Rotational Kinematics and Dynamics

Radian Measure for Angles

Angles in physics are often measured in radians, which relate the arc length of a circle to its radius.

• Definition: One radian is the angle subtended at the center of a circle by an arc whose length equals the radius of the circle.

• Conversion:

• Formula: , where is in radians, is arc length, is radius.

• Example: A full circle is radians ().

Angular Velocity

Angular velocity describes how quickly an object rotates, and in which direction.

• Average Angular Velocity:

• Instantaneous Angular Velocity:

• Vector Direction: Given by the right-hand rule (curl fingers in direction of rotation; thumb points in direction of ).

• Units: radians per second (rad/s).

Angular Acceleration

Angular acceleration measures the rate of change of angular velocity.

• Average Angular Acceleration:

• Instantaneous Angular Acceleration:

• Vector Direction: Same as angular velocity; use the right-hand rule.

• Units: radians per second squared (rad/s2).

Kinematic Equations for Constant Angular Acceleration

Analogous to linear kinematics, these equations describe rotational motion with constant angular acceleration.

Connections Between Linear and Angular Quantities

Linear and angular variables are related for objects in circular motion.

• Linear displacement:

• Linear velocity:

• Linear acceleration:

• Example: The tip of a rotating rod moves faster than a point near the axis.

Moment of Inertia

The moment of inertia () quantifies an object's resistance to changes in rotational motion.

• Discrete Masses:

• Continuous Mass Distribution:

• Depends on: Mass and how it is distributed relative to the axis of rotation.

• Units: kg·m2

• Example: Solid cylinder about center:

Rotational Kinetic Energy

Rotating objects possess kinetic energy due to their motion.

• Formula:

• Example: A spinning disk stores energy in its rotation.

The Parallel Axis Theorem

Used to find the moment of inertia about any axis parallel to one through the center of mass.

• Formula:

• = moment of inertia about center of mass axis; = distance between axes; = total mass.

• Example: Calculating for a rod about one end.

Torque, Work, and Angular Momentum

Torque

Torque () is the rotational equivalent of force, causing objects to rotate.

• Definition:

• Line of Action: The direction along which the force acts.

• Moment Arm: Perpendicular distance from axis to line of action.

• Vector Direction: Right-hand rule; direction of rotation caused by force.

• Units: N·m

Relation Between Torque and Angular Acceleration

Newton's second law for rotation relates torque to angular acceleration.

• Analogous to in linear motion.

Rotation with Translation

Objects can rotate and translate simultaneously, such as rolling wheels.

• Kinetic Energy:

• Forces and Torques: Both must be considered for rolling objects.

• Condition for Rolling Without Slipping:

Work for Rotation and Power

Work and power in rotational motion are analogous to their linear counterparts.

• Work:

• Relation to Rotational Kinetic Energy: Work done by net torque changes rotational kinetic energy.

• Power:

Angular Momentum

Angular momentum () is the rotational analog of linear momentum.

• Discrete:

• Continuous:

• Vector Direction: Right-hand rule; same as .

• Relation to Torque:

• Conservation: If net external torque is zero, is conserved.

• Example: Figure skater spins faster when arms are pulled in (conservation of ).

Fluid Mechanics

Fluid Properties and Density

Fluids (liquids and gases) are characterized by their ability to flow and take the shape of their container.

• Density (): (kg/m3)

• Example: Water at 4°C has kg/m3.

Fluid Pressure

Pressure is the force exerted per unit area by a fluid.

• Atmospheric Pressure: Pressure due to Earth's atmosphere; Pa at sea level.

• Hydrostatic Pressure:

• Units: Pascal (Pa) = N/m2

Pascal’s Law

A change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid.

• Application: Hydraulic lifts and brakes.

Absolute Pressure vs. Gauge Pressure

• Absolute Pressure: Total pressure at a point, including atmospheric pressure.

• Gauge Pressure: Pressure above atmospheric;

Manometers and Barometers

Devices used to measure fluid pressure.

• Manometer: Measures pressure difference using a column of fluid.

• Barometer: Measures atmospheric pressure, typically with mercury.

Buoyancy and Archimedes’ Principle

Objects in fluids experience an upward buoyant force equal to the weight of the fluid displaced.

• Buoyant Force:

• Archimedes’ Principle: The buoyant force equals the weight of the displaced fluid.

• Example: A floating object displaces its own weight in fluid.

Properties of an Ideal Fluid

• Incompressible

• No viscosity (frictionless)

• Steady (laminar) flow

• No turbulence

The Continuity Equation

Expresses conservation of mass for fluid flow.

• Where is cross-sectional area, is fluid speed.

• Example: Water speeds up when flowing from a wide pipe to a narrow one.

Bernoulli’s Equation and Principle

Relates pressure, velocity, and height in a flowing fluid (conservation of energy for fluids).

• Bernoulli’s Principle: Where fluid speed increases, pressure decreases.

• Example: Airplane wings generate lift due to pressure differences.