Glossary
Angular Acceleration
The rate at which an object's angular velocity changes over time. It is directly proportional to the net torque applied and inversely proportional to the rotational inertia.
Example:
When a bicycle wheel speeds up from rest, it experiences angular acceleration.
Angular Momentum
A measure of the amount of rotation an object has, analogous to linear momentum. For a rigid body, it is the product of its rotational inertia and angular velocity (L = Iω).
Example:
A planet orbiting the sun conserves its angular momentum as it moves in its elliptical path.
Axis of Rotation
The imaginary line around which an object rotates. The distribution of mass relative to this axis is crucial for determining rotational inertia.
Example:
For a spinning top, the axis of rotation runs vertically through its center, from its tip to its base.
Axis of Rotation
The imaginary line about which an object rotates. The distribution of mass relative to this axis is crucial for determining rotational inertia.
Example:
For a spinning door, the hinges define the axis of rotation.
Calculus for Rotational Inertia
The mathematical method using integration ($I = \int r^2 dm$) to calculate the rotational inertia of continuous objects (solids) by summing the contributions of infinitesimal mass elements.
Example:
Deriving the rotational inertia of a uniform rod about its end requires using calculus for rotational inertia to integrate over its length.
Center of Mass
The unique point where the weighted average of all the masses in a system is located. For a rigid body, rotational inertia is minimum when the axis of rotation passes through this point.
Example:
When balancing a broom on your finger, you're trying to place your finger directly under its center of mass to keep it stable.
Center of Mass (CM)
The unique point where the weighted average of all the masses of a system is located. For a rigid body, it's the point where the entire mass of the object can be considered to be concentrated for translational motion analysis, and it's the axis about which rotational inertia is minimized.
Example:
A baseball bat's center of mass is closer to the thicker end, which is why it feels balanced differently when held at various points.
Moment of Inertia
Another name for rotational inertia, representing an object's resistance to rotational acceleration. It quantifies how difficult it is to start or stop an object from spinning.
Example:
The moment of inertia of a bicycle wheel is greater than that of a solid disk of the same mass and radius because more of its mass is concentrated at the rim.
Parallel Axis Theorem
A theorem that relates the rotational inertia of a rigid body about any axis to its rotational inertia about a parallel axis passing through its center of mass. It is given by the formula $I' = I_{cm} + Md^2$.
Example:
To find the rotational inertia of a door rotating about its hinges, you could use the Parallel Axis Theorem if you know its inertia about an axis through its center.
Parallel Axis Theorem
A theorem that relates the rotational inertia of a rigid body about any axis to its rotational inertia about a parallel axis passing through its center of mass. It is given by I' = I_cm + Md^2.
Example:
To find the rotational inertia of a door rotating about its top edge, you can use the Parallel Axis Theorem if you know its rotational inertia about an axis through its center of mass.
Point Mass
An idealized object that has mass but no dimensions, often used to simplify calculations in physics. For a *point mass*, rotational inertia is calculated as $I = mr^2$.
Example:
When calculating the rotational inertia of a satellite orbiting Earth, the satellite can often be approximated as a point mass.
Rigid Body
An idealized object that does not deform under the influence of external forces, maintaining a fixed shape and size. In rotational mechanics, all points within a rigid body maintain their relative positions.
Example:
When analyzing the rotation of a spinning top, it's often modeled as a rigid body because its shape doesn't change significantly during motion.
Rotational Inertia
A measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on both the total mass and how that mass is distributed relative to the axis of rotation.
Example:
A gymnast performing a flip tucks their body in to reduce their rotational inertia, allowing them to spin faster.
Rotational Inertia
A measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on the object's mass and how that mass is distributed relative to the axis of rotation.
Example:
A figure skater pulls their arms in to decrease their rotational inertia, allowing them to spin faster.
Rotational Kinetic Energy
The energy an object possesses due to its rotational motion, analogous to translational kinetic energy. It is given by K_rot = (1/2)Iω^2.
Example:
A spinning flywheel stores rotational kinetic energy that can be used to power machinery.
Thin Cylindrical Shells, Disks, Coaxial Rings/Shells
Common idealized shapes representing objects like pipes, wheels, or nested cylinders, for which rotational inertia formulas are derived using calculus, typically about their central axis.
Example:
A car wheel can be approximated as a disk or a combination of thin cylindrical shells for calculating its rotational inertia.
Thin Rods
A common idealized shape used in rotational inertia problems, often assumed to have uniform or non-uniform density, for which rotational inertia can be derived using calculus.
Example:
Calculating the rotational inertia of a spinning baton, which can be approximated as a thin rod, often involves specific formulas derived from integration.
Torque
A rotational force that causes an object to rotate or change its rotational motion. It is the rotational equivalent of linear force.
Example:
Applying a torque to a wrench turns a bolt, causing it to rotate.
Torque
The rotational equivalent of force, causing an object to undergo angular acceleration. It is the product of force and the perpendicular distance from the axis of rotation to the line of action of the force.
Example:
Applying a force to a wrench handle to tighten a bolt creates torque, causing the bolt to rotate.