Question

If the length of rod \(\mathrm{A}\) is \((2.35 \pm 0.01) \mathrm{cm}\) and that of \(\mathrm{B}\) is \((5.68 \pm 0.01) \mathrm{cm}\) then the rod \(\mathrm{B}\) is longer than \(\mathrm{rod} \mathrm{A}\) by \(\ldots\) (a) \((2.43 \pm 0.00) \mathrm{cm}\) (b) \((3.33 \pm 0.02) \mathrm{cm}\) (c) \((2.43 \pm 0.01) \mathrm{cm}\) (d) \((2.43 \pm 0.001) \mathrm{cm}\)

Short answer

(b) \( (3.33 \pm 0.02) \mathrm{cm}\)

Step-by-step solution

Identify the length of each rod with its uncertainty

We have the following information: Length of rod A: \(2.35 \pm 0.01 \mathrm{cm}\) Length of rod B: \(5.68 \pm 0.01 \mathrm{cm}\)

Calculate the difference in length between rod B and rod A

Subtract the length of rod A from the length of rod B to get the difference: Difference = Length of rod B - Length of rod A = \(5.68 \mathrm{cm} - 2.35 \mathrm{cm} = 3.33 \mathrm{cm}\)

Calculate the uncertainty in the difference

To find the uncertainty in the difference, we will add the uncertainties of each individual measurement: Uncertainty of the difference = Uncertainty of rod A + Uncertainty of rod B = \(0.01\mathrm{cm} + 0.01\mathrm{cm} = 0.02\mathrm{cm}\)

Write the final answer with uncertainty

The length difference between rod B and rod A is: \(3.33 \pm 0.02 \mathrm{cm}\) Answer: (b) \( (3.33 \pm 0.02) \mathrm{cm}\)

Key concepts

Precision in Measurements

Precision in measurements refers to the consistency or repeatability of a set of measurements. In other words, if you measure something several times and get the same or very similar results each time, the measurements are said to be precise. Precision is important because it tells you how much you can trust your measurement. For example, in the original exercise dealing with rods A and B, the lengths of the rods have an associated uncertainty of ±0.01 cm. This indicates that each measurement might vary by a little, but overall, they are very consistent.

When considering precision, it’s important to differentiate it from accuracy. Accuracy refers to how close a measurement is to the true value, while precision is about getting the same results repeatedly. Therefore, a measurement can be precise without being accurate, and vice versa.

• Precision gives you confidence in the repeatability of your results.
• It involves the aspect of variation often due to limitations of measuring instruments.

In the case of the rods, an uncertainty of ±0.01 cm suggests good precision, as repeated measurements of these rods would likely yield similar results.

Error Propagation

Error propagation is the process by which uncertainties in measurements affect the results of calculations involving those measurements. When performing any calculation that involves measured quantities, it is crucial to understand how errors in those quantities combine to produce an error in the final outcome.

In the exercise provided, to find how much longer rod B is than rod A, you need to subtract their lengths. Even if both rods are measured with precision, each measurement carries its uncertainty. Therefore, when we calculate the difference between the lengths, we propagate these errors by summing their absolute uncertainties.

• Error in a sum or difference: Add the absolute uncertainties.
• Error in a product or quotient: Add the relative uncertainties.

Hence, given that the lengths of the rods were each measured with an uncertainty of ±0.01 cm, the result of the subtraction (i.e., 3.33 cm) also carries an uncertainty of ±0.02 cm, calculated by adding the individual uncertainties. This shows the importance of considering error propagation in ensuring that any derived quantity accurately reflects the uncertainty of the measurements.

Significant Figures

Significant figures play a crucial role in scientific computations, as they dictate the precision of reported measurements and calculations. A significant figure is comprised of all the digits in a measurement that are known with certainty, plus one final digit that is estimated.

In the context of the original exercise, significant figures are essential when expressing the lengths of rods A and B. The numbers 2.35 cm and 5.68 cm each contain three significant figures. This indicates the precision level of the measurements. When calculating the difference in their lengths (3.33 cm), we match our calculated answer’s precision with the input measurements by retaining the appropriate number of significant figures.

• Significant figures in addition/subtraction: The result should match the least number of decimal places.
• Significant figures in multiplication/division: The result should match the fewest significant figures of any operand.

In essence, using significant figures helps ensure that a calculated result reflects the precision of the measurement inputs, avoiding the inclusion of unjustified precision and therefore providing a more accurate representation of the measurement's reliability and accuracy.