Glossary
Angular Acceleration ($\alpha$)
The rate at which an object's angular velocity changes over time, indicating how quickly its rotation is speeding up or slowing down.
Example:
When a bicycle wheel starts from rest and spins up, it experiences a positive angular acceleration.
Angular acceleration
The rate at which an object's angular velocity changes over time. It indicates how quickly an object speeds up or slows down its rotation, measured in radians per second squared (rad/s²).
Example:
A spinning figure skater pulling their arms in experiences a large angular acceleration as their rotational speed rapidly increases.
Linear Analysis
The examination of the translational motion of an object's center of mass. This involves applying Newton's second law for linear motion ($\Sigma \vec{F} = m\vec{a}$) to determine linear acceleration.
Example:
When analyzing a car skidding to a stop, a linear analysis would focus on the forces (friction, gravity, normal) acting on its center of mass to determine its linear deceleration.
Mass Distribution
Refers to how an object's mass is spread out relative to its axis of rotation, significantly affecting its rotational inertia.
Example:
A hollow cylinder has a larger rotational inertia than a solid cylinder of the same mass and radius because its mass distribution is farther from the center.
Net Torque
The vector sum of all individual torques acting on an object, which determines its angular acceleration.
Example:
When you push on a door at two different points, the combined effect of these pushes creates a net torque that causes the door to swing open.
Net torque
The vector sum of all individual torques acting on an object. A non-zero net torque is required to change an object's angular velocity, causing angular acceleration.
Example:
If you push a door open, the net torque is the sum of the torque you apply and any opposing friction torque from the hinges.
Newton's Second Law for Rotation
States that the net torque acting on an object is directly proportional to its angular acceleration and inversely proportional to its rotational inertia ($\Sigma \tau = I\alpha$). It is the rotational analogue of Newton's second law for linear motion.
Example:
Using Newton's Second Law for Rotation, engineers can calculate the angular acceleration of a turbine blade given the net torque applied by the steam and the blade's rotational inertia.
Newton's Second Law for Rotation
States that the net torque on an object is equal to the product of its rotational inertia and angular acceleration ($\sum au = I \alpha$).
Example:
Using Newton's Second Law for Rotation, you can calculate how quickly a spinning top will slow down if friction applies a constant torque.
Point Mass
An idealized mass concentrated at a single point, often used as a component when calculating the rotational inertia of a system of discrete masses.
Example:
To find the rotational inertia of a barbell, you can treat each weight plate as a point mass at a certain distance from the bar's center.
Rolling Without Slipping
A condition where a rolling object's point of contact with the surface has zero relative velocity, linking its linear and angular motion.
Example:
A car tire on dry pavement typically exhibits rolling without slipping, meaning its linear speed is directly related to its angular speed.
Rotational Analysis
The examination of an object's rotational motion about its axis of rotation. This involves applying Newton's second law for rotation ($\Sigma \tau = I\alpha$) to determine angular acceleration.
Example:
To understand how fast a bicycle wheel spins up, a rotational analysis would consider the torque from the chain and the wheel's rotational inertia.
Rotational Inertia ($I$)
A measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion.
Example:
A figure skater pulls their arms in to decrease their rotational inertia, allowing them to spin much faster.
Rotational inertia
A measure of an object's resistance to changes in its angular velocity. It depends on the object's mass and how that mass is distributed relative to the axis of rotation, measured in kilogram-meters squared (kg·m²).
Example:
A merry-go-round with children spread out on the edges has a larger rotational inertia than if they were all clustered in the center, making it harder to start spinning.
Tangential Acceleration ($a$)
The linear acceleration of a point on a rotating object, directed along the tangent to its circular path.
Example:
As a car tire speeds up, a point on its tread experiences tangential acceleration in the direction of motion.
Tangential Speed ($v$)
The linear speed of a point on a rotating object, measured along the tangent to its circular path.
Example:
The outer edge of a spinning merry-go-round has a greater tangential speed than a point closer to the center.
Torque
A twisting force that causes or tends to cause rotation. It is the rotational equivalent of linear force, calculated as the product of force and the perpendicular distance from the axis of rotation.
Example:
Applying a wrench to a stubborn bolt requires sufficient torque to loosen it.
Torque ($ au$)
The rotational equivalent of force, which causes an object to rotate or change its rotational motion.
Example:
Applying a force to the end of a wrench to tighten a bolt creates a torque that rotates the bolt.