Glossary

10.3 Dynamics of Rotational Motion: Rotational Inertia

3 min read•june 18, 2024

Rotational motion is all about spinning objects. It's like linear motion's cooler cousin, with its own set of rules and equations. Instead of moving in straight lines, we're dealing with things that rotate around an axis.

Inertia plays a big role in rotational motion. Just like how mass resists changes in linear motion, resists changes in spinning. The distribution of mass matters a lot here, affecting how easily an object can start or stop rotating.

Rotational Motion and Inertia

Concept of rotational inertia

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  • measures an object's resistance to rotational depends on the object's mass and mass distribution relative to the
  • Objects with more mass concentrated farther from the axis of rotation have greater rotational inertia (figure skater pulling arms in during a spin)
  • ([I](https://www.fiveableKeyTerm:I)[I](https://www.fiveableKeyTerm:I)) quantitatively measures an object's rotational inertia calculated using I=∑mr2I = \sum mr^2 or I=∫[r](https://www.fiveableKeyTerm:r)2dmI = \int [r](https://www.fiveableKeyTerm:r)^2 dm
    • mm represents mass of each particle or differential mass element
    • rr represents perpendicular distance from axis of rotation to each particle or mass element
  • Rotational inertia depends on object's shape and axis of rotation changing axis can change object's rotational inertia (rolling a cylinder about its central axis vs its end)
  • Hollow objects have more rotational inertia than solid objects of the same mass and size (hollow cylinder vs solid cylinder)
  • The allows calculation of about any axis parallel to an axis through the center of mass

Calculation of torque

  • Torque (τ\tau) is the rotational equivalent of force causing an object to rotate about an axis calculated using τ=rFsin⁡θ\tau = rF\sin\theta or τ=rF⊥\tau = rF_{\perp}
    • rr represents distance from axis of rotation to point where force is applied
    • FF represents magnitude of force
    • θ\theta represents angle between force vector and vector from axis of rotation to point of force application
    • F⊥F_{\perp} represents component of force perpendicular to lever arm
  • Net torque on an object determines its rotational acceleration (α\alpha) relationship between net torque and rotational acceleration: ∑τ=Iα\sum \tau = I\alpha
  • Torque can be positive (counterclockwise) or negative (clockwise) depending on direction of force (turning a doorknob)
  • Forces applied closer to axis of rotation produce less torque than forces applied farther away (pushing on a door near the hinge vs near the handle)

Linear vs rotational motion

  • Linear motion variables and their rotational counterparts:
    1. Displacement (xx) → (θ\theta)
    2. Velocity (vv) → (ω\omega)
    3. Acceleration (aa) → (α\alpha)
    4. Mass (mm) → Moment of inertia (II)
    5. Force (FF) → Torque (τ\tau)
  • Equations for constant acceleration:
    • Linear: x=x0+v0t+12at2x = x_0 + v_0t + \frac{1}{2}at^2, v=v0+atv = v_0 + at, v2=v02+2a(x−x0)v^2 = v_0^2 + 2a(x - x_0)
    • Rotational: θ=θ0+ω0t+12αt2\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2, ω=ω0+αt\omega = \omega_0 + \alpha t, ω2=ω02+2α(θ−θ0)\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)
  • Newton's Second Law:
    • Linear: ∑F=ma\sum F = ma
    • Rotational: ∑τ=Iα\sum \tau = I\alpha
  • Work-energy theorem:
    • Linear: W=ΔKE=12mvf2−12mvi2W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2
    • Rotational: W=ΔKErot=12Iωf2−12Iωi2W = \Delta KE_{rot} = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2

Angular Momentum and Conservation

  • (L) is the rotational analog of linear momentum, calculated as L = I
  • The is the distance from the axis of rotation at which the entire mass of an object can be considered concentrated for rotational calculations
  • states that in the absence of external torques, the total of a system remains constant
  • is the energy associated with an object's rotational motion, calculated as KE_rot = 1/2Iω²

Key Terms to Review (41)

Acceleration: Acceleration is the rate of change of velocity over time. It is a vector quantity, meaning it has both magnitude and direction.
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.
Axis of Rotation: The axis of rotation is an imaginary line about which an object rotates or pivots. This concept is fundamental to understanding rotational motion and its associated dynamics, kinematics, and conservation principles.
Conservation of Angular Momentum: Conservation of angular momentum is a fundamental principle in physics that states the total angular momentum of a closed system remains constant unless an external torque is applied. This principle governs the dynamics of rotational motion, the behavior of colliding extended bodies, and the unique properties of gyroscopic systems.
F⊥: F⊥, or the perpendicular force, is a key concept in the study of rotational dynamics and rotational inertia. It represents the component of a force that is perpendicular to the line of rotation, which is crucial in determining the rotational motion of an object.
I: The term 'I' refers to the moment of inertia, a fundamental concept in the study of rotational motion, wave intensity, electrical resistance, and circuit analysis. Moment of inertia describes an object's resistance to changes in its rotational motion, while intensity is a measure of the energy carried by a wave, and resistance is a measure of an object's opposition to the flow of electric current. Understanding the role of 'I' in these various contexts is crucial for comprehending the underlying principles of physics.
I = ∫r²dm: The equation $$I = \int r^2 dm$$ represents the moment of inertia, which is a measure of an object's resistance to changes in its rotational motion. It involves integrating the mass distribution of an object with respect to its distance from the axis of rotation, where $$r$$ is the perpendicular distance from the axis and $$dm$$ is the infinitesimal mass element. Understanding this equation is crucial in analyzing how different mass distributions affect rotational dynamics, particularly when considering how mass placement influences stability and motion.
I = Σmr²: I = Σmr² is a fundamental equation in the study of rotational motion, which describes the rotational inertia or moment of inertia of an object. Rotational inertia is a measure of an object's resistance to changes in its rotational motion, similar to how linear inertia describes an object's resistance to changes in its linear motion.
Kg⋅m²: kg⋅m² is a unit of rotational inertia, also known as moment of inertia, which is a measure of an object's resistance to changes in its rotational motion. It represents the product of an object's mass and the square of its distance from the axis of rotation.
Law of conservation of angular momentum: The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant. This principle is fundamental in analyzing rotational motion and interactions.
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.
R: In physics, 'r' typically represents the distance from a rotation axis or center of mass to a point of interest. This distance is crucial in understanding rotational motion, electric circuits, and how different components interact within a system. It plays a vital role in calculations for rotational inertia, resistance, and electromotive force, linking the concepts of linear and angular relationships in physics.
Rad/s: Radians per second (rad/s) is a unit of angular velocity, which describes the rate of change of an object's angular position over time. It represents the number of radians an object rotates through in one second, providing a measure of how quickly an object is spinning or rotating around a fixed axis.
Radius of Gyration: The radius of gyration is a measure of the distribution of mass within a rotating object. It represents the distance from the axis of rotation at which the object's mass would be concentrated to have the same rotational inertia as the actual object. The radius of gyration is a crucial parameter in understanding the dynamics of rotational motion and the concept of rotational inertia.
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.
V = v₀ + at: The equation v = v₀ + at describes the relationship between an object's initial velocity (v₀), its acceleration (a), and its final velocity (v) after a certain time (t) has elapsed. This equation is fundamental in understanding the dynamics of rotational motion and the concept of rotational inertia.
V² = v₀² + 2a(x - x₀): The equation v² = v₀² + 2a(x - x₀) is a fundamental relationship in physics that describes the motion of an object under constant acceleration. It relates the final velocity (v), initial velocity (v₀), acceleration (a), and the change in position (x - x₀) of an object.
W = ΔKE = ½mv²ᶠ - ½mv²ᵢ: This equation represents the work-energy principle, stating that the work done on an object is equal to the change in its kinetic energy. It connects the concepts of work and energy by showing how the force applied over a distance results in changes to the motion of an object, particularly in rotational motion where mass distribution plays a key role.
W = ΔKEᵣₒₜ = ½Iω²ᶠ - ½Iω²ᵢ: The work done on a rotating object is equal to the change in its rotational kinetic energy, which can be calculated as the difference between the final and initial rotational kinetic energies.
X = x₀ + v₀t + ½at²: The equation x = x₀ + v₀t + ½at² describes the position of an object in motion under constant acceleration over time. It connects initial position, initial velocity, acceleration, and time to determine the final position of the object. Understanding this equation is crucial for analyzing both linear and rotational motion, as it helps describe how objects move and change their positions in relation to forces acting on them.
α: α, or alpha, is a variable used to represent various physical quantities in different contexts. In the fields of rotational dynamics and thermal expansion, α holds specific meanings and plays important roles in understanding the underlying principles.
θ (Theta): Theta (θ) is a fundamental mathematical symbol used to represent an angle in various contexts, including rotation, wave interference, and resolution limits. It is a Greek letter that serves as a variable or parameter to quantify and analyze angular relationships and phenomena.
θ = θ₀ + ω₀t + ½αt²: This equation represents the angular position of an object undergoing rotational motion, where θ is the angular position, θ₀ is the initial angular position, ω₀ is the initial angular velocity, t is the time, and α is the angular acceleration. This equation is used to describe the dynamics of rotational motion, particularly in the context of rotational inertia.
Στ = Iα: The equation Στ = Iα describes the relationship between torque (Στ), rotational inertia (I), and angular acceleration (α) in rotational motion. It indicates that the net torque acting on an object is equal to the product of its rotational inertia and its angular acceleration, highlighting how mass distribution and rotational forces affect an object's motion around an axis.
τ (Tau): Tau (τ) is a fundamental physical quantity that represents the torque, or rotational force, acting on an object. It is a vector quantity that describes the tendency of a force to cause rotational motion around a specific axis or point. Tau is a crucial concept in the study of rotational dynamics and is essential for understanding the behavior of rigid bodies undergoing rotational motion.
τ = rF⊥: The equation τ = rF⊥ represents the relationship between the torque (τ) acting on an object, the perpendicular distance (r) from the axis of rotation to the line of action of the force, and the component of the force (F⊥) that is perpendicular to the line connecting the axis and the point of application of the force. This equation is a fundamental concept in the dynamics of rotational motion and understanding rotational inertia.
τ = rFsinθ: τ = rFsinθ is a fundamental equation in the study of rotational dynamics, which describes the relationship between the torque (τ) acting on an object, the distance (r) from the axis of rotation to the point of application of the force (F), and the angle (θ) between the force and the line connecting the axis to the point of application.
ω: Omega (ω) is a Greek letter that represents angular velocity, a fundamental concept in rotational motion, wave energy, and electrical circuits. It describes the rate of change of angular displacement, the dynamics of rotational inertia, the intensity of waves, and the electromotive force in electrical systems.
ω = ω₀ + αt: The angular velocity of a rotating object can be expressed as the sum of its initial angular velocity (ω₀) and the product of its angular acceleration (α) and the time elapsed (t). This equation describes the relationship between these key rotational motion variables.
ω² = ω₀² + 2α(θ - θ₀): This equation represents the angular acceleration of a rotating object, where $ω^2$ is the final angular velocity, $ω_0^2$ is the initial angular velocity, $α$ is the angular acceleration, $θ$ is the final angular position, and $θ_0$ is the initial angular position. This equation is used to describe the dynamics of rotational motion and is closely related to the concept of rotational inertia.
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