Lesson Explainer: Measuring Lengths Physics

In this explainer, we will learn how to correctly use rulers and other instruments to measure lengths.

Lengths and distances can be measured using various tools. One common tool for measuring lengths is the ruler.

A ruler is a straight strip of material with carefully spaced markings. These markings always begin at zero and increase in value.

On a ruler, some of the markings are numbered and some are not. The ruler shown has numbered marks in units of centimetres.

Marks that are not numbered divide the ruler into smaller equal segments. On the ruler above, there are lines midway between the numbered marks, as well as finer lines. The lines separated by the smallest spacing indicate the smallest difference in length a ruler can measure. This distance is called the ruler’s resolution.

The diagram above shows a magnified view of part of the ruler. There are ten equal divisions between the two centimetre marks. Therefore, the ruler measures lengths to the nearest one-tenth of a centimetre.

Consider using a ruler to measure the length of a pencil. There are several things to remember doing to make sure that the measurement is accurate.

The length of the pencil must be parallel to the edge of the ruler. One end of the pencil must be aligned with the ruler’s zero marker. We then look at the pencil’s other end to see which ruler mark it most closely aligns with. The distance from the zero marker to that mark is the measured length of the pencil.

Let’s look at some example questions.

Example 1: Explaining an Incorrect Length Measurement

Nader uses a centimetre ruler to measure the length of a straight line, as shown in the diagram. He determines that the length of the line is 2.8 cm. Which of the following statements explains why this answer is incorrect?

  1. The ruler is not parallel to the line; thus, the line is actually longer than 2.8 cm.
  2. The ruler is not parallel to the line; thus, the line is actually shorter than 2.8 cm.
  3. He has measured the line from the wrong end of the ruler. The line is actually 9.2 cm long.
  4. The maximum resolution of the ruler is 1 cm; thus, the length of the line should be recorded as 3 cm.
  5. Measurements using a ruler should always be rounded up; thus, the length of the line should be recorded as 3 cm.

Answer

Nader is measuring the length of the line marked “𝑋” using a ruler. He has aligned the object so that it runs parallel to the ruler’s edge, but he has not lined up the end of the object with the zero marker of the ruler.

Instead, the rightmost end is aligned with the 12.0 cm mark, causing the left end of the object to reach the 2.8 cm mark.

If Nader had positioned the end of the straight line at 0 cm, he would have measured the length correctly. The true length of the straight line is 12.0 cm minus 2.8 cm, or 9.2 cm.

The answer option that best explains why Nader’s measurement is incorrect is C.

Example 2: Measuring a Length Incorrectly Using a Ruler

Sarah uses a centimetre ruler to measure the length of a straight line, as shown in the diagram.

She determines that the length of the line is 18.2 cm. Which of the following statements explains why this answer is incorrect?

  1. The ruler is not parallel to the line. The line is actually shorter than 18.2 cm.
  2. The ruler is not parallel to the line. The line is actually longer than 18.2 cm.
  3. The maximum resolution of the ruler is 1 cm; thus, the length of the line should be recorded as 18 cm.
  4. Measurements using a ruler should always be rounded up; thus, the length of the line should be recorded as 19 cm.
  5. The zero reading is not vertically aligned with the line’s start point.

Answer

Sarah has lined up one end of the straight line with the zero marker of the ruler, which is correct, but the line and ruler are not parallel.

This is why Sarah’s reading is incorrect, but we see that both answer options A and B give this reason. Option A goes on to say that the straight line is actually shorter than 18.2 cm, while option B claims that it is longer than that.

To measure a line length of 18.2 cm, Sarah would have to read the ruler as follows.

If we highlight the length along the ruler’s edge between the dashed red lines, we know that the length is 18.2 cm.

Because the dashed red lines are perpendicular to the straight line “𝑋,” we can create a right triangle using that line, the purple line, and the dashed red line on the right.

Since the 18.2 cm line is the hypotenuse of this triangle, we know it is the longest of all sides. This means that the straight line that Sarah is measuring is shorter than 18.2 cm, so answer option A is correct.

Example 3: Analyzing the Measurement of a Curved Object’s Length

Sameh uses a ruler to measure the length of a line, as shown in the diagram.

He determines that the length of the line is 6.0 cm. Which of the following statements explains why this answer is incorrect?

  1. The line is curved, so its length cannot be easily measured with a ruler. The line is actually longer than 6.0 cm.
  2. The line is curved, so its length cannot be easily measured with a ruler. The line is actually shorter than 6.0 cm.
  3. The 0 cm reading on the ruler is not parallel to the end of the line.
  4. The 10 cm reading on the ruler is not parallel to the end of the line.
  5. The endpoint of the curve lies on the 6.2 cm reading.

Answer

The line marked “𝑋” is curved, so no matter how the ruler is positioned, it can never be parallel to the full length of the line.

Part of the correct response must be that the line is curved and so cannot be easily measured by a straight ruler.

Both answer options A and B include this statement. To decide between them, we need to know whether the curved line is actually longer or shorter than 6.0 cm.

One way to think of this is to consider how the line would change if it were made straight. Currently, the line is bowed and, therefore, fits between the 0.0 cm and 6.0 cm marks on the ruler. If the line were straightened, it would extend beyond those bounds; it would be longer than 6.0 cm.

Therefore, we choose option A as our answer.

Given a ruler, we may want to measure a length that is longer than the ruler itself.

To measure a length like this accurately, we need to have enough rulers and must arrange them properly.

Here, we have lined up the zero marker on the second ruler with the maximum length of the first one. We have also ensured that the two rulers are parallel.

With this arrangement, we can measure the length of the line by adding the total measured range of the first ruler, 12.0 cm, to the length indicated by the second ruler beginning from zero, 6.2 cm. The line’s measured length is, therefore, 18.2 cm.

Example 4: Explaining Why a Length Measurement Using Multiple Rulers Is Incorrect

Adel uses two centimetre rulers to measure the length of a straight line, as shown in the diagram. He determines that the length of the line is 17.1 cm. Which of the following statements correctly explains why this answer is incorrect?

  1. He has positioned the second ruler at the end of the first one, and there is a space in between. Thus, the line is actually longer than 17.1 cm.
  2. He has positioned the second ruler at the end of the first one, and there is a space in between. Thus, the line is actually shorter than 17.1 cm.
  3. The ruler is not parallel to the line.
  4. The maximum resolution of the ruler is 1 cm. Thus, the length of the line should be recorded as 17 cm.
  5. Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 18 cm.

Answer

We can see that the total length of the first ruler is 12.0 cm and that the straight line extends 5.1 cm beyond the zero marker of the second ruler. Adding 12.0 cm to 5.1 cm gives 17.1 cm, but this does not account for the full length of the line that we are measuring.

Because the zero marker of the second ruler is not aligned with the 12.0 cm mark of the first ruler, the line length reading of 17.1 cm is not correct. The straight line is actually longer than this by some amount.

Answer option A explains this reason best.

Rulers are helpful tools for measuring lengths of a few centimetres up to lengths of metres. For smaller lengths, those on the order of millimetres, another device called a micrometer is typically used. The diagram below shows a sketch of this instrument.

As shown below, we measure lengths using micrometers by adjusting a knob that pins the object being measured in place.

Micrometers measure lengths of millimetres, but they do this with a precision on the order of one-millionth of a metre, also called a micrometer.

Example 5: Determining Which Measurement Device Measures Wire Thickness

Which of the following items can be used to measure the width of a wire?

  1. A metre ruler
  2. A protractor
  3. A micrometer
  4. A thermometer
  5. A balance

Answer

Listed here are five measurement tools, and we want to choose the one capable of measuring the width of a wire.

We know that this width will be given by a length, a distance between two points. This means that the thermometer (which measures temperature), the balance (which measures weight), and the protractor (which measures angles) are not the items that we will choose.

Therefore, options A (a metre ruler) and C (a micrometer) remain. A metre ruler is a ruler that can measure lengths up to one metre. It is typically marked in centimetres, with tightly bunched, hard-to-read marks in between each centimetre representing millimetres.

A micrometer is specially designed to measure lengths on the order of one millimetre, with a precision of about one-millionth of a metre. Using a micrometer rather than a metre ruler would generally give a more accurate reading of the thickness of a wire. Therefore, we will choose answer option C.

Key Points

  • Rulers and micrometers are tools used for measuring object lengths.
  • To give a correct reading, the ruler must
    • be parallel to the length being measured,
    • have its zero marker lined up with one end of the length being measured,
    • be combined end to end for lengths greater than that of one ruler so that the zero marker of the additional ruler lines up with the maximum marked length of the first ruler.
  • Rulers are useful for measuring lengths of one or more centimetres up to several metres, whereas micrometers are used for measuring lengths on the order of one millimetre.

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