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Chapter 1: Problem 72

Which of the following are dimensionally correct? (A) \(h=\frac{2 T \cos \theta}{\rho r g}\) (B) \(v=\sqrt{\frac{p}{\rho}}\) (C) \(\frac{d v}{d t}=\frac{\pi p r^{4} t}{\delta \eta l}\) (D) \(T=\sqrt{\frac{m g l}{I}}\)

Short Answer

Expert verified
Options (A), (B), and (C) are dimensionally correct, while option (D) is not.

Step by step solution

01

Option (A)

Given expression: \(h=\frac{2 T \cos \theta}{\rho r g}\) Checking dimensions: LHS: Dimensions of \(h\) are [L]. RHS: Dimensions of \(\frac{2 \times T \times \cos \theta}{\rho \times r \times g}\) are \(\frac{\Theta}{[\text{ML}^{-3}][\text{L}][\text{LT}^{-2}]}\), which simplifies to [L]. LHS and RHS dimensions are consistent, so option (A) is dimensionally correct.
02

Option (B)

Given expression: \(v=\sqrt{\frac{p}{\rho}}\) Checking dimensions: LHS: Dimensions of \(v\) are [LT^(-1)]. RHS: Dimensions of \(\sqrt{\frac{p}{\rho}}\) are \(\sqrt{\frac{\text{[ML}^{-1}\text{T}^{-2}]}{\text{[ML}^{-3}}}\), which simplifies to [LT^(-1)]. LHS and RHS dimensions are consistent, so option (B) is dimensionally correct.
03

Option (C)

Given expression: \(\frac{dv}{dt}=\frac{\pi p r^{4} t}{\delta \eta l}\) Checking dimensions: LHS: Dimensions of \(\frac{dv}{dt}\) are [LT^(-2)]. RHS: Dimensions of \(\frac{\pi \times p \times r^{4} \times t}{\delta \times \eta \times l}\) are \(\frac{\text{[ML}^{-1}\text{T}^{-2} \times \text{[L]}^{4} \times \text{T}}{\text{[ML}^{-1}\text{T}^{-1} \times \text{L}}\), which simplifies to [LT^(-2)]. LHS and RHS dimensions are consistent, so option (C) is dimensionally correct.
04

Option (D)

Given expression: \(T=\sqrt{\frac{m g l}{I}}\) Checking dimensions: LHS: Dimensions of \(T\) are [螛]. RHS: Dimensions of \(\sqrt{\frac{m \times g \times l}{I}}\) are \(\sqrt{\frac{\text{[M]} \times \text{[LT}^{-2}\text{]} \times \text{[L]}}{\text{[ML}^{2}\text{]}}\), which simplifies to [T^(-1)]. LHS and RHS dimensions are not consistent, so option (D) is not dimensionally correct. Based on the analysis, options (A), (B), and (C) are dimensionally correct, while option (D) is not.

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