Glossary
Angular Displacement
The angle through which an object rotates about an axis. It is a vector quantity, with its direction indicating the axis of rotation.
Example:
A merry-go-round completing half a turn undergoes an angular displacement of π radians.
Angular Displacement (Δθ)
The angle through which an object rotates about an axis, measured in radians. It describes the change in angular position.
Example:
A merry-go-round completing half a turn has an angular displacement of π radians.
Area Under the Curve (Torque-Angle Graph)
On a torque-angle graph, the area enclosed between the torque curve and the angular position axis represents the total work done by the torque.
Example:
If a robot arm's motor torque varies as it rotates, calculating the area under the curve of its torque-angle graph will give the total work performed by the motor.
Conservation of Energy (Rotational)
In an isolated system, the total mechanical energy (including rotational kinetic energy) remains constant, meaning energy can transform between forms but is neither created nor destroyed.
Example:
A rolling ball descending a ramp demonstrates the Conservation of Energy, as its gravitational potential energy converts into both translational and rotational kinetic energy.
Constant Torque (Work done by)
When the applied torque remains uniform throughout the rotation, the work done simplifies to the product of the constant torque and the total angular displacement.
Example:
If a motor applies a constant torque to a fan blade, the work done is simply that torque multiplied by how many radians the blade spins.
Energy Conservation (in rotational systems)
The principle that the total mechanical energy (sum of kinetic and potential energy, including rotational kinetic energy) of an isolated rotational system remains constant if only conservative forces and torques do work.
Example:
A spinning ice skater pulling their arms in demonstrates energy conservation, as their rotational kinetic energy increases while their angular momentum is conserved.
Graphical Work Analysis
A method to determine the work done by a torque by finding the area under its curve when plotted against angular position. Positive areas indicate positive work, and negative areas indicate negative work.
Example:
If a torque vs. angular position curve shows a triangular shape, calculating the area of that triangle directly gives the work done.
Negative Work (Rotational)
Work done when the torque and angular displacement are in opposite directions, resulting in energy being transferred out of the system and typically decreasing its rotational kinetic energy.
Example:
Friction acting on a spinning top does negative work, gradually slowing it down until it stops.
Positive Work (Rotational)
Work done when the torque and angular displacement are in the same direction, resulting in energy being transferred into the system and typically increasing its rotational kinetic energy.
Example:
When a child pushes a swing in the direction it's already moving, they do positive work, making the swing go higher.
Radians (rad)
The standard SI unit for measuring angles, especially crucial for calculations involving rotational motion, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
Example:
When calculating the work done by a torque, it's essential to convert degrees to radians to ensure the formula W = τΔθ yields correct results.
Rotational Inertia (I)
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 solid disk has less rotational inertia than a hoop of the same mass and radius, making the disk easier to spin up.
Rotational Kinetic Energy
The energy an object possesses due to its rotation, dependent on its moment of inertia and angular velocity. Work done by torque can change this energy.
Example:
A spinning flywheel stores a significant amount of rotational kinetic energy, which can be used to power machinery.
Rotational Kinetic Energy (KE_rot)
The energy an object possesses due to its rotation, dependent on its rotational inertia and angular speed. It is given by the formula KE = (1/2)Iω².
Example:
A spinning flywheel stores a significant amount of rotational kinetic energy, which can be used to power machinery.
Torque
A rotational force that can transfer energy to or from an object when applied over an angular displacement. It is the rotational equivalent of linear force, causing an object to rotate or change its rotational motion.
Example:
When you use a wrench to tighten a bolt, the force you apply to the wrench handle creates a torque that rotates the bolt.
Torque (τ)
The rotational equivalent of force, causing an object to rotate or change its rotational motion. It is calculated as the product of force and the perpendicular distance from the pivot.
Example:
When you use a wrench to tighten a bolt, the force you apply to the end of the wrench creates a torque that turns the bolt.
Torque-Angle Graph
A graphical representation showing how the magnitude of a torque changes with respect to the angular position of a rotating object. The horizontal axis represents angular position, and the vertical axis represents torque.
Example:
Analyzing a Torque-Angle Graph for a winding spring can show how the torque required to turn it increases as the spring gets tighter.
Work Done by Torque
The energy transferred to or from a rigid system by a torque, depending on the magnitude of the torque and the angular displacement it rotates through. It is calculated by integrating torque with respect to angular displacement.
Example:
Lifting a bucket by winding a rope around a pulley involves work done by torque on the pulley, transferring energy to the bucket.
Work Done by a Torque (W)
The energy transferred to or from a rotating system by a torque acting over an angular displacement. It is a scalar quantity measured in joules.
Example:
If a motor applies a constant torque to spin a fan blade, the work done by the torque is the energy transferred to the fan, increasing its rotational speed.
Work-Energy Theorem (Rotational)
States that the net work done by all torques on a rotating object equals the change in its rotational kinetic energy. It links work and energy changes in rotational systems.
Example:
If a spinning figure skater pulls their arms in, the internal work done changes their rotational inertia, and the Work-Energy Theorem explains the resulting change in their rotational kinetic energy.
Work-Energy Theorem for Rotational Motion
States that the net work done by all torques on a rigid body equals the change in its rotational kinetic energy.
Example:
If a brake applies a torque to a spinning wheel, the work-energy theorem for rotational motion can be used to find how much the wheel's rotational kinetic energy decreases.
Work-Torque Relationship (W = τΔθ)
The fundamental equation stating that the work done by a constant torque is the product of the torque's magnitude and the angular displacement. This relationship quantifies energy transfer in rotational motion.
Example:
To calculate the energy needed to spin a heavy flywheel through a certain angle, you would use the Work-Torque Relationship by multiplying the applied torque by the angle it turns.