6: Momentum
- Page ID
- 104135
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter introduces and explores the fundamental conservation laws of momentum and angular momentum, which, along with energy conservation, form the core principles of physics. We begin with linear momentum, defined as mass times velocity, and learn that it is conserved when no external forces act. Momentum conservation explains interactions like collisions, categorized as elastic (kinetic energy conserved) or inelastic. Moving beyond linear motion, we explore angular momentum for rotating objects, introducing torque as the rotational analog of force and rotational inertia as analogous to mass. The chapter covers how torque and force determine changes in angular momentum and extends Newton’s Laws to rotation. Static equilibrium, where net force and net torque are zero, is examined for its applications in balance and stability. Finally, a summary of linear and angular analogs reinforces the interconnectedness of translational and rotational motion, providing a foundation for understanding the behavior of interacting systems in various physical contexts.
- 6.1: Overview
- We expand our understanding of interactions by introducing momentum and angular momentum, complementing our focus on energy. The chapter explores how forces impact changes in linear and rotational motion, applying concepts like torque and impulse to analyze interactions. This approach broadens our ability to describe the effects of forces in various physical systems.
- 6.2: Linear Momentum
- We define linear momentum as mass times velocity and introduce impulse, the product of force and time. The impulse-momentum theorem states that impulse equals momentum change. Internal forces within a system do not alter total momentum, while external forces do. Conservation of momentum applies if no net external force acts. We categorize systems as closed (isolated from external forces) or open, allowing exchanges. This model underpins analysis of interactions like collisions.
- 6.3: Applications of Momentum Conservation
- We apply momentum conservation to analyze interactions called collisions. Collisions may conserve kinetic energy (elastic) or lose it to thermal/internal energy (inelastic). For one-object systems, momentum changes if external forces act, as seen with a cart rebounding off a wall. For multi-object systems, momentum conservation applies to both elastic and inelastic collisions, with momentum charts and diagrams aiding in analysis.
- 6.4: Angular Motion
- We extend momentum concepts to angular motion, introducing rotational analogs of force (torque) and inertia (rotational inertia). Angular motion is described by angular momentum, which, like linear momentum, is conserved in the absence of external torques. We define angular velocity, using the right-hand rule for its direction, and establish relationships among angular displacement, velocity, and acceleration. This framework enables analysis of rotating systems and their dynamics.
- 6.5: Rotational Inertia
- We introduce rotational inertia, the rotational analog of mass, which depends on mass distribution relative to the rotation axis. For rotating bodies, rotational kinetic energy combines mass distribution and angular velocity. Calculating rotational inertia involves summing or integrating mass elements at distances from the axis. Various shapes, like rods and disks, have unique rotational inertia values, and composite objects’ inertia is the sum of their components.
- 6.6: Torque
- We introduce torque as the rotational analog of force, dependent on force magnitude, direction, and distance from the pivot. Torque causes angular acceleration, with greater torque from forces applied farther from the pivot. Using the right-hand rule, we determine torque direction. Only forces perpendicular to the position vector contribute to torque. Extended force diagrams help calculate net torque, essential for understanding rotational motion and equilibrium.
- 6.7: Static Equilibirium
- We examine static equilibrium, where an object remains at rest with zero net force and zero net torque. Achieving static equilibrium involves balancing forces and torques. To solve problems, we select a pivot point, draw an extended force diagram, and balance torques and force components. This analysis is crucial for understanding structures like beams or bridges under load, where forces and torques must be precisely balanced to maintain stability.
- 6.8: Angular Momentum
- We introduce angular momentum as the rotational equivalent of linear momentum, defined by rotational inertia and angular velocity. Angular momentum is conserved if no external torque acts on a system. The direction of angular momentum follows the right-hand rule. Angular impulse changes angular momentum, with torque causing this change. Applications, like figure skaters pulling arms in to spin faster, demonstrate how rotational inertia and angular velocity affect angular momentum.
- 6.9: Summary of Linear and Angular Analogs
- This section provides a summary table that aligns linear motion and rotational motion concepts. Key variables, like position, velocity, force, and momentum, are paired with their rotational counterparts, helping clarify analogies between translational and rotational dynamics. The table serves as a quick reference for comparing and understanding the relationships across both types of motion.
- 6.10: Wrap-up
- This chapter wraps up the three fundamental conservation laws: energy, momentum, and angular momentum. These principles allow us to predict system behavior before and after interactions. While effective, our models have limitations at very small scales or high speeds, where quantum mechanics and relativity become relevant. These conservation concepts, however, remain central across all physics, even as we refine our models for more complex realms.