10: Fixed-Axis Rotation Introduction
- Page ID
- 94605
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We begin to address rotational motion in this chapter, starting with fixed-axis rotation. Fixed-axis rotation describes the rotation around a fixed axis of a rigid body; that is, an object that does not deform as it moves. We will show how to apply all the ideas we’ve developed up to this point about translational motion to an object rotating around a fixed axis. In the next chapter, we extend these ideas to more complex rotational motion, including objects that both rotate and translate, and objects that do not have a fixed rotational axis.
- 10.1: 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.
- 10.2: Uniform Circular Motion
- Uniform circular motion is motion in a circle at constant speed. Centripetal acceleration is the acceleration pointing towards the center of rotation that a particle must have to follow a circular path. Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. An object executing uniform circular motion can be described with equations of motion.
- 10.3: Centripetal Force
- Centripetal force is a “center-seeking” force that always points toward the center of rotation so it is perpendicular to linear velocity. Rotating and accelerated frames of reference are noninertial. Inertial forces, such as the Coriolis force, are needed to explain motion in such frames.
- 10.4: 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.
- 10.5: 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.
- 10.6: 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.
- 10.7: 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.
- 10.8: 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.
- 10.9: 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.
- 10.10: 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.
- 10.11: 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
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