Physics XI - Chapter 07: System of Particles and Rotational Motion
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- Rigid Body: A body in which the distance between any two particles remains constant.
- Center of Mass: Point where entire mass can be assumed concentrated for translational motion.
- For discrete system: x_cm = Σmᵢxᵢ/Σmᵢ
- For continuous body: x_cm = ∫x dm/∫dm
- Rotational Motion Analogues:
- Linear displacement → Angular displacement (θ)
- Linear velocity → Angular velocity (ω = dθ/dt)
- Linear acceleration → Angular acceleration (α = dω/dt)
- Mass (m) → Moment of Inertia (I)
- Force (F) → Torque (τ = r × F)
- Linear momentum (p = mv) → Angular momentum (L = Iω)
- Torque (τ): τ = r × F = rF sinθ. SI unit: N·m.
- Moment of Inertia (I): I = Σmᵢrᵢ² (measure of rotational inertia).
- Thin ring about axis: MR²
- Disc about axis: ½MR²
- Solid sphere about diameter: ⅖MR²
- Hollow sphere about diameter: ⅔MR²
- Rod about center: ML²/12
- Rod about end: ML²/3
- Theorems:
- Parallel Axes: I = I_cm + Md²
- Perpendicular Axes: I_z = I_x + I_y (for laminar bodies)
- Radius of Gyration (k): I = Mk²
- Rotational Kinematics:
- ω = ω₀ + αt
- θ = ω₀t + ½αt²
- ω² = ω₀² + 2αθ
- Rotational Dynamics:
- τ = Iα (Newton's second law for rotation)
- τ = dL/dt
- Angular Momentum (L): L = r × p = Iω. SI unit: kg·m²/s.
- Conservation Laws:
- If τ_ext = 0, then L = constant
- If F_ext = 0, then p = constant
- Rotational Kinetic Energy: K_rot = ½Iω²
- Work done by Torque: W = τθ
- Power in Rotation: P = τω
- Rolling Motion: Combination of translation and rotation.
- Pure rolling: v = ωR (no slipping)
- Total KE = ½Mv² + ½Iω²
- Equilibrium Conditions:
- Translational: ΣF = 0
- Rotational: Στ = 0
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Chapter Summary
This chapter extends our understanding of mechanics from point particles to systems of particles and rigid bodies, introducing the powerful concepts of rotational motion. We begin with the center of mass - a special point that moves as if all the mass were concentrated there and all external forces acted at that point. This concept allows us to separate the translational motion of the entire system from its internal motions, providing a crucial simplification in analyzing complex systems.
The core of this chapter revolves around establishing the beautiful analogies between linear and rotational motion. Just as mass resists linear acceleration, moment of inertia resists angular acceleration. The rotational analogues of Newton's laws lead us to torque as the cause of angular acceleration and angular momentum as the rotational equivalent of linear momentum. The conservation of angular momentum explains fascinating phenomena like why ice skaters spin faster when they pull their arms in, and why divers can control their rotation in mid-air.
We explore practical applications through rolling motion - the combination of translation and rotation that occurs when objects like wheels, spheres, and cylinders move without slipping. The theorems of parallel and perpendicular axes provide powerful tools for calculating moments of inertia of complex shapes. Mastering this chapter is essential for understanding everything from simple spinning tops to complex mechanical systems, planetary motions, and gyroscopic effects. The principles learned here form the foundation for advanced topics in rotational dynamics and have wide applications in engineering, astronomy, and sports science.