Problem 86
Question
The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate by using the formula \(p=\frac{F}{l^{2}}\).If the maximum errors in the measurement of force and length are \(4 \%\) and \(2 \%\) respectively, then the maximum error in the measurement of pressure is (a) \(1 \%\) (b) \(2 \%\) (c) \(8 \%\) (d) \(10 \%\)
Step-by-Step Solution
Verified Answer
The maximum error in the measurement of pressure is 8\%.
1Step 1: Understand the Formula
The formula given is \( p = \frac{F}{l^{2}} \), where \( p \) is pressure, \( F \) is the force, and \( l \) is the length of a side of the plate.
2Step 2: Identify Maximum Errors
The errors given are: maximum error in force \( F \) is \( 4\% \) and maximum error in length \( l \) is \( 2\% \).
3Step 3: Express Pressure Error with Differentials
The relative error in pressure \( p \) can be found using differentials: \( \frac{dp}{p} = \frac{dF}{F} - 2\frac{dl}{l} \).
4Step 4: Substitute Given Errors
Substitute the given maximum errors of force and length into the formula: \( \frac{dp}{p} = 0.04 - 2(0.02) \).
5Step 5: Simplify the Expression
Calculate the expression for the maximum error: \( \frac{dp}{p} = 0.04 - 0.04 = 0.00 \).
6Step 6: Find the Maximum Error in Percentage
Since the expression simplifies to zero and seems counterintuitive to measurement error accumulation, a re-evaluation leads to recognize an error in the theoretical context; due to practical considerations in error propagation, sum linear components instead: effective linearized maximum error derived as \( 0.04 + 0.04 = 0.08 \), therefore 8\%.
Key Concepts
Pressure MeasurementMeasurement ErrorRelative Error
Pressure Measurement
Understanding pressure measurement involves calculating how much force is being applied over a given area. Think of pressure as the way force spreads across a surface. When you measure pressure using the formula \( p = \frac{F}{l^2} \), you apply a force \( F \) to an area defined by \( l^2 \), where \( l \) is the length of the sides of the square plate.
Pressure is a critical factor in various scientific and engineering applications, such as pneumatic systems, weather prediction, and even medicine for blood pressure monitoring.
To measure pressure accurately, both the force exerted onto a surface and the dimensions of the surface area must be known precisely. However, measurements may have inherent errors due to equipment limitations or observational inconsistencies. This reinforces the need to understand error propagation to ensure readings are reliable.
Pressure is a critical factor in various scientific and engineering applications, such as pneumatic systems, weather prediction, and even medicine for blood pressure monitoring.
To measure pressure accurately, both the force exerted onto a surface and the dimensions of the surface area must be known precisely. However, measurements may have inherent errors due to equipment limitations or observational inconsistencies. This reinforces the need to understand error propagation to ensure readings are reliable.
Measurement Error
Measurement error refers to the difference between the measured value and the true value of a quantity. When measuring force and length to calculate pressure, there can be minor inaccuracies that lead to measurement errors.
These errors are often expressed as percentages, like in the problem where the maximum error in force \( F \) is 4% and in length \( l \) is 2%. Errors can stem from various factors such as instrument precision, external conditions, or user mistakes.
Understanding error propagation is essential because it helps identify how small errors in input measurements influence the final calculated values. To manage errors, two practices are often recommended:
These errors are often expressed as percentages, like in the problem where the maximum error in force \( F \) is 4% and in length \( l \) is 2%. Errors can stem from various factors such as instrument precision, external conditions, or user mistakes.
Understanding error propagation is essential because it helps identify how small errors in input measurements influence the final calculated values. To manage errors, two practices are often recommended:
- Double-checking measurements with multiple tools or methods.
- Maintaining equipment properly to avoid systemic errors.
Relative Error
Relative error is a way to provide context to measurement inaccuracies. It is the ratio of the measurement error to the true value, often converted into a percentage. This concept shows how significant an error is in terms of the size of the measured quantity.
In this exercise, the formula \( \frac{dp}{p} = \frac{dF}{F} - 2\frac{dl}{l} \) is used to express how relative errors in force and length propagate into the pressure measurement.
When evaluating multiple errors, like in force and length, it's important to account for their combined effect. Initially, an incorrect simplification led to 0% error, which was then corrected to reflect practical considerations in error sum: the combined maximum error being 8%. Rules of thumb in managing relative errors include:
In this exercise, the formula \( \frac{dp}{p} = \frac{dF}{F} - 2\frac{dl}{l} \) is used to express how relative errors in force and length propagate into the pressure measurement.
When evaluating multiple errors, like in force and length, it's important to account for their combined effect. Initially, an incorrect simplification led to 0% error, which was then corrected to reflect practical considerations in error sum: the combined maximum error being 8%. Rules of thumb in managing relative errors include:
- Always consider how multiple measurement errors interact.
- Calculate the total impact to minimize inaccuracies in critical calculations.
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