Glossary
Angular Acceleration
The rate at which angular velocity changes over time, indicating how quickly the rotation speed is increasing or decreasing.
Example:
When a figure skater pulls their arms in, their angular acceleration increases their spin rate dramatically.
Angular Acceleration (α)
The rate of change of an object's angular velocity, indicating how quickly its rotation speed is changing.
Example:
When a bicycle wheel speeds up from rest, it experiences positive angular acceleration.
Angular Displacement
The angle, measured in radians, through which a point or line has been rotated about a specific axis.
Example:
A car tire completes 10 full rotations, resulting in an angular displacement of 20π radians.
Angular Displacement (Δθ)
The change in the angle of a rotating object, measured from a reference line, indicating how far it has rotated.
Example:
A car tire completing one full rotation undergoes an angular displacement of 2π radians.
Angular Momentum (L)
The rotational equivalent of linear momentum, measuring an object's tendency to continue rotating.
Example:
A planet orbiting the sun possesses angular momentum due to its orbital motion.
Angular Velocity
The rate of change of angular position with respect to time, indicating how fast an object is rotating.
Example:
A ceiling fan spinning at a high speed has a large angular velocity.
Angular Velocity (ω)
The rate at which an object's angular position changes, indicating how fast it is rotating and in what direction.
Example:
A spinning record player rotates with a constant angular velocity.
Center of Mass/Gravity
The unique point where the weighted average of all the masses of a system is located, effectively where the entire mass or weight of an object can be considered to act.
Example:
A tightrope walker adjusts their body to keep their center of mass directly above the rope, maintaining balance.
Conservation of Angular Momentum
A fundamental principle stating that the total angular momentum of a system remains constant if no external torque acts on it.
Example:
When a diver tucks their body mid-air, their conservation of angular momentum causes them to spin faster.
Constant Angular Acceleration Equations
A set of kinematic equations used to describe rotational motion when the angular acceleration remains constant over time.
Example:
To find out how many rotations a flywheel makes while speeding up uniformly, you would use the constant angular acceleration equations.
Equilibrium (Rotational)
A state where the net torque acting on an object is zero, resulting in no change in its angular velocity (either remaining at rest or rotating at a constant angular velocity).
Example:
A balanced seesaw demonstrates rotational equilibrium because the torques on both sides cancel out.
Graphical Analysis
The interpretation of graphs (e.g., angular displacement vs. time, angular velocity vs. time) to understand the relationships between rotational kinematic quantities.
Example:
By performing graphical analysis on an angular velocity-time graph, a student can determine the angular acceleration from its slope.
Kinematic Equations (Rotational)
A set of equations that describe the motion of rotating objects with constant angular acceleration, analogous to linear kinematic equations.
Example:
Using the kinematic equations for rotation, one can predict how long it takes for a spinning top to come to rest given its initial angular velocity and constant angular deceleration.
Linear Motion Analogy
The conceptual parallel between linear kinematic quantities (displacement, velocity, acceleration) and their rotational counterparts, allowing for similar mathematical relationships.
Example:
Just as linear velocity is the rate of change of linear position, angular velocity is the rotational equivalent in the linear motion analogy.
Moment of Inertia (I)
The rotational equivalent of mass, quantifying an object's resistance to changes in its rotational motion, dependent on its mass distribution and axis of rotation.
Example:
A figure skater pulls their arms in to decrease their moment of inertia, allowing them to spin faster.
Newton's Second Law for Rotation
States that the net torque acting on an object is directly proportional to its moment of inertia and angular acceleration ($τ_{net} = Iα$).
Example:
This law explains why a larger net torque is required to rapidly accelerate a massive Ferris wheel compared to a small bicycle wheel.
Radians
The standard unit for measuring angles in physics, defined as the angle subtended at the center of a circle by an arc equal in length to the radius.
Example:
When calculating the distance a bicycle wheel travels, its angular displacement must be in radians for the formulas to work correctly.
Rigid Systems
Objects or systems that maintain their shape and size during motion, meaning all points within the object maintain their relative positions.
Example:
A spinning CD is a good example of a rigid system because all parts of the disk rotate together without deforming.
Rotational Kinematics
The study of how objects move in circles or rotate, focusing on angular displacement, velocity, and acceleration without considering the forces causing the motion.
Example:
Understanding the rotational kinematics of a spinning planet helps predict its position at a future time.
Rotational Kinetic Energy (K_rot)
The energy an object possesses due to its rotation, dependent on its moment of inertia and angular velocity.
Example:
A spinning flywheel stores a significant amount of rotational kinetic energy that can be used to power machinery.
Rotational Statics
The study of objects that are in rotational equilibrium, meaning the net torque acting on them is zero and they are not undergoing any angular acceleration.
Example:
An architect designing a stable bridge must ensure all components are in rotational statics to prevent unwanted rotation or collapse.
Rotational Work-Energy Theorem
States that the net work done by torques on a rotating object equals the change in its rotational kinetic energy.
Example:
The rotational work-energy theorem can be used to calculate the final angular speed of a grinding wheel after a motor applies a certain amount of work to it.
Sign Convention
A rule for assigning positive or negative values to angular quantities (displacement, velocity, acceleration) based on their direction of rotation, typically counterclockwise as positive.
Example:
If a wheel spins clockwise, its angular velocity would be assigned a negative value according to the standard sign convention.
Torque (τ)
The rotational equivalent of force, torque is what causes an object to rotate or change its rotational motion.
Example:
Applying a force to the end of a long lever arm to loosen a stubborn bolt generates significant torque.