10.4 Rotational Kinetic Energy: Work and Energy Revisited
3 min read•june 18, 2024
and energy are key concepts in physics, describing how objects move in circular paths. They help us understand everything from spinning wheels to orbiting planets, connecting force, motion, and energy in rotational systems.
The equations for rotational work and kinetic energy mirror their linear counterparts, but with angular variables. This similarity allows us to apply familiar principles like conservation of energy to rotating objects, revealing the interplay between linear and rotational motion.
Rotational Work and Energy
Equation for rotational work
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- Rotational work () is the work done by a () acting over an ()
- Equation:
- Torque is the product of the force () and the perpendicular distance () from the axis of rotation to the line of action of the force
- Equation:
- Torque causes an object to rotate around an axis (door hinge, wheel axle)
- Angular displacement is the angle through which an object rotates, measured in (rev, deg)
- Substituting the equation for torque into the rotational work equation yields
- Rotational work depends on force, distance from axis, and angular displacement (opening a door, pedaling a bicycle)
- The applies to rotational motion, relating the work done to the change in
Calculation of rotational kinetic energy
- () is the kinetic energy associated with the rotational motion of an object
- Equation:
- () is a measure of an object's resistance to rotational acceleration (also called )
- Depends on the object's mass and its distribution relative to the axis of rotation
- For a point mass: , where is the mass and is the distance from the axis of rotation
- Objects with mass concentrated far from the axis have larger moments of inertia (figure skater with arms extended vs tucked in)
- Depends on the object's mass and its distribution relative to the axis of rotation
- () is the rate of change of angular displacement, measured in per second
- Equation:
- Represents how quickly an object is rotating (spinning wheel, orbiting planet)
- The allows calculation of about any axis parallel to an axis through the center of mass
Conservation of energy in linear vs rotational motion
- The states that energy cannot be created or destroyed, only converted from one form to another
- In a closed system, the total energy (sum of kinetic, potential, and other forms) remains constant
- For systems involving both linear and rotational motion, the total kinetic energy () is the sum of translational kinetic energy () and rotational kinetic energy ()
- Equation:
- Translational KE depends on mass and linear velocity (sliding block)
- Rotational KE depends on moment of inertia and (spinning top)
- Equation:
- When no external forces are acting on the system, the total energy remains constant
- Equation: , where is the potential energy
- Energy can be converted between forms (rolling ball down ramp, pendulum swinging)
- At top of ramp/swing: high potential energy, low kinetic energy
- During motion: potential energy converts to kinetic energy
- At bottom: low potential energy, high kinetic energy (split between translational and rotational)
Angular Momentum and Related Concepts
- is the rotational equivalent of linear momentum, representing the tendency of a rotating object to maintain its rotation
- describes the rate of change of angular velocity over time
- Centripetal force is the force that keeps an object moving in a circular path, directed toward the center of rotation
Key Terms to Review (32)
$ ext{ω}$: $ ext{ω}$ is a Greek letter that represents angular velocity, a measure of how quickly an object rotates around a fixed axis. It is a fundamental concept in the study of rotational motion and is closely related to other important physical quantities such as angular acceleration and rotational kinetic energy.
$
Delta K_{total} +
Delta U = 0$: $
Delta K_{total} +
Delta U = 0$ is a fundamental principle in the context of rotational kinetic energy and work-energy analysis. It states that the total change in kinetic energy plus the change in potential energy of a system is equal to zero, meaning that the energy is conserved during a process.
$ rac{d heta}{dt}$: $ rac{d heta}{dt}$ represents the angular velocity of an object, which is the rate of change of the angular displacement ($ heta$) with respect to time (t). It describes how quickly an object is rotating around a fixed axis or point.
$\tau$: $\tau$ is a symbol used to represent torque, which is the rotational equivalent of force. Torque is a measure of the rotational force that causes an object to rotate about an axis, fulcrum, or pivot. This term is crucial in understanding the concepts of angular acceleration, rotational kinetic energy, and RL circuits.
$\theta$: $\theta$ is a symbol commonly used to represent an angle in various physical contexts, especially in rotational dynamics and oscillatory motion. In the context of rotational kinetic energy, $\theta$ helps describe the orientation of an object and its angular displacement. In relation to a simple pendulum, $\theta$ is crucial for understanding the angular position of the pendulum relative to its resting vertical position, influencing calculations of potential and kinetic energy throughout its swing.
$K_{total}$: $K_{total}$, or total kinetic energy, refers to the sum of both translational kinetic energy and rotational kinetic energy of an object or system. It provides a comprehensive measure of the energy due to motion in all forms, highlighting the importance of rotational motion alongside linear motion in various physical scenarios. Understanding $K_{total}$ is crucial when analyzing systems involving objects that both translate and rotate, as it allows for a complete assessment of their energy dynamics.
$K_r$: $K_r$ is the rotational kinetic energy of an object, which represents the energy an object possesses due to its rotational motion around a fixed axis. This term is crucial in understanding the work and energy relationships involved in rotational dynamics.
$K_t$: $K_t$ is the rotational kinetic energy of an object, which is the energy an object possesses due to its rotational motion. It is a crucial concept in the study of rotational dynamics and the conservation of energy in rotating systems.
$r_\perp$: $r_\perp$ represents the perpendicular distance from the axis of rotation to a given point in a rotating object. This term is crucial when analyzing rotational motion, as it determines how far a point on the object is from the axis, directly affecting both linear and angular quantities such as velocity and acceleration. Understanding $r_\perp$ helps in calculating the moment of inertia and the rotational kinetic energy, connecting it with work done on the system and energy conservation principles.
$W_r$: $W_r$ represents the work done by a net torque on a rotating object, playing a vital role in understanding rotational dynamics and energy transfer. This concept connects directly to how energy is transformed and conserved during rotational motion, similar to linear motion but with angular equivalents. Essentially, $W_r$ helps quantify how much work is exerted through torque over a specific angle, linking it to the rotational kinetic energy of the object involved.
Angular acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It is a vector quantity, often measured in radians per second squared ($\text{rad/s}^2$).
Angular Acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It describes the rotational equivalent of linear acceleration, representing the change in the speed of rotation or the change in the direction of rotation of an object around a fixed axis.
Angular Displacement: Angular displacement is a measure of the change in the angular position of an object about a fixed axis or point of rotation. It describes the amount of rotation an object undergoes, typically expressed in units of radians or degrees.
Angular momentum: Angular momentum is the rotational analog of linear momentum, representing the quantity of rotation of an object. It is a vector quantity given by the product of an object's moment of inertia and its angular velocity.
Angular Momentum: Angular momentum is a measure of the rotational motion of an object around a fixed axis. It describes the object's tendency to continue rotating and the amount of torque required to change its rotational state. This concept is fundamental in understanding the dynamics of rotating systems and is crucial in various areas of physics, from the motion of satellites to the behavior of subatomic particles.
Angular velocity: Angular velocity is the rate of change of the rotation angle with respect to time. It is usually measured in radians per second (rad/s).
Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object rotating around a fixed axis or point. It describes the speed of rotational motion and is a vector quantity, indicating both the magnitude and direction of the rotation.
Law of conservation of energy: The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. This fundamental principle means that the total energy in a closed system remains constant over time, even as energy changes forms, such as from potential to kinetic energy. It plays a crucial role in understanding how different types of energy interact and helps to analyze various physical systems.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation.
Moment of Inertia: The moment of inertia is a measure of an object's resistance to rotational acceleration. It quantifies how an object's mass is distributed about its axis of rotation and determines the object's rotational dynamics, including angular acceleration, angular momentum, and rotational kinetic energy.
Parallel Axis Theorem: The parallel axis theorem is a principle used in rotational motion that allows us to calculate the moment of inertia of a rigid body about any axis parallel to an axis through its center of mass. This theorem states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass plus the product of the mass of the object and the square of the distance between the two axes. This concept is crucial in understanding how objects behave when they rotate around different axes.
Radians: Radians are a unit of angular measure used in mathematics and physics. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius.
Radians: Radians are a unit of angular measurement that describe the angle subtended by a circular arc. They provide a way to quantify the amount of rotation around a central point, and are particularly useful in the study of rotational motion and energy.
Rotational inertia: Rotational inertia, also known as the moment of inertia, is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on the object's mass distribution relative to the axis of rotation.
Rotational Inertia: Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It quantifies how difficult it is to change the rotational motion of an object around a fixed axis or point.
Rotational kinetic energy: Rotational kinetic energy is the energy possessed by a rotating object due to its angular motion. It is given by the formula $KE_{rot} = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
Rotational Kinetic Energy: Rotational kinetic energy is the energy possessed by an object due to its rotational motion. It is the energy an object has by virtue of being in a state of rotation, and it depends on the object's rotational inertia and angular velocity.
Rotational Work: Rotational work is the work done by a torque acting on an object undergoing rotational motion. It is the product of the torque and the angular displacement of the object, and it represents the energy transferred to or from the object during the rotational motion.
SI unit of torque: The SI unit of torque is the newton-meter (Nm), which measures the rotational force applied to an object. Torque quantifies the tendency of a force to rotate an object about an axis.
Torque: Torque is the rotational equivalent of force, representing the ability to cause an object to rotate about a specific axis or pivot point. It is the product of the force applied and the perpendicular distance between the axis of rotation and the line of action of the force, and it plays a crucial role in the study of rotational motion and equilibrium.
Work-energy theorem: The work-energy theorem states that the net work done by forces on an object is equal to the change in its kinetic energy. It is a fundamental principle connecting the concepts of work and energy.
Work-Energy Theorem: The Work-Energy Theorem states that the work done by all forces acting on an object equals the change in its kinetic energy. This relationship highlights how work and energy are interchangeable; when work is done on an object, it results in a change in that object's energy state. Understanding this theorem is crucial because it connects the concept of work with energy, showing how forces impact motion and energy transformations.