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 significant figures.

Short answer

Answer: The total force on the object is \(5.12 \mathrm{~N}\).

Step-by-step solution

Determine the number of significant figures in each force

First, we need to find the number of significant figures in each given force. The first force has 5 significant figures (\(2.0031 \mathrm{~N}\)), and the second force has 3 significant figures (\(3.12 \mathrm{~N}\)).

Add the forces

Since the forces are acting in the same direction, we can simply add them together to find the total force on the object: $$ 2.0031\ \mathrm{N} + 3.12\ \mathrm{N} = 5.1231\ \mathrm{N} $$

Round the answer

The total force needs to be reported to the correct number of significant figures. Since the second force had the least number of significant figures (3), we need to round our answer to 3 significant figures as well. The total force on the object is \(5.12 \mathrm{~N}\) (rounded to 3 significant figures).

Key concepts

Adding Forces in Physics

When forces are applied to an object in the same direction, they combine to produce a cumulative effect on the object. This process is commonly known as adding forces. It is an essential concept in physics, particularly in the study of dynamics where the behavior of moving objects is analyzed.

Imagine pushing a sled across the snow with a friend. If you both push in the same direction, the sled moves more effortlessly because your combined efforts add up to move the sled. In physics, when two forces, say, 2.0031 N and 3.12 N, are applied in the same direction on an object, the resulting force is simply the sum of both forces. Using arithmetic, the forces add up to 5.1231 N. However, in scientific calculations, the precision of the final answer must be limited to that of the least precise force to maintain the integrity of the data. As such, the total force is reported with the same number of significant figures as the least precise measurement, in this case, down to 3 significant figures which makes the total force 5.12 N after rounding.

Measuring Force

Measuring force is a fundamental concept in physics that involves quantifying the push or pull acting upon an object. Force measurement is crucial for understanding the interactions between objects and their subsequent motions.

One common unit of force is the newton (N), named after Sir Isaac Newton due to his groundbreaking work on forces. Precision in force measurements is critical, so instruments like spring scales, load cells, and dynamometers are often utilized. For instance, if a force of 2.0031 N is measured using a sensitive instrument, its accuracy might go up to four significant figures beyond the decimal. Exact measurements enable precise calculations, which are vital in fields ranging from engineering to space travel. When reporting measurements, recognizing significant figures ensures the accuracy and reliability of the data communicated.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form, which is particularly useful in physics and engineering. This notation simplifies calculations and provides a way to express the number of significant figures.

A number is written in scientific notation as the product of a number between 1 and 10 and a power of 10. For example, the speed of light in a vacuum is approximately 299,792,458 meters per second, which in scientific notation is written as \(2.99792458 \times 10^8\) meters per second. This number has 9 significant figures, and each is essential when performing high-precision calculations in physics. Scientific notation is highly beneficial when adding forces with very different magnitudes, as it allows for easier observation of significant figures and minimizes the risk of arithmetic errors during calculations.