Unit 7: Torque and Rotational Motion

📚Study Guide: Torque and Rotational Motion

Unit 7: Torque and Rotational Motion

Rotational motion extends the ideas of kinematics and dynamics from straight-line motion to spinning objects. Every linear quantity has a rotational analog: position becomes angular position, velocity becomes angular velocity, acceleration becomes angular acceleration, force becomes torque, mass becomes rotational inertia, and momentum becomes angular momentum. Understanding these analogies is the single best way to organize this unit. Torque is the rotational equivalent of force; it depends not only on the magnitude of the applied force but also on the distance from the pivot point (lever arm) and the angle at which the force is applied. The mathematical expression τ = rF sinθ reveals that maximum torque occurs when the force is perpendicular to the lever arm. Rotational inertia (or moment of inertia) quantifies an object's resistance to angular acceleration, but unlike mass, it depends on how mass is distributed relative to the axis of rotation. A dumbbell rotated about its center has a larger rotational inertia than a solid sphere of the same mass and radius, because more mass is farther from the axis. Newton's Second Law for rotation states that the net torque equals rotational inertia times angular acceleration: τ_net = Iα. You will apply this to systems such as pulleys with mass, rotating doors, and compound objects. The unit also covers rolling motion without slipping, where the translational speed of the center of mass is related to the angular speed by v = rω. A rolling object possesses both translational kinetic energy (½mv²) and rotational kinetic energy (½Iω²), and you must include both when applying energy conservation. Finally, angular momentum L = Iω and its conservation law provide powerful tools for analyzing collisions involving rotation, spinning ice skaters pulling in their arms, and gyroscopic stability. The AP Exam often combines translational and rotational dynamics in a single problem, requiring you to write F = ma for the linear motion and τ = Iα for the rotation, while linking them through constraints like strings wound around pulleys.

Key Concepts

• Torque: The rotational effect of a force. τ = rF sinθ, where r is the lever arm (perpendicular distance from axis to line of force). Torque causes angular acceleration.
• Rotational Inertia (I): Resistance to change in rotational motion. Depends on mass and its distribution: I = Σmr² for point masses. More mass far from the axis means larger I.
• Newton's Second Law for Rotation: τ_net = Iα. The net torque about an axis equals the rotational inertia about that axis times the angular acceleration.
• Rotational Kinematics: The same mathematical structure as linear kinematics: ω = ω₀ + αt, θ = ω₀t + ½αt², etc.
• Rolling Without Slipping: The condition v_cm = rω links linear and angular motion. If an object rolls without slipping, the point in contact with the ground is instantaneously at rest.
• Rotational Kinetic Energy: KE_rot = ½Iω². Rolling objects have both translational and rotational kinetic energy.
• Angular Momentum: L = Iω. Conserved when net external torque is zero. This explains why a spinning skater speeds up when pulling arms inward.

Vocabulary

• Torque (τ): A force's ability to cause rotation. Unit: N·m (not joule).
• Lever Arm: The perpendicular distance from the axis of rotation to the line of action of the force.
• Rotational Inertia (Moment of Inertia, I): The sum of mr² for all particles in an object. It measures resistance to angular acceleration.
• Angular Acceleration (α): The rate of change of angular velocity. Unit: rad/s².
• Rolling Without Slipping: A condition of pure rolling where the translational speed equals the tangential speed: v = rω.
• Angular Momentum (L): The rotational analog of linear momentum, L = Iω. Conserved in the absence of net external torque.
• Precession: The slow rotation of the axis of a spinning object around a vertical axis, caused by an external torque.

Essential Formulas

• τ = r*F*sinθ = I*α
• I = Σ(m*r²) (point masses)
• KE_rot = ½*I*ω²
• L = I*ω
• v = r*ω; a = r*α; x = r*θ
• ω = ω0 + α*t
• θ = ω0*t + ½*α*t²
• I_i*ω_i = I_f*ω_f (conservation of angular momentum)

Common Mistakes

• Using the Wrong Lever Arm: The lever arm is the perpendicular distance from the pivot to the line of action of the force, not the distance to the point where the force is applied unless the force is perpendicular.
• Forgetting Rolling Objects Have Both Translational and Rotational KE: When a sphere rolls down a hill, use KE_total = ½mv² + ½Iω². Do not use just ½mv².
• Confusing Linear and Angular Variables: Do not substitute linear velocity v into a formula that requires angular velocity ω. Convert using v = rω.
• Assuming Angular Momentum Is Conserved When External Torque Exists: Gravity can exert a torque if the force does not act through the pivot. Only claim L is conserved if τ_net_ext = 0.

AP Exam Strategies

• Choose the Axis to Eliminate Unknown Torques: If a force acts through your chosen pivot, its lever arm is zero and it produces no torque. This simplifies your τ_net equation.
• Draw an Extended Free-Body Diagram: Show the object and draw forces where they actually act. This helps you identify lever arms correctly.
• For Rolling, Use v_cm = rω and Energy Methods: Rolling without slipping problems are often easiest solved with conservation of energy, including both translational and rotational terms.
• Distinguish Between Linear and Angular Quantities: When a problem involves both translation and rotation, write F = ma for the CM motion and τ = Iα for the rotation separately.

Real-World Applications

• Figure Skating Spin: A skater pulls in their arms, reducing rotational inertia I. Since angular momentum is conserved, ω increases dramatically.
• Gyroscopes: The conservation of angular momentum makes gyroscopes stable reference platforms used in aircraft and spacecraft navigation.
• Tightening Bolts: Mechanics use long wrenches to increase the lever arm r, thereby increasing torque for the same applied force.