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Chapter 1: Problem 43

Determine the number of significant figures in the following measured numbers: (a) \(1.007 \mathrm{~m}\), (b) \(8.03 \mathrm{~cm}\) (c) \(16.272 \mathrm{~kg}\) (d) \(0.015 \mu\) s (microseconds).

Short Answer

Expert verified
(a) 4, (b) 3, (c) 5, (d) 2

Step by step solution

01

Understanding Significant Figures

Significant figures are the digits in a number that carry meaning regarding its precision. They include all non-zero digits, zeros between significant figures, and trailing zeros in the decimal portion.
02

Identifying Significant Figures in 1.007 m

In the number 1.007, all the digits are significant. The non-zero digits (1 and 7) and the zeros between them are significant. Thus, it has 4 significant figures.
03

Identifying Significant Figures in 8.03 cm

For the number 8.03, both the non-zero digits (8 and 3) and the zero between them are significant. Therefore, this number has 3 significant figures.
04

Identifying Significant Figures in 16.272 kg

In 16.272, all digits are non-zero and therefore significant, giving a total of 5 significant figures.
05

Identifying Significant Figures in 0.015 渭s

The number 0.015 has two significant digits: the '1' and '5'. The leading zeros do not count as significant figures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Precision in Measurements
When we talk about measurements, precision is a key factor. Maintaining precision ensures that the data we use is as accurate as possible. Significant figures play a pivotal role in this because they represent all the known digits in a measurement plus one estimated digit. This tells us how precise a measurement is, giving us confidence in the reproducibility and reliability of the data. Always remember that more significant figures mean higher precision. This is crucial when doing any scientific calculations or experiments. For example, measuring 16.272 kg is more precise compared to 16 kg, as it has more significant figures, meaning it provides more detail about the weight.
Non-Zero Digits Significance
Non-zero digits in a number are always significant because they are undeniably part of the value of the measurement. They tell us exactly what is being measured without any ambiguity. Whether they are at the beginning, middle, or end of a number, non-zero digits signal whole parts of a measure that are known with certainty. For example, in numbers like 1.007 m and 8.03 cm, digits 1, 7, 8, and 3 are significant. They carry the measurement's true value and are always counted when determining the total number of significant figures.
Trailing Zeros in Decimals
Trailing zeros in decimal numbers play a crucial role in indicating precision. When zeros appear after a decimal point and are located at the end of a number, they are significant because they show the exactness of the measurement. These zeros confirm the measurement isn't simply an approximation. Instead, it signifies that the value neither extends all the way to the next digit nor stops short. For example, in the number 1.007, the two zeros are considered significant because they show that the measurement is precise to four digits.
Leading Zeros Insignificance
Not all zeros in measurements hold significance. Leading zeros, for instance, are zeros that come before any non-zero digit in a decimal. They don't affect the measurement's precision or the number of significant figures. Instead, they serve solely as placeholders to show the decimal point's position. In the number 0.015 渭s, the leading zeros are not considered significant. They are there just to orient us within the decimal system. Therefore, the significant figures in this case are only the '1' and '5,' which tell us the actual measured quantity.

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Most popular questions from this chapter

Tony's Pizza Palace sells a medium 9.0 -in. (diameter) pizza for \(\$ 7.95,\) and a large 12 -in. pizza for \(\$ 13.50 .\) Which pizza is the better buy?

In a radioactivity experiment, a solid lead brick (with same measurements as a patio brick, 2.00 in. \(\times 4.00\) in. \(\times 8.00\) in., except with a density that is 11.4 times that of water) is to be modified to hold a solid cylindrical piece of plastic. To accomplish this, the machinists are told to drill a cylindrical hole \(2.0 \mathrm{~cm}\) in diameter through the center of the brick parallel to the longest side of the brick. (a) What is the mass of lead (in kilograms) removed from the brick? (b) What percentage of the original lead remains in the brick? (c) Assuming the cylindrical hole is completely filled with plastic (with a density twice that of water), determine the overall (average) density of the brick/plastic combination after fabrication is complete.

A spherical shell is formed by taking a solid sphere of radius \(20.0 \mathrm{~cm}\) and hollowing out a spherical section from the shell's interior. Assume the hollow section and the sphere itself have the same center location (that is, they are concentric). (a) If the hollow section takes up 90.0 percent of the total volume, what is its radius? (b) What is the ratio of the outer area to the inner area of the shell?

A person weighs 170 lb. (a) What is his mass in kilograms? (b) Assuming the density of the average human body is about that of water (which is true), estimate his body's volume in both cubic meters and liters. Explain why the smaller unit of the liter is more appropriate (convenient) for describing a volume of this size.

Newton's second law of motion (Section 4.3 ) is expressed by the equation \(F=m a,\) where \(F\) represents force, \(m\) is mass, and \(a\) is acceleration. (a) The SI unit of force is, appropriately, called the newton (N). What are the units of the newton in terms of base quantities? (b) An equation for force associated with uniform circular motion (Section 7.3) is \(F=m v^{2} / r,\) where \(v\) is speed and \(r\) is the radius of the circular path. Does this equation give the same units for the newton?
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