When problems overwhelm our ability to model them, dimensional analysis allows us to cut through the mental block and establish some quantitative insight without thinking.

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 \( c \)

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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 \( v .\)

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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 \( G .\)

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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 \( \Delta P = P_1- P_2 .\)

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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 \( \mu . \)

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