Center of Mass & Momentum Conservation – Key Formulas & Concepts | Entrance Exam Physics Notes

CENTER OF MASS (COM) — SHORT NOTES

1. Definition

The Center of Mass of a system is the point where the entire mass of the system can be considered to be concentrated for translational motion.

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2. COM for a System of Particles

$${\vec{R}}_{CM}=\frac{m_1{\vec{r}}_1+m_2{\vec{r}}_2+\cdots }{m_1+m_2+\cdots }$$

For 1D:

$$x_{CM}=\frac{\sum m_i{x}_i}{\sum m_{\mathrm{i<span\; class="vlist"></span>}}}$$

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3. COM for Continuous Bodies

$$x_{CM} = \frac{\int x \, dm}{\int dm}$$

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4. Standard COM Positions

• Uniform Rod (length L): at ( L/2 )

• Uniform Ring: at center

• Uniform Sphere: at center

• Right triangle lamina: at intersection of medians (2:1 ratio)

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5. Velocity of COM

$${\vec{v}}_{CM}=\frac{\sum m_i{\vec{v}}_i}{\sum m_{\mathrm{i<span\; class="vlist-s"></span>}}}$$

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6. Acceleration of COM

$$\vec{a}_{CM} = \frac{\sum \vec{F}_{ext}}{M}$$

Only external forces change COM motion; internal forces do not.

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7. Motion of COM

• COM moves like a particle having total mass of system and subject only to external forces.

• Internal explosions or internal forces do not change COM motion.

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8. Important Concept

If no external force acts:

• COM remains at rest, or

• Moves with constant velocity.

Used in collision problems, rocket motion, bodies breaking apart.

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CONSERVATION OF MOMENTUM — SHORT NOTES

1. Linear Momentum

$$\vec{p} = m\vec{v}$$

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2. Newton’s Second Law (Momentum Form)

$${\vec{F}}_{net}=\frac{d\vec{p}}{dt}$$

If $F_{net}=0$→ momentum remains constant.

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3. Law of Conservation of Momentum

If the net external force on a system is zero, its total momentum remains constant.

$$\sum {\vec{p}}_{initial}=\sum {\vec{p}}_{fina\mathrm{l}}$$

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4. Types of Collisions & Momentum

✔ Elastic Collision

• Momentum conserved

• Kinetic energy conserved

• Example: gas molecules

✔ Inelastic Collision

• Momentum conserved

• Kinetic energy not conserved

✔ Perfectly Inelastic Collision

• Bodies stick together

$$v=\frac{m_1{u}_1+m_2{u}_2}{m_1+m_2}$$

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5. Explosion / Recoil

Momentum before = Momentum after
If system starts from rest:

$$0=m_1{v}_1+m_2{v}_{\mathrm{2}}$$

Used in:

• Gun recoil

• Rockets

• Breaking objects

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6. Impulse

If force acts for short time:

$$\text{Impulse}=F\mathrm{\Delta }t=\mathrm{\Delta }p$$

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7. Center of Mass Interpretation

If external force = 0:

• COM velocity is constant

• Total momentum is constant

Thus momentum conservation is equivalent to COM moving uniformly.

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8. Common Entrance Exam Applications

• Bullet + block problems

• Firecracker exploding mid-air

• Collisions on frictionless surfaces

• Recoil velocity

• Multi-stage rockets

• Block + wedge problems (horizontal momentum conserved)