Problem 91 - Master Resource Book in JEE Main Physics

Question

A physical quantity \(P\) is related to four observables \(a, b, c\) and \(d\) are as follows \(P=a^{3} b^{2} / \sqrt{c} d\) The percentage errors of measurement in \(a, b, c\) and \(d\) are \(1 \%, 3 \%, 4 \%\) and \(2 \%\) respectively. What is the percentage error in the quantity \(P\), if the value of \(P\) calculated using the above relation turns out to be 3.763, to what value should you round-off the result? [NCERT] (a) \(13 \%\) and \(3.8\) (b) \(1.3 \%\) and \(0.38\) (c) \(1.3 \%\) and \(3.8\) (d) \(3.896\) and 13

Step-by-Step Solution

Answer

13% and 3.8 (Option a)

Step 1: Understand the formula

The given physical quantity is expressed as \( P = \frac{a^3 b^2}{\sqrt{c} d} \). This means any error in the measurement of \(a, b, c, \) and \(d\) will affect \(P\).

Step 2: Apply error propagation

For multiplication and division, the percentage error in \( P \) is given by the formula: \( \frac{\Delta P}{P} \times 100 \% = 3 \times \frac{\Delta a}{a} + 2 \times \frac{\Delta b}{b} + \frac{1}{2} \times \frac{\Delta c}{c} + \frac{\Delta d}{d} \). Substitute the given percentage errors into this formula.

Step 3: Calculate individual contributions

Plugging in the errors: \(3 \times 1\% + 2 \times 3\% + \frac{1}{2} \times 4\% + 2\%\). Each term contributes as follows:- \(3 \times 1\% = 3\%\)- \(2 \times 3\% = 6\%\)- \(\frac{1}{2} \times 4\% = 2\%\)- \(2\%\)

Step 4: Sum the contributions

Add the individual contributions to get the total percentage error in \(P\): \[ 3\% + 6\% + 2\% + 2\% = 13\% \]

Step 5: Round-off the calculated value

The calculated value of \(P\) is 3.763, with a total error of 13\%. Round 3.763 to 3.8 (one decimal place) to appropriately reflect the error margin.

Key Concepts

Percentage Error Calculation

Percentage error calculation is a vital step when assessing how uncertainties in measurements affect derived quantities. When dealing with a formula like \( P = \frac{a^3 b^2}{\sqrt{c} d} \), each measurement error propagates through the formula to affect the error in \( P \). The total percentage error of \( P \) can be calculated by summing the individual contributions of each variable's error. When variables are multiplied or divided in a formula, the percentage errors are added up as weighted contributions:

• The factor from \( a \) is \( 3 \times \frac{\Delta a}{a} = 3 \times 1\% \).
• The factor from \( b \) is \( 2 \times \frac{\Delta b}{b} = 2 \times 3\% \).
• The factor from \( c \), since it's under a square root, is \( \frac{1}{2} \times \frac{\Delta c}{c} = \frac{1}{2} \times 4\% \).
• The factor from \( d \) is \( \frac{\Delta d}{d} = 2\% \).

After calculating these individual contributions, they are added up to get the total percentage error, which in this example equals \( 13\% \). Understanding how to compute this helps in predicting how reliable a physical quantity calculation is.

Physical Quantities and Measurement

Physical quantities are fundamental to physics, serving as the building blocks for describing phenomena in our universe. Physical quantities such as length, mass, and time are measured using different units. In the given exercise, the physical quantity \( P \) is derived from measurements of other quantities \( a, b, c, \) and \( d \). Accurate measurement of these quantities is critical. Each measurement has an associated error or uncertainty, expressing reliability or precision.

• Measurements should be taken as accurately as possible given the tool's capacity and environmental conditions.
• Uncertainties arise due to instrument limitations, human error, and environmental factors.
Precision in measurements translates directly into precision in derived quantities like \( P \), underscoring the importance of understanding measurement principles for sound calculations.

Rounding Off in Measurements

Rounding off in measurements is essential to accurately convey precision. In context, measurements and their derived quantities often exceed the precision that is scientifically meaningful, leading us to round them. Once percentage error or uncertainty is determined, it provides guidance on how many decimal places are appropriate.
For example, in the exercise, \( P \) was calculated as 3.763, with a percentage error of \( 13\% \). Therefore, it is rounded to one decimal place as 3.8.

• Rounding is performed to match the precision indicated by the uncertainty or error percentage.
• A smaller percentage error means the measurement is more precise, allowing for more decimals to be retained after rounding.
Rounding helps convey the level of confidence in the measurement's accuracy and prevents overstatement of precision in scientific reporting.