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# 11: Rotational Kinematics, Angular Momentum, and Energy

• 11.1: Prelude to Fixed-Axis Rotation Introduction
In previous chapters, we described motion (kinematics) and how to change motion (dynamics), and we defined important concepts such as energy for objects that can be considered as point masses. Point masses, by definition, have no shape and so can only undergo translational motion. However, we know from everyday life that rotational motion is also very important and that many objects that move have both translation and rotation.
• 11.2: Rotational Variables
The angular position of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference. The angular velocity of a rotating body about a fixed axis is defined as ω(rad/s), the rotational rate of the body in radians per second. If the system’s angular velocity is not constant, then the system has an angular acceleration. The instantaneous angular acceleration is the time derivative of angular velocity.
• 11.3: Rotation with Constant Angular Acceleration
The kinematics of rotational motion describes the relationships among rotation angle, angular velocity and acceleration, and time. For constant angular acceleration, the angular velocity varies linearly, so the average angular velocity is 1/2 the initial plus final angular velocity over a given time period. A graphical analysis involves finding the area under an angular velocity-vs.-time or angular acceleration-vs.-time graph to get the change in angular displacement and velocity, respectively.
• 11.4: Relating Angular and Translational Quantities
The linear kinematic equation have the rotational counterparts in which x = θ, v = ω, a = α. A system undergoing uniform circular motion has a constant angular velocity, but points at a distance r from the rotation axis have a linear centripetal acceleration. A system undergoing nonuniform circular motion has an angular acceleration and therefore has both a linear centripetal and linear tangential acceleration at a point a distance r from the axis of rotation.
• 11.5: Moment of Inertia and Rotational Kinetic Energy
The rotational kinetic energy is the kinetic energy of rotation of a rotating rigid body or system of particles. The moment of inertia for a system of point particles rotating about a fixed axis is the sum of the product between the mass of each point particle and the distance of the point particles to the rotation axis. In systems that are both rotating and translating, conservation of mechanical energy can be used if there are no nonconservative forces at work.
• 11.6: Calculating Moments of Inertia
Moments of inertia can be found by summing or integrating over every ‘piece of mass’ that makes up an object, multiplied by the square of the distance of each ‘piece of mass’ to the axis. The parallel axis theorem makes it possible to find an object's moment of inertia about a new axis of rotation once it is known for a parallel axis. The moment of inertia for a compound object is simply the sum of the moments of inertia for each individual object that makes up the compound object.
• 11.7: Torque
The magnitude of a torque about a fixed axis is calculated by finding the lever arm to the point where the force is applied and multiplying the perpendicular distance from the axis to the line upon which the force vector lies by the magnitude of the force. The sign of the torque is found using the right hand rule. The net torque can be found from summing the individual torques about a given axis.
• 11.8: Newton’s Second Law for Rotation
Newton’s second law for rotation says that the sum of the torques on a rotating system about a fixed axis equals the product of the moment of inertia and the angular acceleration. In the vector form of Newton’s second law for rotation, the torque vector is in the same direction as the angular acceleration. If the angular acceleration of a rotating system is positive, the torque on the system is also positive, and if the angular acceleration is negative, the torque is negative.
• 11.9: Work and Power for Rotational Motion
The incremental work in rotating a rigid body about a fixed axis is the sum of the torques about the axis times the incremental angle. The total work done to rotate a rigid body through an angle θ about a fixed axis is the sum of the torques integrated over the angular displacement. The work-energy theorem relates the rotational work done to the change in rotational kinetic energy: W_AB = K_B − K_A. The power delivered to a system that is rotating about a fixed axis is the torque times the angul
• 11.10: Fixed-Axis Rotation Introduction (Exercises)
• 11.11: Fixed-Axis Rotation Introduction (Summary)
• 11.12: Conservation of Energy
Energy is conserved in rotational motion just as in translational motion.
• 11.13: Quantities of Rotational Kinematics
The angle of rotation is a measurement of the amount (the angle) that a figure is rotated about a fixed point— often the center of a circle.
• 11.14: Angular Acceleration
Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.
• 11.15: Rotational Kinematics
The motion of rolling without slipping can be broken down into rotational and translational motion.
• 11.16: Dynamics
Rotational inertia is the tendency of a rotating object to remain rotating unless a torque is applied to it.
• 11.17: Rotational Kinetic Energy
The rotational kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy.
• 11.18: Prelude to Angular Momentum
A helicopter can be used to illustrate the concept of angular momentum. The lift blades spin about a vertical axis through the main body and carry angular momentum. The body of the helicopter tends to rotate in the opposite sense in order to conserve angular momentum. The small rotors at the tail of the aircraft provide a counter thrust against the body to prevent this from happening, and the helicopter stabilizes itself.
• 11.19: Rolling Motion
In rolling motion without slipping, a static friction force is present between the rolling object and the surface. The linear velocity, acceleration, and distance of the center of mass are the angular variables multiplied by the radius of the object. In rolling motion with slipping, a kinetic friction force arises between the rolling object and the surface. Energy conservation can be used to analyze rolling motion since energy is conserved in rolling motion without slipping.
• 11.20: Angular Momentum
The angular momentum of a single particle about a designated origin is the vector product of the position vector in the given coordinate system and the particle’s linear momentum. The net torque on a system about a given origin is the time derivative of the angular momentum about that origin. A rigid rotating body has angular momentum directed along the axis of rotation.
• 11.21: Conservation of Angular Momentum
In the absence of external torques, a system’s total angular momentum is conserved. The angular velocity is inversely proportional to the moment of inertia, so if the moment of inertia decreases, the angular velocity must increase to conserve angular momentum. Systems containing both point particles and rigid bodies can be analyzed using conservation of angular momentum. The angular momentum of all bodies in the system must be taken about a common axis.
• 11.22: Precession of a Gyroscope
When a gyroscope is set on a pivot near the surface of Earth, it precesses around a vertical axis, since the torque is always horizontal and perpendicular to the angular momentum vector. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque, and it rotates about a horizontal axis, falling over just as we would expect.
• 11.23: Angular Momentum (Exercises)
• 11.24: Angular Momentum (Summary)
• 11.25: Conservation of Angular Momentum
The law of conservation of angular momentum states that when no external torque acts on an object, no change of angular momentum will occur.
• 11.26: Vector Nature of Rotational Kinematics
The direction of angular quantities, such as angular velocity and angular momentum, is determined by using the right hand rule.
• 11.27: Problem Solving
Identify the problem and solve the appropriate equation or equations for the quantity to be determined.
• 11.28: Linear and Rotational Quantities
The description of motion could be sometimes easier with angular quantities such as angular velocity, rotational inertia, torque, etc.