22: Dimensions
- Page ID
- 8545
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 22.1: Mass, Length and Time
- This page explains that mechanical quantities are defined by three fundamental dimensions: mass, length, and time. Each physical quantity, like force (expressed as MLT−2), is linked to these dimensions, with a distinction made between dimensions and units (e.g., force in the MKS system). Additionally, a fourth dimension, quantity of electricity, is introduced for electromagnetic quantities.
- 22.2: Table of Dimensions
- This page explores the dimensions and MKS units of mechanical quantities, distinguishing between straightforward and derived units. It also examines the philosophical debate on whether angles are dimensionless, given they are ratios of lengths, versus being dimensioned due to the need for specific units like radians or degrees, encouraging further discussion among readers.
- 22.3: Checking Equations
- This page emphasizes the importance of checking dimensional consistency in complex physical calculations to identify potential errors. It explains that while ensuring dimensions balance is critical, it doesn't guarantee overall accuracy due to possible oversight of dimensionless constants.
- 22.4: Deducing Relationships
- This page explores dimensional analysis to establish relationships among various physical quantities, including the period of a simple pendulum, torsion constant of a cylinder, orbital period of a planet, and viscous drag on a sphere in fluid.
- 22.5: Dimensionless Quantities
- This page highlights the importance of dimensionless quantities in fluid dynamics, focusing on the Reynolds number, which represents the ratio of internal to external forces on a body in a fluid. It discusses the role of speed, viscosity, and size in determining this ratio, emphasizing kinematic viscosity.
- 22.6: Different Fundamental Quantities
- This page explores the use of fundamental units in high-energy particle physics, focusing on energy (E), speed (V), and angular momentum (J) instead of conventional mass, length, and time. It explains the derivation of dimensions and the expression of quantities in "natural" units like GeV, c, and ħ, outlining the relationships between these units.
- 22.7: Appendix A
- This page provides a variety of exercises in classical mechanics, covering topics such as static and dynamic equilibrium, rotational dynamics, buoyancy, and the effects of forces on systems. Problems range in difficulty and examine different scenarios, including motion on inclined planes, stability of shapes, and rotational behavior of objects like yo-yos and pendulums.
- 22.8: Appendix B
- This page covers various principles and exercises in physics, focusing on dynamics, mechanics, and energy conservation. Topics include angular momentum, rotational dynamics, stability conditions, and equilibrium in rotating frames. Practical applications involve the motion of rods, cylinders, chains, and other objects, deriving relationships among essential variables.