/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 The energy released during an ex... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 22

The energy released during an explosion, \(E,\) is a function of the time after detonation \(t,\) the blast radius \(R\) at time \(t,\) and the ambient air pressure \(p,\) and density \(\rho .\) Determine, by dimensional analysis, the general form of the expression for \(E\) in terms of the other variables.

Short Answer

Expert verified
The energy released \(E\) in an explosion can be expressed as \(E = Kp^2/R蟻 \) where K is a dimensionless constant.

Step by step solution

01

Identify the Dimensions

To begin, it's important to establish the dimensions of each of the given variables. These dimensions are as follows: \n\n - Energy (E) has dimensions of \( [ML^2T^{-2}]\) where M is Mass, L is Length and T is Time.\n - Time (t) has dimensions of \([T]\).\n - Blast radius (R) has dimensions of \([L]\).\n - Ambient air pressure (p) has dimensions of \( [ML^{-1}T^{-2}]\).\n - Ambient air density (蟻) has dimensions of \( [ML^{-3}]\).
02

Formulate the General Form

We know from the principle of dimensional homogeneity that the dimensions on both sides of an equation must be the same. Therefore, the dependency of E on the other variables must produce the same dimensions as E. So, we can postulate a general form of the relationship as follows: \n\n \(E = Kt^ap^bR^c蟻^d\) \n\nwhere K is a dimensionless constant, and a, b, c, and d are the powers to be determined.
03

Resolve the Powers

We equate the dimensions on both sides. For the LHS, it is \([ML^2T^{-2}]\). The RHS becomes \([T^a][ML^{-1}T^{-2}]^b[L]^c[ML^{-3}]^d\)\n Eequating dimensions, we get 3 equations: \n \(a - 2b + d = -2\), \(-b + c - 3d = 2\), \(b + d = 1\)\n Solving these equations, we get: a = 0, b = 2, c = -1, d = -1.
04

Form the final expression

Now, replacing these values of a, b, c, and d in the proposed general equation, we get \n\n\( E = Kp^2/R蟻 \)

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