College Physics, 11th Edition
Authors: Raymond A. Serway and Chris Vuille
ISBN-13: 978-1305952300
See our solution for Question 1P from Chapter 1 from College Physics 11th Edition by Raymod Serway and Chris Vuille.
Problem 1P
Chapter:
Problem:
Step-by-Step Solution
Given Information
We are given with following equations
\[T=2 \pi \sqrt{\dfrac{\ell}{g}}\]We need to check if the equation is dimensionally consistent
Part 1: Standard Dimensions
The dimensions of given variables are
\[\begin{array}{l}l = {\left[ L \right]^1}\\\\2\pi = {\left[ {} \right]^0}\\\\g = \left[ L \right]{\left[ T \right]^{ - 2}}\\\\T = {\left[ T \right]^1}\end{array}\]
Part 2: Check for given expression
\[\begin{array}{l}T = 2\pi \sqrt {\dfrac{\ell }{g}} \\\\{\left[ T \right]^1} = {\left[ {} \right]^0}\sqrt {\dfrac{{\left[ L \right]}}{{\left[ L \right]{{\left[ T \right]}^{ - 2}}}}} \\\\{\left[ T \right]^1} = \sqrt {\dfrac{1}{{1{{\left[ T \right]}^{ - 2}}}}} \\\\{\left[ T \right]^1} = {\left[ {{{\left[ T \right]}^{ - 2}}} \right]^{1/2}}\\\\{\left[ T \right]^1} = {\left[ T \right]^1}\end{array}\]Thus, the equation is dimensionally consistent
We are given with following equations
\[T=2 \pi \sqrt{\dfrac{\ell}{g}}\]We need to check if the equation is dimensionally consistent
Part 1: Standard Dimensions
The dimensions of given variables are
\[\begin{array}{l}l = {\left[ L \right]^1}\\\\2\pi = {\left[ {} \right]^0}\\\\g = \left[ L \right]{\left[ T \right]^{ - 2}}\\\\T = {\left[ T \right]^1}\end{array}\]
Part 2: Check for given expression
\[\begin{array}{l}T = 2\pi \sqrt {\dfrac{\ell }{g}} \\\\{\left[ T \right]^1} = {\left[ {} \right]^0}\sqrt {\dfrac{{\left[ L \right]}}{{\left[ L \right]{{\left[ T \right]}^{ - 2}}}}} \\\\{\left[ T \right]^1} = \sqrt {\dfrac{1}{{1{{\left[ T \right]}^{ - 2}}}}} \\\\{\left[ T \right]^1} = {\left[ {{{\left[ T \right]}^{ - 2}}} \right]^{1/2}}\\\\{\left[ T \right]^1} = {\left[ T \right]^1}\end{array}\]Thus, the equation is dimensionally consistent