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Classical Mechanics

Dimensional Analysis

Excel in math and science

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Concept Quizzes

Buckingham Pi Theorem

Buckingham Pi Theorem (Dimensional Analysis)

Using the Buckingham \(\pi\) theorem, find the formula for the radius of a black hole in term of the black hole’s mass \(m, \) the gravitational constant \( G, \) and the speed of light \( c \)

\[ r \approx \frac{m^2G}{c}\] \[ r \approx \frac{mG}{c^2}\] \[ r \approx \frac{G}{mc^2}\] \[ r \approx \frac{G^2}{mc^2}\]

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As shown in the above figure, a liquid with density \( \rho \) and viscosity \( \mu \) flows through a pipe with diameter \( d. \) In a section of the pipe with length \( L, \) the liquid flows with a speed of \( v. \) Using the Buckingham \(\pi\) theorem, determine the pressure difference \( \Delta P = P_1 - P_2 \) in terms of the fluid properties \( d, L, \rho, \mu \) and \( v .\)

Details

the function \(f\) is unitless, so that it outputs a pure number.

\[ \Delta P \approx \rho v^2 \times f\left(\frac{d}{L},\frac{\mu}{d \rho v}\right) \] \[ \Delta P \approx \rho v^2 \times f\left( d L,\frac{\mu L}{\rho v}\right) \] \[ \Delta P \approx v^2 \times f\left(\frac{d}{L},\frac{\rho}{d \mu v}\right) \] \[ \Delta P \approx \rho v^2 \times f\left(Ld,\frac{\mu}{d \rho v}\right) \]

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The sand timer in the above figure has a due time of \( T. \) The radius of the hole is \( r, \) the initial height of the sand is \( H, \) and the density of the sand is \( \rho. \) Using the Buckingham \(\pi\) theorem, determine \( T \) in terms of the properties \( r, H, \rho, \) and the gravitational constant \( G .\)

\[ T \approx \sqrt{\rho G} f\left(\frac{r}{H}\right)\] \[ T \approx \frac{1}{\sqrt{\rho G}} f\left(\frac{r}{H}\right)\] \[ T \approx \frac{G}{\sqrt{\rho }} f\left(\frac{1}{rH}\right)\] \[ T \approx \frac{G}{\sqrt{\rho}} f\left(rH\right)\]

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As shown in the above figure, a liquid with density \( \rho \) flows through a pipe with diameter \( d. \) In a section of the pipe with length \( L, \) the liquid flows with a speed of \( v_1 \) at the center and \( v_2 \) at the edge of the flow. The pressures are \( P_1 \) at the center and \( P_2 \) at the edge. Using the Buckingham \(\pi\) theorem, express the viscosity \( \mu \) of the fluid in terms of the liquid properties \( d, L, \rho, \Delta v = v_1 - v_2 \) and \( \Delta P = P_1- P_2 .\)

\[ \mu \approx \frac{\rho}{\Delta v d} \times f(\frac{d}{L},\frac{\Delta P \rho}{ (d \Delta v)^2})\] \[ \mu \approx \rho d \Delta v \times f(\frac{d}{L},\frac{\Delta P}{\rho (\Delta v)^2})\] \[ \mu \approx \frac{\rho d }{\Delta v} \times f(\frac{d}{L},\frac{\Delta P}{d (\Delta v)^2})\] \[ \mu \approx d \Delta v \times f(\frac{d}{L},\frac{\Delta P}{\rho (\Delta v)^2})\]

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The drag force \( F \) depends on four quantities: two parameters of the cone which are the speed of the cone \( v \) and the size of cone \( r,\) and two parameters of the air which are the density of the air \( \rho \) and the viscosity of the air \( \mu. \) Find the independent dimensionless groups that can be produced with \( F, v, r, \rho \) and \( \mu . \)

\[ \frac{F\rho}{v^2 r^2 } \text{ and } \frac{\nu v}{r} \] \[ \frac{r^2 F}{v^2} \text{ and } \frac{\rho}{vr \nu} \] \[ \frac{F}{v r^2 \rho} \text{ and } \frac{\nu v}{r} \] \[ \frac{F}{v^2 r^2 \rho} \text{ and } \frac{\nu}{vr} \]

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by Brilliant Staff

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 5 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Dimensional Analysis

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 5 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Concept Quizzes

Buckingham Pi Theorem (Dimensional Analysis)

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.