NEET
- High weightage: Centre of mass appears in 2-3 questions in every JEE Main, JEE Advanced, and NEET paper.
- Foundation for rotational dynamics: Concepts like torque, moment of inertia, and angular momentum build directly on COM.
- Real-world applications: Rocket motion, planetary motion, and sports physics all involve centre of mass.
- Connects to other topics: COM links to work-energy (KE of a system = translational KE of COM + internal KE), momentum conservation, and collisions.
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Frequently Asked Questions — Centre of Mass
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