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Chapter 1: Problem 161

Equation of \(\ell_{1}=\ell_{0}\left[1+\alpha\left(\mathrm{T}_{2}-\mathrm{T}_{1}\right)\right]\) find out the dimensions of the coefficient of linear expansion \(\alpha\) suffix. (a) \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{1} \mathrm{~K}^{1}\) (b) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{1} \mathrm{~K}^{1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{0} \mathrm{~K}^{1}\) (d) \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0} \mathrm{~K}^{-1}\)

Short Answer

Expert verified
The dimensions of the coefficient of linear expansion α are \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0} \mathrm{~K}^{-1}\) (option d).

Step by step solution

01

Identify the Dimensions of each term in the equation

The equation is given by: \(\ell_{1} = \ell_{0} \left[1 + \alpha \left( T_2 - T_1 \right) \right]\) Where, \(\ell_{1}\) and \(\ell_{0}\) are lengths, so they have dimension \(\mathrm{L}\). \(T_{1}\) and \(T_{2}\) are temperatures, so they have the dimension of temperature, usually represented by \(\mathrm{K}\ (Kelvin)\).
02

Rewrite the equation considering only the dimensions

Considering only the dimensions, the equation can be rewritten as: \([\mathrm{L}] = [\mathrm{L}] (1 + [\alpha] [\mathrm{K}])\)
03

Solve for the dimensions of α

Since the dimensions must be equal on both sides of the equation, we can solve for the dimensions of α: \([\alpha] [\mathrm{K}] = \frac{[\mathrm{L}] - [\mathrm{L}]}{[\mathrm{L}]}\) The dimensions of α can be written as: \([\alpha] = \frac{1}{[\mathrm{K}]}\) The dimensions of α are \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0} \mathrm{~K}^{-1}\), which corresponds to option (d).

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