Motion of System of Particles and Rigid Body — Revision Notes

⚡ 30-Second Revision

Center of Mass (CM): — $vec{R}_{CM} = \frac{sum m_ivec{r}_i}{M}$, $vec{F}_{ext} = Mvec{A}_{CM}$
Rotational Kinematics: — $omega = omega_0 + alpha t$, $heta = omega_0 t + \frac{1}{2}alpha t^2$, $omega^2 = omega_0^2 + 2alpha\theta$
Linear-Angular Relations: — $v = romega$, $a_t = ralpha$
Torque: — $vec{\tau} = vec{r} \times vec{F}$, $au = Ialpha$
Moment of Inertia (I): — $I = sum m_i r_i^2$, $I = int r^2 ,dm$

- Parallel Axis Theorem: $I = I_{CM} + Md^2$ - Perpendicular Axis Theorem: $I_z = I_x + I_y$ (for planar bodies)

Angular Momentum (L): — $vec{L} = Ivec{omega}$ (rigid body), $vec{L} = vec{r} \times vec{p}$ (particle)

- Conservation: If $vec{\tau}_{ext} = 0$, then $L = \text{constant} implies I_1omega_1 = I_2omega_2$

Rotational Kinetic Energy: — $K_{rot} = \frac{1}{2}Iomega^2$
Rolling Motion (no slip): — $v_{CM} = Romega$

- Total KE: $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}omega^2$ - Acceleration on Incline: $a = \frac{gsin\theta}{1 + \frac{I_{CM}}{MR^2}}$

2-Minute Revision

This chapter extends mechanics to extended objects. Start with the Center of Mass (CM), which simplifies the translational motion of a system of particles; remember $vec{F}_{ext} = Mvec{A}_{CM}$. For rigid bodies, both translational and rotational motion occur.

Understand the rotational analogues: angular displacement ($heta$), velocity ($omega$), and acceleration ($alpha$). Key relations are $v=romega$ and $a_t=ralpha$. **Torque ($au$)** is the rotational force, $au = Ialpha$.

**Moment of Inertia ($I$)** is rotational mass, depending on mass distribution. Master the Parallel ($I = I_{CM} + Md^2$) and Perpendicular ($I_z = I_x + I_y$) Axis Theorems for calculating $I$ for various axes.

**Angular Momentum ($L = Iomega$)** is conserved if net external torque is zero ($I_1omega_1 = I_2omega_2$), a frequently tested concept. Rotational Kinetic Energy is $rac{1}{2}Iomega^2$. Finally, Rolling Motion is a combination of translation and rotation.

The no-slip condition is $v_{CM} = Romega$. Total kinetic energy is the sum of translational and rotational KE. The acceleration of a rolling body down an incline depends on its moment of inertia factor ($I_{CM}/MR^2$).

Focus on problem-solving applications of these core formulas and theorems.

5-Minute Revision

The 'Motion of System of Particles and Rigid Body' chapter is a cornerstone of NEET Physics. Begin by solidifying your understanding of the Center of Mass (CM). For discrete particles, $X_{CM} = \frac{sum m_i x_i}{sum m_i}$. Remember that the CM's motion is governed by external forces only, $vec{F}_{ext} = Mvec{A}_{CM}$. This is crucial for problems involving explosions or projectile motion of extended objects.

Next, delve into Rotational Motion. Establish the clear analogies between linear and rotational quantities: $vec{s} leftrightarrow vec{\theta}$, $vec{v} leftrightarrow vec{omega}$, $vec{a} leftrightarrow vec{alpha}$, $m leftrightarrow I$, $vec{F} leftrightarrow vec{\tau}$, $vec{p} leftrightarrow vec{L}$, $K_{lin} leftrightarrow K_{rot}$. The equations of rotational kinematics are identical in form to linear ones. Key relations are $v=romega$ and $a_t=ralpha$.

**Torque ($vec{\tau} = vec{r} \times vec{F}$)** is the rotational equivalent of force, causing angular acceleration ($au = Ialpha$). **Moment of Inertia ($I$)** is critical; it's the rotational inertia and depends on mass distribution.

Memorize $I$ for standard shapes (e.g., disc $I_{CM} = \frac{1}{2}MR^2$, solid sphere $I_{CM} = \frac{2}{5}MR^2$, rod $I_{CM} = \frac{1}{12}ML^2$). Crucially, master the **Parallel Axis Theorem ($I = I_{CM} + Md^2$) and Perpendicular Axis Theorem ($I_z = I_x + I_y$)** to calculate $I$ about any axis.

For example, a rod pivoted at one end has $I = \frac{1}{3}ML^2$.

**Angular Momentum ($vec{L} = Ivec{omega}$)** is a conserved quantity if the net external torque is zero. This means $I_1omega_1 = I_2omega_2$. Practice problems like an ice skater pulling in arms or a disc dropped onto another. Rotational Kinetic Energy is $K_{rot} = \frac{1}{2}Iomega^2$.

Finally, Rolling Motion combines translation and rotation. The condition for pure rolling (no slipping) is $v_{CM} = Romega$. The total kinetic energy is $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}omega^2$. For a body rolling down an inclined plane, its acceleration is $a = \frac{gsin\theta}{1 + I_{CM}/MR^2}$. Remember, bodies with smaller $I_{CM}/MR^2$ (e.g., solid sphere) accelerate faster. Practice applying these formulas to various scenarios and comparing different rolling bodies.

Prelims Revision Notes

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Center of Mass (CM):

* For discrete particles: $vec{R}_{CM} = \frac{sum m_ivec{r}_i}{M_{total}}$. * For continuous body: $vec{R}_{CM} = \frac{int vec{r} ,dm}{int dm}$. * Motion of CM: $vec{F}_{ext} = M_{total}vec{A}_{CM}$. Internal forces do not affect CM motion. * CM of symmetrical, homogeneous bodies lies at their geometric center.

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Rotational Kinematics:

* Angular displacement ($heta$), angular velocity ($omega = d\theta/dt$), angular acceleration ($alpha = domega/dt$). * Relations: $v = romega$, $a_t = ralpha$, $a_c = romega^2 = v^2/r$. * Equations (for constant $alpha$): $omega = omega_0 + alpha t$, $heta = omega_0 t + \frac{1}{2}alpha t^2$, $omega^2 = omega_0^2 + 2alpha\theta$.

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**Torque ($vec{\tau}$):**

* Rotational analogue of force. $vec{\tau} = vec{r} \times vec{F}$. Magnitude $au = rFsinphi$. * Newton's 2nd Law for rotation: $vec{\tau}_{ext} = Ivec{alpha}$.

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Moment of Inertia (I):

* Rotational analogue of mass. $I = sum m_i r_i^2$ (discrete), $I = int r^2 ,dm$ (continuous). * Parallel Axis Theorem: $I = I_{CM} + Md^2$. * Perpendicular Axis Theorem (planar bodies): $I_z = I_x + I_y$. * **Standard $I_{CM}$ values:** * Ring/Hollow cylinder: $MR^2$ * Disc/Solid cylinder: $rac{1}{2}MR^2$ * Solid sphere: $rac{2}{5}MR^2$ * Hollow sphere: $rac{2}{3}MR^2$ * Rod (perpendicular to length): $rac{1}{12}ML^2$

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**Angular Momentum ($vec{L}$):**

* For a particle: $vec{L} = vec{r} \times vec{p} = vec{r} \times (mvec{v})$. * For a rigid body: $vec{L} = Ivec{omega}$. * Conservation of Angular Momentum: If $vec{\tau}_{ext} = 0$, then $L = \text{constant} implies I_1omega_1 = I_2omega_2$.

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Rotational Kinetic Energy: — $K_{rot} = \frac{1}{2}Iomega^2$.

1
Rolling Motion (without slipping):

* Condition: $v_{CM} = Romega$. * Total Kinetic Energy: $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}omega^2 = \frac{1}{2}Mv_{CM}^2(1 + \frac{I_{CM}}{MR^2})$. * Acceleration on inclined plane: $a = \frac{gsin\theta}{1 + \frac{I_{CM}}{MR^2}}$. * Time to roll down: $t = sqrt{\frac{2s}{a}}$.

Key Strategy: Understand the analogies, memorize standard $I$ values, and master the application of conservation laws and theorems. Practice problem-solving for rolling motion and angular momentum conservation.

Vyyuha Quick Recall

Can Rotating Things Move Around Rapidly?

Center of Mass ($vec{R}_{CM}$)
Rotational Kinematics ($omega, alpha, \theta$)
Torque ($vec{\tau} = Ivec{alpha}$)
Moment of Inertia ($I$, Parallel/Perpendicular Axis Theorems)
Angular Momentum ($vec{L} = Ivec{omega}$, Conservation)
Rolling Motion ($v_{CM} = Romega$, $K_{total}$)