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Chapter 7: Problem 86

A student drops two spherical balls of different diameters and different densities. She has a stroboscopic photograph showing the positions of each ball as a function of time. However, she wants to express the velocity of each as a function of time in dimensionless form. Develop the dimensionless group. The equation of motion for each ball is $$m g-\frac{C_{D}}{2} \rho A V^{2}=m \frac{d V}{d t}$$ where \(m\) is ball mass, \(g\) is acceleration of gravity, \(C_{D}\) is a dimensionless and constant drag coefficient, \(\rho\) is air mass density, \(A\) is ball cross-sectional area \(\left(=\pi \mathrm{D}^{2} / 4\right)\) with \(D\) ball diameter, \(V\) is ball velocity, and \(t\) is time.

Short Answer

Expert verified
The dimensionless group for velocity is \( \Pi_1 = \frac{V}{\sqrt{gD}} \) and that for time is \( \Pi_2 = \frac{t}{\sqrt{D/g}} \)

Step by step solution

01

Identify the Variables and Constants

The given variables are mass of the ball (m), acceleration due to gravity (g), dimensionless drag coefficient (Cd), air mass density (ρ), ball cross-sectional area (A), ball diameter (D), ball velocity (V), and time (t).
02

Identify the Dimensional Groups

The givens have the dimensions as follows: mass (m) [M], acceleration due to gravity (g) [L/T^2], air mass density (ρ) [M/L^3], ball cross-sectional area (A) [L^2], ball diameter (D) [L], and ball velocity (V) [L/T]. Dimensionless drag coefficient (Cd) and time (t) [T] do not have dimensions.
03

Formulate the Dimensionless Groups

The two dimensionless groups needed can be developed using the Buckingham Pi theorem. We have five dimensions (M, T, and L) and seven dimensionally independent physical quantities (m, g, ρ, A, D, V, t) that gives us seven - five = two dimensionless groups. \nThe first will be the unitless velocity (which we will represent as Pi_1), and the second being the unitless time (denoted as Pi_2). Following the Buckingham Pi theorem we obtain: \( \Pi_1 = \frac{V}{\sqrt{gD}} \)\n\( \Pi_2 = \frac{t}{\sqrt{D/g}} \)\nThese are the required dimensionless group for velocity and time respectively.

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