Chapter 10: Rotational Kinematics

Introduction to Rotational Motion

Rotational motion is a fundamental concept in physics, describing the movement of objects around a fixed axis. Unlike translational motion, which involves movement from one point to another, rotational motion focuses on how objects spin or rotate.

• Translational motion refers to the movement of an object's center of mass through space.

• Rotational motion involves the rotation of an object about an axis, with its center of mass remaining stationary.

• Rotational motion has its own kinematics, dynamics, energy, and momentum analogous to those in translational motion.

Rigid Bodies and Mass Distribution

In rotational motion, we often deal with rigid bodies, which are objects that do not deform during rotation. The mass of a rigid body is distributed throughout its volume, affecting its rotational properties.

• Rigid body: An object with a fixed shape that does not deform as it rotates.

• Center of mass: The point where the mass of the body is considered to be concentrated for translational motion.

• For rotational motion, the distribution of mass relative to the axis of rotation is crucial.

Angular Quantities

Rotational motion is described using angular quantities, which are analogous to linear quantities in translational motion.

• Angular displacement (): Measures how much an object has rotated, typically in radians.

• Angular velocity (): Describes how fast an object rotates, in radians per second.

• Angular acceleration (): The rate of change of angular velocity, in radians per second squared.

Conversion Between Degrees and Radians

• One full rotation: radians

• To convert degrees to radians:

Relationship Between Linear and Angular Quantities

Linear and angular quantities are related through the radius of rotation.

• Linear velocity ():

• Linear acceleration ():

• Points farther from the axis of rotation travel a greater distance and have higher linear velocity for the same angular velocity.

Rotational Kinematic Equations

Rotational kinematics can be described using equations analogous to those for linear motion, substituting angular quantities for linear ones.

Sign Convention in Rotational Motion

Unlike linear motion, rotational motion uses clockwise and counterclockwise directions. A sign convention is necessary to distinguish positive and negative directions.

• Counterclockwise rotation is typically defined as the positive direction.

• Clockwise rotation is defined as the negative direction.

Chapter 8: Rotational Dynamics

Introduction to Rotational Dynamics

Rotational dynamics studies the forces and torques that cause rotational motion. The rotational analog of force is torque, and the rotational analog of mass is moment of inertia.

• Torque (): The rotational equivalent of force, causing angular acceleration.

• Moment of inertia (): The rotational equivalent of mass, representing an object's resistance to changes in rotational motion.

Torque and Lever Arm

Torque depends on both the magnitude of the force and its distance from the axis of rotation (lever arm).

• Torque formula: , where is the lever arm and is the angle between the force and lever arm.

• If the force is parallel to the lever arm (), no torque is produced.

Rotational Newton's Second Law

Newton's second law for rotation relates torque to angular acceleration and moment of inertia.

• Just as for linear motion, torque is proportional to angular acceleration and moment of inertia.

Moment of Inertia

The moment of inertia depends on the mass distribution and the axis of rotation. Objects with mass farther from the axis have larger moments of inertia.

• Point mass:

• Extended bodies:

• Common moments of inertia:

  • Solid disk (about center):

  • Thin rod (about center):

Parallel Axis Theorem

When rotating about an axis not passing through the center of mass, the parallel axis theorem is used.

• is the moment of inertia about the center of mass, is total mass, is the distance between axes.

Examples and Applications

• Dumbbell system: For two masses and separated by distance , the moment of inertia about an axis halfway between them is .

• Pulley with friction: If a torque is applied to a pulley, and frictional torque is present, use to solve for angular acceleration.

• Bucket on a pulley: When a mass hangs from a cord wrapped around a pulley, both the linear acceleration of the mass and the angular acceleration of the pulley can be found using Newton's laws and rotational dynamics.

Sample Table: Moments of Inertia for Common Shapes

Key Equations Summary

• Angular displacement:

• Angular velocity:

• Torque:

• Rotational Newton's second law:

• Moment of inertia (point mass):

• Parallel axis theorem:

Example Application: Pulley System with Friction

• Given: Pulley mass kg, radius m, frictional torque N·m, angular speed rad/s in s.

• Find: Moment of inertia using and .

Additional info: Some equations and values were inferred from context and standard physics curriculum.