Linear vs Angular Motion

Polar Coordinates and Angular Position

Rotational motion is often described using polar coordinates, which specify a point's distance from a reference axis and its angle from that axis.

• r: Distance from the origin (radius)

• θ: Angular position (angle from x-axis)

• Position vector:

Example: A point at distance r from the origin and angle θ from the x-axis has coordinates (r, θ).

Comparing Linear and Angular Quantities

Many concepts in linear motion have direct analogs in rotational motion.

• Displacement: x (linear) → θ (angular)

• Velocity: v (linear) → ω (angular)

• Acceleration: a (linear) → α (angular)

• Mass: m (linear) → I (moment of inertia, rotational)

• Force: F (linear) → τ (torque, rotational)

Rotational Kinematics

Angular Displacement, Velocity, and Acceleration

Rotational kinematics describes the motion of objects rotating about a fixed axis.

• Angular displacement: (radians)

• Angular velocity:

• Angular acceleration:

Kinematic equations for constant angular acceleration:

Example: A wheel starts from rest () and accelerates at for : .

Arc Length and Angular Velocity

Relationship Between Arc Length and Angle

The arc length s swept out by a rotating object is related to its radius and angular displacement.

• Linear velocity:

• Angular velocity:

Example: For a point on a wheel of radius 0.5 m rotating at , .

Uniform Circular Motion

• In uniform circular motion, is constant.

• Linear speed is constant, but direction changes.

• Acceleration is always directed toward the center (centripetal acceleration).

Angular Momentum

Definition and Conservation

Angular momentum is a measure of the rotational motion of an object.

• Angular momentum:

• Moment of inertia: (sum over all mass elements)

• Conservation: If no external torque acts, angular momentum is conserved.

Example: A spinning ice skater pulls in her arms, reducing I and increasing ω to conserve L.

Kinetic Energy in Rotational Motion

• Rotational kinetic energy:

• Total kinetic energy:

Moments of Inertia

Definition and Calculation

The moment of inertia quantifies how mass is distributed relative to the axis of rotation.

• For a solid disk:

• For a thin rod about center:

Parallel Axis Theorem

Allows calculation of moment of inertia about any axis parallel to one through the center of mass.

• Where is the distance between axes.

Torque

Definition and Calculation

Torque is the rotational analog of force, causing changes in rotational motion.

• SI unit: Newton-meter (N·m)

• Torque depends on both the magnitude of the force and its distance from the axis of rotation.

Example: Applying a force of 10 N at a distance of 0.5 m from the axis: .

Torque and Angular Acceleration

• Newton's second law for rotation:

• Analogous to in linear motion.

Expanding Analogies: Linear vs Rotational Quantities

Worked Examples

Example 1: Pulley System

A rope is attached to a pulley; a mass hangs from the rope. Use conservation of energy to find the velocity of the mass.

• Potential energy lost:

• Kinetic energy gained:

• Relate and via

Example 2: Torque on a Rod

Calculate the torque produced by a force applied at the end of a rod.

• If force is perpendicular,

Additional Relationships

• Linear to angular force:

• Linear to angular momentum:

Additional info: These notes cover core concepts from chapters on rotational motion, including kinematics, energy, torque, and moment of inertia, suitable for college-level introductory physics.