Question

The length of a rod is \((10.15 \pm 0.06) \mathrm{cm}\) what is the length of two such rods? (a) \((20.30 \pm 0.06) \mathrm{cm}\) (b) \((20.30 \pm 1.6) \mathrm{cm}\) (c) \((10.30 \pm 0.12) \mathrm{cm}\) (d) \((20.30 \pm 0.12) \mathrm{cm}\)

Short answer

The short answer is: (d) \((20.30 \pm 0.12) \mathrm{cm}\).

Step-by-step solution

Add the lengths of the two rods

First, let's add the lengths of the two rods together: \(L_{total} = 10.15 cm + 10.15 cm = 20.30 cm\)

Add the uncertainties in the lengths

Now we'll add the uncertainties in the lengths of the two rods: \(\Delta L_{total} = 0.06 cm + 0.06 cm = 0.12 cm\)

Combine the total length and uncertainty

Now, let's combine the total length and the total uncertainty: \(L_{total} \pm \Delta L_{total} = 20.30 cm \pm 0.12 cm\)

Compare with the given options

We compare our result with the given options: (a) \((20.30 \pm 0.06) \mathrm{cm}\) (b) \((20.30 \pm 1.6) \mathrm{cm}\) (c) \((10.30 \pm 0.12) \mathrm{cm}\) (d) \((20.30 \pm 0.12) \mathrm{cm}\) The total length we calculated matches option (d): \((20.30 \pm 0.12) {\rm cm}\) Hence, the correct answer is (d) \((20.30 \pm 0.12) {\rm cm}\).

Key concepts

Significant Figures

Significant figures are crucial in measurements as they indicate the precision of the measurement. They help us understand how accurately a measurement was made. For example, the length of the rod given in the exercise is 10.15 cm with an uncertainty of ±0.06 cm. Here, the number 10.15 has four significant figures, reflecting its precision level.

• The digits 1, 0, 1, and 5 are all considered significant.
• The trailing zero is counted because it is between significant digits.
• The uncertainty, 0.06, also has two significant figures affecting the measurement precision.

By using significant figures, we ensure that calculations such as addition and multiplication carry through the same level of precision. This helps avoid implying more precision than actually measured. When the measurement of the total length of both rods is 20.30 cm, we maintain the same level of precision in the significant figures.

Error Propagation

Error propagation is the method used to determine the uncertainty in the result of a calculated measurement. It involves knowing how errors can combine when mathematical operations are performed.
Suppose you are adding two measured quantities, as shown in the exercise where lengths are added. The error propagation rule for addition states that:

• We simply add the absolute uncertainties together.

For the lengths of the rods, both with an uncertainty of ±0.06 cm, the combined uncertainty is calculated as:\[ \Delta L_{total} = 0.06 \text{ cm} + 0.06 \text{ cm} = 0.12 \text{ cm} \]
This principle ensures that the uncertainty in the outcome truly reflects the uncertainties of the original measurements. Thus, the total length of two rods will have a calculated uncertainty of ±0.12 cm.

Addition of Uncertainties

When dealing with measurement calculations involving addition, it's essential to properly handle uncertainties. The addition of uncertainties concept clarifies how the impact on total uncertainty is computed.

• If you add or subtract measurements, the absolute uncertainties are added.
• This rule applies regardless of whether the quantities themselves are positive or negative.

In our exercise, since we're adding two identical lengths of 10.15 cm each, we must add their respective uncertainties (±0.06 cm).
The correct way to express the total length is:\[ L_{total} \pm \Delta L_{total} = 20.30 \text{ cm} \pm 0.12 \text{ cm} \]
By adding the individual uncertainties, we ensure the result accounts for measurement imprecision, respecting scientific measurement rules. Properly managing the uncertainty ensures that the calculated total reflects possible variations in original measurements.