10.7 Newton’s Second Law for Rotation
3 min read•june 24, 2024
Rotational dynamics applies Newton's Second Law to spinning objects. It connects , , and , explaining why a figure skater spins faster with arms tucked in or why it's easier to turn a longer wrench.
Understanding rotational dynamics helps us grasp everyday phenomena like bicycle pedaling or door opening. It shows how changing an object's mass distribution or the force applied affects its rotation, making complex motions more intuitive.
Rotational Dynamics
Application of Newton's Second Law for Rotation
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- relates net torque to and :
- represents the net torque acting on the object (door handle)
- represents the object's moment of inertia, determined by its mass distribution and (figure skater spinning with arms extended or pulled in)
- represents the object's resulting angular acceleration (spinning tire)
- To calculate angular acceleration, rearrange the equation:
- Determine the net torque acting on the object by summing all torques (forces applied to a wrench)
- Calculate the object's moment of inertia based on its shape, mass, and axis of rotation ( vs )
- Divide the net torque by the moment of inertia to find the angular acceleration (spinning wheel after applying brake)
Effects of torque and moment of inertia
- Angular acceleration is directly proportional to net torque
- Increasing net torque while keeping moment of inertia constant increases angular acceleration proportionally (pedaling a bicycle harder)
- Decreasing net torque while keeping moment of inertia constant decreases angular acceleration proportionally (reducing force on a rotating door)
- Angular acceleration is inversely proportional to moment of inertia
- Increasing moment of inertia while keeping net torque constant decreases angular acceleration proportionally (extending arms while spinning)
- Decreasing moment of inertia while keeping net torque constant increases angular acceleration proportionally (pulling arms in while spinning)
- Changes in mass distribution affect moment of inertia and angular acceleration under a given net torque (moving weights closer to or farther from the axis of rotation)
Vector notation in rotational dynamics
- Torque, angular acceleration, and moment of inertia are vector quantities in rotational dynamics
- Torque is a vector pointing along the axis of rotation, following the right-hand rule (turning a doorknob clockwise)
- Angular acceleration is a vector pointing along the axis of rotation, in the same direction as net torque (wheel spinning faster)
- Moment of inertia is a scalar quantity depending on mass distribution and axis of rotation (hoop vs disc)
- The vector form of Newton's Second Law for Rotation:
- Angular acceleration vector direction is determined by net torque vector direction (counterclockwise torque produces counterclockwise acceleration)
- Angular acceleration vector magnitude is determined by net torque vector magnitude divided by moment of inertia (larger torque or smaller moment of inertia produces larger acceleration)
Advanced Concepts in Rotational Dynamics
- is conserved in the absence of external torques, relating to the and angular velocity of a system
- Rotational inertia, also known as moment of inertia, determines an object's resistance to rotational acceleration
- The allows calculation of rotational inertia about any axis parallel to an axis through the center of mass
- The relates the moment of inertia of a planar object to its moments of inertia about two perpendicular axes in its plane
Key Terms to Review (26)
Angular acceleration: Angular acceleration is the rate of change of angular velocity over time. It describes how quickly an object is rotating or spinning.
Angular Acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It describes the rotational analog of linear acceleration, quantifying the change in the rotational motion of an object around a fixed axis or point.
Angular momentum: Angular momentum is a measure of the quantity of rotation of an object and is a vector quantity. It is given by the product of the moment of inertia and angular velocity.
Angular Momentum: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is the measure of an object's rotational inertia and its tendency to continue rotating around a specific axis. Angular momentum is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in understanding the behavior of rotating systems across various topics in physics.
Axis of Rotation: The axis of rotation is the imaginary line around which an object or system rotates. It is the fixed point or line that an object pivots or spins around as it undergoes rotational motion.
Euler: Euler is a fundamental concept in physics, particularly in the context of rotational dynamics. It refers to the mathematical framework developed by the renowned Swiss mathematician Leonhard Euler, which provides a comprehensive approach to describing the motion and behavior of rigid bodies undergoing rotation.
Hollow Cylinder: A hollow cylinder is a three-dimensional geometric shape that is cylindrical in nature, with a hollow or empty space running through the center. This shape is commonly used in various engineering and scientific applications, particularly in the context of rotational dynamics and Newton's Second Law for Rotation.
I: The letter 'I' is a personal pronoun that represents the speaker or writer. It is a fundamental part of language and communication, used to express one's own thoughts, actions, and experiences. In the context of physics, the letter 'I' can have various meanings and applications depending on the specific topic or concept being discussed.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a fixed axis. It depends on the mass distribution 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 is a scalar quantity that depends on the mass and distribution of an object's mass about a given axis of rotation. The moment of inertia is a crucial concept in the study of rotational dynamics, as it determines how an object will respond to applied torques.
N·m: N·m, or Newton-meter, is a unit of torque or rotational force in the International System of Units (SI). It represents the product of force (in newtons) and the perpendicular distance (in meters) from the axis of rotation to the line of action of the force, and is used to quantify the rotational effect of a force.
Newton’s second law for rotation: Newton’s second law for rotation states that the net torque acting on an object is equal to the product of its moment of inertia and angular acceleration. Mathematically, it is expressed as $\tau = I \alpha$.
Newton's Second Law for Rotation: Newton's Second Law for Rotation is a fundamental principle in classical mechanics that describes the relationship between the angular acceleration of a rotating object and the net torque acting upon it. It provides a mathematical framework for understanding the dynamics of rotational motion.
Parallel Axis Theorem: The parallel axis theorem is a fundamental principle in rotational dynamics that relates the moment of inertia of an object about any arbitrary axis to its moment of inertia about a parallel axis passing through the object's center of mass. This theorem is crucial in understanding the rotational motion and energy of rigid bodies.
Perpendicular Axis Theorem: The perpendicular axis theorem is a fundamental principle in rotational dynamics that relates the moment of inertia of an object about an arbitrary axis to its moment of inertia about an axis perpendicular to the original axis. This theorem is particularly useful in the context of Newton's Second Law for Rotation, as it allows for the calculation of rotational quantities such as angular acceleration and torque.
Rad/s²: The unit of angular acceleration, which measures the rate of change of angular velocity over time. It represents the number of radians per second squared, indicating the acceleration of an object rotating around a fixed axis.
Radius Vector: The radius vector is a position vector that describes the location of a particle or object relative to a fixed point of reference, typically the origin of a coordinate system. It is a fundamental concept in the study of rotational motion and the application of Newton's Second Law to rotational dynamics.
Rigid Body: A rigid body is an idealized object that maintains its shape and size regardless of the forces acting upon it. It is a fundamental concept in classical mechanics that simplifies the analysis of the motion and behavior of objects.
Rotational Equilibrium: Rotational equilibrium is a state where the net torque acting on an object is zero, resulting in the object's rotational motion remaining constant or the object remaining at rest. This concept is fundamental in understanding the behavior of rigid bodies under the influence of external forces.
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 is the rotational equivalent of linear inertia, which is a measure of an object's resistance to changes in its linear motion.
Solid Cylinder: A solid cylinder is a three-dimensional geometric shape that consists of a circular base and a curved surface connecting the base to another identical circular base. It is a fundamental shape in physics and engineering, with important applications in the study of rotational motion and dynamics.
Torque: Torque is a measure of the rotational force applied to an object, which causes it to rotate about an axis. It is influenced by the magnitude of the force applied, the distance from the axis of rotation, and the angle at which the force is applied, making it crucial for understanding rotational motion and equilibrium.
Vector Cross Product: The vector cross product is a binary operation on two vectors that results in a third vector which is perpendicular to both of the original vectors. It is a fundamental concept in physics, particularly in the study of rotational motion and angular momentum.
α: Alpha (α) is a Greek letter that is commonly used to represent various physical quantities and variables in the context of rotational motion and dynamics. It is a fundamental term that is essential for understanding the concepts of rotational variables and Newton's Second Law for rotation.
τ = Iα: The equation τ = Iα, where τ represents torque, I represents moment of inertia, and α represents angular acceleration, is a fundamental relationship in rotational dynamics. This equation describes the connection between the torque applied to an object and the resulting angular acceleration it experiences.
τnet: τnet, or the net torque, is the sum of all the individual torques acting on an object around a specific axis of rotation. It is a crucial concept in the study of rotational dynamics and is central to understanding Newton's Second Law for Rotation.