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Assertion : Specific gravity of a fluid ...

Assertion : Specific gravity of a fluid is a dimensionless quantity.
Reason : It is the ratio of fluid to the density of water

A

a. If both assertion and reason are true and reason is the correct explanation of assertion

B

b. If both assertion and reason are true but reason is not the correct explanation of assertion

C

c. If assertion is true but reason is false

D

d. If both assertion and reason are false

Text Solution

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The correct Answer is:
To solve the question regarding the assertion and reason, we will analyze both statements step by step. ### Step 1: Understanding the Assertion The assertion states that "the specific gravity of a fluid is a dimensionless quantity." - Specific gravity (SG) is defined as the ratio of the density of a fluid to the density of a reference fluid, typically water. Mathematically, it can be expressed as: \[ SG = \frac{\text{Density of fluid}}{\text{Density of water}} \] ### Step 2: Analyzing the Dimensions To determine if specific gravity is dimensionless, we need to analyze the dimensions involved in the formula. - The density of the fluid has the dimension of mass per unit volume, which is: \[ \text{Density} = \frac{M}{L^3} \] where \(M\) is mass and \(L\) is length. - Similarly, the density of water also has the same dimension: \[ \text{Density of water} = \frac{M}{L^3} \] ### Step 3: Calculating the Specific Gravity Substituting the dimensions into the specific gravity formula, we get: \[ SG = \frac{\frac{M}{L^3}}{\frac{M}{L^3}} = \frac{M/L^3}{M/L^3} = 1 \] ### Step 4: Conclusion on the Assertion Since the dimensions cancel out, we find that specific gravity is indeed a dimensionless quantity. Therefore, the assertion is **true**. ### Step 5: Understanding the Reason The reason states that "it is the ratio of the fluid to the density of water." - This statement is accurate as it correctly describes how specific gravity is calculated. It is the ratio of the density of the fluid to the density of water. ### Step 6: Conclusion on the Reason Since the reason accurately explains the assertion, it is also **true**. ### Final Conclusion Both the assertion and the reason are true, and the reason provides a correct explanation for the assertion. Therefore, the correct answer is that both the assertion and reason are true, and the reason is the correct explanation of the assertion. ---
To solve the question regarding the assertion and reason, we will analyze both statements step by step. ### Step 1: Understanding the Assertion The assertion states that "the specific gravity of a fluid is a dimensionless quantity." - Specific gravity (SG) is defined as the ratio of the density of a fluid to the density of a reference fluid, typically water. Mathematically, it can be expressed as: \[ SG = \frac{\text{Density of fluid}}{\text{Density of water}} ...
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