Class 11 Physics System of Particles and Rotational Motion Notes

System of Particles

A system of particles is a collection of many particles, each with mass $m_i$mi​ and position vector $\vec{r_i}$ri​​.

Total Mass of the System:

$$M = \sum_{i} m_i$$M=i∑​mi​

Position of Center of Mass (COM):

$$\vec{R}_{\text{COM}} = \frac{\sum_i m_i \vec{r_i}}{\sum_i m_i}$$RCOM​=∑i​mi​∑i​mi​ri​​​

• The center of mass behaves as if all mass is concentrated at that point.
• Motion of the system can be analyzed by translational motion of COM plus motion relative to COM.

---

🔹 Linear Momentum of System

$$\vec{P} = \sum_i m_i \vec{v_i} = M \vec{V}_{\text{COM}}$$P=i∑​mi​vi​​=MVCOM​

• Total external force $\vec{F}_{\text{ext}}$Fext​ = Rate of change of total momentum:

$$\vec{F}_{\text{ext}} = \frac{d\vec{P}}{dt}$$Fext​=dtdP​

---

🔹 Torque (Moment of Force)

Torque is the rotational analogue of force.$$\vec{\tau} = \vec{r} \times \vec{F}$$τ=r×F

Where:

• $\vec{r}$r = position vector from axis of rotation
• $\vec{F}$F = applied force
• Torque produces rotational motion about an axis.

---

🔹 Angular Momentum

Angular momentum $\vec{L}$L of a particle about an axis:$$\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v}$$L=r×p​=r×mv

• For a system of particles:

$$\vec{L}_{\text{total}} = \sum_i \vec{r_i} \times m_i \vec{v_i}$$Ltotal​=i∑​ri​​×mi​vi​​

• Conservation of Angular Momentum:
  If no external torque acts, total angular momentum is constant.

---

🔹 Moment of Inertia (Rotational Inertia)

Moment of inertia $I$I is the rotational analogue of mass.$$I = \sum_i m_i r_i^2$$I=i∑​mi​ri2​

Where $r_i$ri​ is the perpendicular distance of the particle from the axis of rotation.

Common Formulas for Rigid Bodies:

• Solid cylinder (axis through center): $I = \frac{1}{2}MR^2$I=21​MR2
• Solid sphere: $I = \frac{2}{5}MR^2$I=52​MR2
• Thin rod (axis through center, perpendicular): $I = \frac{1}{12}ML^2$I=121​ML2

---

🔹 Parallel Axis Theorem

$$I = I_{\text{COM}} + Md^2$$I=ICOM​+Md2

• $I$I = moment of inertia about any axis
• $I_{\text{COM}}$ICOM​ = moment of inertia about COM axis
• $d$d = distance between axes

---

🔹 Perpendicular Axis Theorem

$$I_z = I_x + I_y$$Iz​=Ix​+Iy​

• For a flat lamina lying in xy-plane
• $I_x, I_y$Ix​,Iy​ = moments of inertia about x and y axes in plane
• $I_z$Iz​ = moment of inertia perpendicular to plane

---

🔹 Rotational Kinematics (Equations of Motion)

Analogous to linear motion:

Linear	Rotational
$v = u + at$v=u+at	$\omega = \omega_0 + \alpha t$ω=ω0​+αt
$s = ut + \frac{1}{2}at^2$s=ut+21​at2	$\theta = \omega_0 t + \frac{1}{2}\alpha t^2$θ=ω0​t+21​αt2
$v^2 = u^2 + 2as$v2=u2+2as	$\omega^2 = \omega_0^2 + 2\alpha \theta$ω2=ω02​+2αθ

Where:

• $\omega$ω = angular velocity
• $\alpha$α = angular acceleration
• $\theta$θ = angular displacement

---

🔹 Rotational Kinetic Energy

$$K = \frac{1}{2}I\omega^2$$K=21​Iω2

• Total energy of a rolling body = Translational + Rotational:

$$K_{\text{total}} = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$$Ktotal​=21​Mv2+21​Iω2

---

🔹 Work and Power in Rotation

• Work done by torque:

$$W = \tau \theta$$W=τθ

• Power:

$$P = \tau \omega$$P=τω