Rotational Dynamics: Torque and Moment of Inertia

• Rotational dynamics is the study of the motion of rotating objects and the forces that cause them.

Torque

• Torque refers to a twisting force that induces rotation.
• Torque is a vector.
• Magnitude of torque is calculated by τ = rFsinθ (r = distance from axis of rotation to force application point, F = force magnitude, θ = angle between force vector and lever arm).
• Counterclockwise rotation results in positive torque, while clockwise rotation causes negative torque.
• Net torque is the sum of all torques acting on an object.
• The SI unit for torque is the newton-meter (N⋅m).

Moment of Inertia

• Moment of inertia (I) measures an object's resistance to changes in rotation rate.
• This depends on mass distribution and the axis of rotation.
• For a single particle of mass m at distance r from the rotation axis, I = mr².
• For a system of particles, moment of inertia is the sum of each particle's moments of inertia: I = Σ mr².
• Moment of inertia for continuous objects involves integration.
• Parallel Axis Theorem: I = Icm + Md² (Icm = moment of inertia about the center of mass, M = total mass, d = distance between axes).

Rotational Kinetic Energy

• Rotational kinetic energy (KEr) is the kinetic energy from an object's rotation.
• It is given by KEr = (1/2)Iω² (I = moment of inertia, ω = angular velocity).
• Total kinetic energy of a rolling object sums translational and rotational kinetic energies.

Angular Momentum

• Angular momentum (L) measures an object's tendency to continue rotating.
• For a single particle, L = r × p = r × mv (r = position vector, p = linear momentum, m = mass, v = velocity).
• Magnitude of angular momentum is L = rmvsinθ (θ = angle between r and v).
• For a rigid object rotating around a fixed axis, L = Iω (I = moment of inertia, ω = angular velocity).
• Angular momentum is conserved if net external torque is zero.

Rolling Motion

• Rolling motion combines translational and rotational motion.
• For an object rolling without slipping, vcm = rω (vcm = center of mass velocity, r = object's radius, ω = angular velocity).
• Center of mass acceleration relates to angular acceleration as acm = rα.

Work and Power in Rotational Motion

• Work done by torque is given by W = τθ (τ = torque, θ = angular displacement).
• Power delivered by torque is P = τω (τ = torque, ω = angular velocity).

Relationship Between Torque and Angular Acceleration

• Net torque equals the product of moment of inertia and angular acceleration: Στ = Iα.
• This is the rotational equivalent of Newton's second law (F = ma).

Conservation of Angular Momentum

• If net external torque on a system equals zero, then total angular momentum stays constant.
• Li = Lf (Li and Lf are initial and final angular momenta).
• A spinning skater increases rotation by pulling arms closer due to conservation of angular momentum.

Examples of Rotational Motion

• Spinning top
• Rotating wheel
• Planet orbiting a star
• Spinning figure skater
• Rolling ball

Explore rotational dynamics, focusing on torque and moment of inertia. Learn how torque, a twisting force, affects rotation and how moment of inertia resists changes in rotation rate. Understand the formulas and units of measurement for these concepts.