Introduction
Torque τ = r × F is the rotational analog of force. It causes changes in angular momentum, producing rotational acceleration. Understanding torque is essential for analyzing rotating systems, from simple levers to complex machinery and celestial mechanics.
Definition of Torque
Torque about point O due to force F applied at position r: τ = r × F. Magnitude: τ = rF sinθ = r_⊥ F = r F_⊥, where r_⊥ = r sinθ is the moment arm (perpendicular distance from O to line of force), and F_⊥ is force component perpendicular to r. Direction: perpendicular to plane containing r and F, by right-hand rule. Units: N·m (same as work but different physical meaning).
Moment Arm and Leverage
The moment arm is the perpendicular distance from pivot to force line of action. Larger moment arm means greater torque for same force - principle of leverage. Torque can be increased by: increasing force, increasing moment arm, or optimizing angle (θ = 90° gives maximum torque for given r and F). This principle underlies all mechanical advantage devices.
\nIntroduction
Torque τ = r × F is the rotational analog of force. It causes changes in angular momentum, producing rotational acceleration. Understanding torque is essential for analyzing rotating systems, from simple levers to complex machinery and celestial mechanics.
Definition of Torque
Torque about point O due to force F applied at position r: τ = r × F. Magnitude: τ = rF sinθ = r_⊥ F = r F_⊥, where r_⊥ = r sinθ is the moment arm (perpendicular distance from O to line of force), and F_⊥ is force component perpendicular to r. Direction: perpendicular to plane containing r and F, by right-hand rule. Units: N·m (same as work but different physical meaning).
Moment Arm and Leverage
The moment arm is the perpendicular distance from pivot to force line of action. Larger moment arm means greater torque for same force - principle of leverage. Torque can be increased by: increasing force, increasing moment arm, or optimizing angle (θ = 90° gives maximum torque for given r and F). This principle underlies all mechanical advantage devices.
\nNet Torque and Equilibrium
Net torque is vector sum of all torques about a point: τ_net = Σ τ_i. For static equilibrium: both F_net = 0 (no translation) and τ_net = 0 (no rotation). For a body to be in complete equilibrium, vector sum of forces and sum of torques about any point must both vanish. Torque equilibrium condition is independent of choice of reference point when force equilibrium holds.
Rotational Dynamics Equation
For fixed-axis rotation: τ = Iα, where I is moment of inertia and α is angular acceleration. This is the rotational analog of F = ma. The moment of inertia I plays the role of mass, resisting angular acceleration. For a particle: I = mr2, so τ = mr2α. This equation governs rotation of rigid bodies about fixed axes.
Applications
Applications: Levers and simple machines (torque multiplication); Wrenches (longer handles increase torque); Doors (knobs far from hinges minimize required force); See-saws and balances (torque equality); Rotating machinery (torque determines angular acceleration); Gyroscopic effects (torque perpendicular to angular momentum causes precession).
\nSolved Example: Seesaw Equilibrium
A uniform seesaw (plank) of mass 20 kg and length 4 m is pivoted at its center. Child A (30 kg) sits 1.5 m left of pivot. Where should Child B (40 kg) sit to balance? If Child B sits at end (2 m from pivot), what force is needed at other end to maintain balance? Solution: For balance, torques must balance. τ_A = 30×9.8×1.5 = 441 N·m clockwise. For equilibrium: τ_B = τ_A → 40×9.8×d = 441 → d = 441/392 = 1.125 m. Child B must sit 1.125 m right of pivot. If Child B sits at end (2 m): τ_B = 40×9.8×2 = 784 N·m counterclockwise. Net torque without additional force: 784 - 441 = 343 N·m CCW. To balance, need additional torque 343 N·m CW. If adult pushes down at left end (2 m from pivot): F×2 = 343 → F = 171.5 N downward needed at left end.
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