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Classical Mechanics

Dimensional Analysis

Concept Quizzes

Buckingham Pi Theorem (Dimensional Analysis)

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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by Brilliant Staff

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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by Brilliant Staff

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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by Brilliant Staff

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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by Brilliant Staff

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