Question

Subtract \(0.2 \mathrm{~J}\) from \(7.36 \mathrm{~J}\) and express the result with correct number of significant figures. (a) \(7.160 \mathrm{~J}\) (b) \(7.016 \mathrm{~J}\) (c) \(7.16 \mathrm{~J}\) (d) \(7.2 \mathrm{~J}\)

Short answer

The correct answer is (d) \(7.2 \mathrm{~J}\).

Step-by-step solution

Identify the numbers

In this problem, we need to subtract \(0.2 \mathrm{~J}\) from \(7.36 \mathrm{~J}.\)

Subtract the numbers

Now, we will perform the subtraction as follows: \[7.36 \mathrm{~J} - 0.2 \mathrm{~J} = 7.16 \mathrm{~J}\]

Determine significant figures

For subtraction, the rule for significant figures is that the result should have the same number of decimal places as the number with the least number of decimal places. In this problem, \(7.36 \mathrm{~J}\) has 2 decimal places, and \(0.2 \mathrm{~J}\) has 1 decimal place. Therefore, the result must be expressed with 1 decimal place.

Round the result

Now we will round the result to 1 decimal place, as follows: \[7.16 \mathrm{~J} \approx 7.2 \mathrm{~J}\] The correctly rounded answer with the appropriate number of significant figures is: (d) \(7.2 \mathrm{~J}\)

Key concepts

Subtraction in Physics

Subtraction in physics follows the same basic rules as subtraction in mathematics. However, it often requires careful consideration of significant figures. When performing subtraction, you align the numbers by their decimal points, carefully account for all significant figures, and perform the operation. The purpose of paying attention to significant figures is to ensure that your result is as precise as the measurement allows. This is critical in physics, where results need to be as precise as possible to reflect the reliability of the measured data. Always double-check to ensure that units, such as Joules (J) in this exercise, are consistent throughout your calculations.
Understanding this foundational skill is paramount for solving more complex physics problems efficiently.

Decimal Places

When working with decimal numbers, it's important to understand what decimal places are. Decimal places are the positions of numbers to the right of the decimal point. In physics and scientific calculations, decimal places help express the precision of a measurement. The more decimal places a number has, the more precise it is. In subtraction operations, like in the exercise provided, the number of decimal places in your result should match the number with the least decimal places from your original numbers. For example, if subtracting 0.2 J (1 decimal place) from 7.36 J (2 decimal places), your answer should be rounded to reflect 1 decimal place. This reflects the certainty (or approximation) of the least precise measurement.

Rounding Numbers

Rounding numbers is an important step in calculations that involve decimal numbers to ensure consistency with significant figures. Rounding refers to adjusting the number to the desired decimal places or significant figures based on specific rules. Typically, if a number is 5 or greater, you round up the final digit. If it's less than 5, you round down. In our exercise, initially calculating 7.16 J needed rounding to 7.2 J to match the least precise measurement's decimal places, adhering to the rules of significant figures. This process ensures that the result reflects the proper level of precision.

Joule (J)

The Joule, symbolized as J, is the SI unit of energy. It's a standard measure used in physics to quantify the amount of work done or energy transferred. One joule is equal to the energy transferred when a force of one newton acts over a distance of one meter. In calculations, especially those involving conversion and significant figures, maintaining units like Joules is critical. This ensures understanding and accuracy, as SI units are globally recognized and provide a consistent framework for communicating measurements and calculations in scientific contexts.