Question

Two different forces, acting on the same object, are measured. One force is \(2.0031 \mathrm{~N}\) and the other force, in the same direction, is \(3.12 \mathrm{~N}\). These are the only forces acting on the object. Find the total force on the object to the correct number of stgnificant figures.

Short answer

Question: Calculate the sum of two forces acting on an object in the same direction, with the forces being \(2.0031 \mathrm{~N}\) and \(3.12 \mathrm{~N}\). Answer: The total force acting on the object, considering the correct number of significant figures, is \(5.12 \mathrm{~N}\).

Step-by-step solution

Identify the number of significant figures in each force

To find the total force, we will first identify the number of significant figures in each of the two forces. In the first force, \(2.0031 \mathrm{~N}\), we have 5 significant figures, and in the second force, \(3.12 \mathrm{~N}\), we have 3 significant figures.

Add the forces

Since the forces are in the same direction, we will simply add the magnitudes of the two forces: \(2.0031 \mathrm{~N} + 3.12 \mathrm{~N}\).

Determine the significant figures in the result

When adding numbers, the result should have the same number of decimal places as the number with the least decimal places. In this case, the first force has 4 decimal places, while the second one has 2 decimal places. Therefore, the result should have 2 decimal places.

Add the forces and round off to the correct number of decimal places

Adding the forces, we get: \(2.0031 + 3.12 = 5.1231\). Now, we round off the result to 2 decimal places (least decimal places), so we have \(5.12 \mathrm{~N}\).

Write the final answer

The total force acting on the object, considering the correct number of significant figures, is \(5.12 \mathrm{~N}\).

Key concepts

Adding Significant Figures

Understanding how to add significant figures is essential in physics, especially when measuring quantities like forces. When combining measurements, accuracy and precision matter. Let's take the example of two forces of magnitudes 2.0031 N and 3.12 N.

When adding these forces, the rule is straightforward: The number of decimal places in the result should equal the least number of decimal places in any of the numbers being added. In our sample problem, this means limiting our final answer to two decimal places because the second force, 3.12 N, has the fewest decimal places. After performing the addition 2.0031 + 3.12, we initially get 5.1231 N, but considering significant figures, we round this to 5.12 N to reflect the precision of our least precise measurement.

Measuring Forces

When measuring forces in physics, every digit counts. Accurate force measurement is crucial since it impacts how we understand interactions and calculate further values in problems. Precision is given in the form of significant figures—each digit represents how precisely we've measured something.

Measuring forces often involves sensitive equipment that can provide readings to a high degree of accuracy. Using our exercise as an example, forces are measured to be 2.0031 N and 3.12 N. Here, the equipment likely has different resolutions, or the circumstances of measurement necessitated different levels of precision. The key is to report a final answer that respects the least precise measurement, hence both honesty and accuracy are maintained in our final reported value.

Significant Figures Rules

Rules for significant figures are at the heart of reliable scientific communication, ensuring consistency and clarity in data reporting. Here are some core rules:

• Non-zero digits are always significant.
• Leading zeros are not significant as they only indicate the position of the decimal point.
• Captive zeros (zeros between non-zero digits) are significant.
• Trailing zeros in a number with a decimal point are significant.
• When adding or subtracting, the result should have no more decimal places than the least precise measurement.
• When multipling or dividing, the number of significant figures in the result should match the least number of significant figures in any single number used in the calculation.

In our textbook example, we add 2.0031 N (5 significant figures) to 3.12 N (3 significant figures). The addition rule necessitates that the final answer adopts the precision of the least precise value, hence the answer is reported as 5.12 N, reflecting 3 significant figures.