Problem 79
Question
A physical quantity is represented by \(X=\mathrm{M}^{a} \mathrm{~L}^{b} \mathrm{~T}^{-c}\). If percentage errors in the measurements of \(M, L\) and \(T\) are \(\alpha \%, \beta \%\) and \(\gamma \%\) respectively, then total percentage error is (a) \((\alpha a+\beta b-\gamma c) \%\) (b) \((\alpha a+\beta b+\gamma c) \%\) (c) \((\alpha a-\beta b-\gamma c) q\) (d) 096
Step-by-Step Solution
Verified Answer
The total percentage error is (a) \((\alpha a + \beta b - \gamma c)\%\).
1Step 1: Identify the Expression for Total Percentage Error
The expression for the total percentage error of a physical quantity expressed in terms of base quantities uses the principle that the percentage error of a quantity with a power relation is the sum of the products of the absolute power of each base quantity and its percentage error.
2Step 2: Apply Error Formula to Given Expression
The physical quantity is given as \(X = M^a L^b T^{-c}\). The percentage error in \(X\) can be calculated using the formula: Sum of \(\alpha \times a\), \(\beta \times b\) and \(-\gamma \times c\), which leads to the expression \((\alpha a + \beta b - \gamma c)\%\).
3Step 3: Match Result to Options
From the calculation in Step 2, the expression we derived \((\alpha a + \beta b - \gamma c)\%)\) matches option (a) in the list of provided answers.
Key Concepts
Error MeasurementPercentage ErrorPhysical Quantities
Error Measurement
Error measurement is an essential concept when dealing with physical quantities. It allows us to determine how close a measured or calculated value is to the true or accepted value. The error can arise due to imperfections in measurement tools or procedures and is vital for evaluating the accuracy and reliability of experiments.
There are two main types of errors:
There are two main types of errors:
- Systematic Errors: These are consistent, repeatable errors associated with faulty equipment or flawed experiment design.
- Random Errors: These are unpredictable variations that can occur in any physical measurement.
Percentage Error
When working with measurements, understanding the discrepancy between an observed value and its true value is crucial. This is where percentage error comes into play. It provides a way to express the error in terms of how large it is relative to the correct value. This is helpful when comparing errors of different magnitudes or when dealing with quantities that have significantly different scales.
The percentage error can be described by the equation:\[\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right) \times 100\%\]
This formula helps in conveying how much the measured values deviate proportionally from the true values, giving a better understanding of measurement accuracy.
In the context of the exercise, when dealing with a physical quantity that is expressed as a product of powers of base quantities, the total percentage error is obtained by summing the individual errors each weighted by their respective exponents. Thus, the result shows how each factor contributes to the overall error percentage.
The percentage error can be described by the equation:\[\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right) \times 100\%\]
This formula helps in conveying how much the measured values deviate proportionally from the true values, giving a better understanding of measurement accuracy.
In the context of the exercise, when dealing with a physical quantity that is expressed as a product of powers of base quantities, the total percentage error is obtained by summing the individual errors each weighted by their respective exponents. Thus, the result shows how each factor contributes to the overall error percentage.
Physical Quantities
Physical quantities are fundamental to the study and application of physics. They describe aspects of the physical world, allowing us to measure and understand phenomena. Each physical quantity is generally expressed through a combination of base quantities like mass (M), length (L), and time (T), raised to certain powers.
In an algebraic representation, a physical quantity can be shown as:\[X = M^a L^b T^{-c}\]
Here, \(a\), \(b\), and \(c\) are the powers which represent the dimensional dependence of the quantity on mass, length, and time respectively. Correct dimensional analysis involves ensuring the consistency of these relationships according to established physical laws.
The accurate measurement and manipulation of physical quantities form the basis of scientific experimentation, where understanding their errors is critical. As illustrated, by knowing the percentage errors of the base quantities, one can determine the total percentage error of the physical quantity, ensuring more precise and reliable measurement outcomes.
In an algebraic representation, a physical quantity can be shown as:\[X = M^a L^b T^{-c}\]
Here, \(a\), \(b\), and \(c\) are the powers which represent the dimensional dependence of the quantity on mass, length, and time respectively. Correct dimensional analysis involves ensuring the consistency of these relationships according to established physical laws.
The accurate measurement and manipulation of physical quantities form the basis of scientific experimentation, where understanding their errors is critical. As illustrated, by knowing the percentage errors of the base quantities, one can determine the total percentage error of the physical quantity, ensuring more precise and reliable measurement outcomes.
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