In figure 2.1 (shown below), when the pendulum swings from point A to point B and then swings back from point B to point A, it is said to have completed one oscillation.

⁠

The movement for one oscillation will be: A —> O —> B —> O —> A.

Time period: the time taken for a pendulum to complete one oscillation. Its symbol is T and the SI unit is the second.

You should not use t to denote the time period. It must be T.

t is used to indicate the total time for some number of oscillations

T is used to indicate the time for one oscillation

Frequency: represents the number of oscillations per second. Its SI unit is s-1, which is called the Hertz (Hz).

For example, a frequency of 5 Hz means that there are 5 oscillations per second.

Key Concepts

1. Measurement of Length

There are various types of length that you will frequently encounter: distance, displacement, height, radius, diameter, etc. All of these are technically just measurements of length, so they all have the same unit. The same applies to other quantities — for instance, cross-sectional area and surface area have the same unit because both are measurements of area.

The length of an object can be measured by using any of the following instruments: ruler, measuring tape, vernier calipers, and micrometer screw gauge.

It is necessary to avoid parallax error while taking measurements with a ruler. Parallax error occurs when your line of sight is not perpendicular to the ruler, as shown below in figure 1.3:

⁠

⁠

To avoid parallax error, you have to ensure that your line of sight is perpendicular (at 90 degrees) to the ruler.

Note that in figure 3, the wrong case shows that the line of sight is not at 90 degrees to the ruler — in fact, that is exactly why that approach is wrong.

In some cases, it becomes very challenging to measure short lengths, such as measuring the thickness of a single sheet of paper. To make useful measurements in such cases, you should measure the total thickness of a large number of sheets (50 or more) and then divide that by the number of sheets. This will minimize the amount of error in your measurement.

You can use the following formula to calculate the thickness of one sheet of paper, using the total thickness of 50 sheets.

⁠

⁠

The thickness in this case will be the average thickness of 1 sheet of paper. This takes into account the fact that the thickness of each sheet may not be exactly the same.

The same method can be applied to find the average value in all such cases, where direct measurement is not possible, for example, while measuring the time period of a pendulum.

⁠

Measuring tapes can measure length of up to 100 cm (1 m).

It can measure to the nearest 0.1 cm.

Some measuring tapes come in the form of retractable steel tapes whereas others come in the form of cloth tapes. The cloth tape is the one that is used to measure the waist of a person.

The cloth tape is very flexible, so it is useful for measuring linear lengths (e.g. length of a pen) and circular lengths (circumference of the waist).

It is very simple to use a measuring tape. The only key precaution is to ensure that parallax error is avoided.

Parallax error is avoided by ensuring that the line of sight remains perpendicular to the ruler, just as shown in figure 1.3 above.

⁠

The vernier caliper, shown in figure 1.4, is another instrument that can be used to measure short lengths.

The caliper has two sets of jaws: external jaws and internal jaws. The jaws on the lower side are the external jaws and the jaws on the upper side are the internal jaws. As the name indicates, external jaws are used to measure the external diameter, whereas the internal jaws are used to measure the internal diameter.

For example, if you have a bottle cap, you may want to measure its internal or external diameter. Depending on which measurement you need to take, you would use either the internal jaws or external jaws.

⁠

Parts/Components of the caliper

External jaws: The pair of jaws on the lower side, labelled as 1. The external jaws are used to measure external diameters. For example, you could use the external jaws to measure the diameter of a tennis ball.

Internal jaws: The pair of jaws on the upper side, labelled as 2. The internal jaws are used to measure internal diameters. For example, you could use the internal jaws to measure the internal diameter of a bottle cap.

Main scale: The long calibrated scale, labelled as 3. It gives readings in cm. The main scale is fixed.

Vernier scale: The shorter scale, labelled as 4. It gives readings to the tenth of a mm (0.1 mm). The vernier scale is movable — it is the part that is moved to the point where the object fits firmly between the jaws.

Lock screw: the small screw, labelled as 5. It is used to “lock” the jaws into place so that they do not move while a measurement is being taken. This is similar to the lock on the micrometer screw gauge, as shown in figure 1.4.

Depth probe: the tail at the end of the instrument, labelled as 6. It is used to measure depth. ponents of the vernier caliper are described below. The numbering used in the figure above applies to the following descriptions.

Using a vernier caliper

The object is placed between the jaws and the vernier scale is moved until the object is held firmly between the jaws. If an external measurement is being made (e.g. diameter of a tennis ball), the object is placed between the external jaws. If an internal measurement is being made, the object is placed between the internal jaws.

Once the jaws grip the object firmly, the lock screw is tightened so that the vernier scale does not move. This ensures an accurate measurement.

After tightening the jaw, the two readings (main scale and vernier scale) are recorded and the measurement is calculated.

Reading a vernier caliper

The measurement of a vernier caliper, like the micrometer, is made up of two parts: the reading shown by the main scale and the reading shown by the vernier scale.

You need to note the main scale reading and the vernier scale reading, and then add the two to obtain the final measurement.

Vernier caliper measurement = main scale reading + vernier scale reading

Uses of a vernier caliper

Calipers are used to measure very short lengths — typically these are diameters.

The vernier caliper is useful for measuring internal and external diameters. It may also be used to measure the depth of a beaker, using the depth probe.

It gives measurements to the nearest tenth of a mm (0.1 mm) — which is equivalent to the hundredth of a cm (0.01 cm).

The caliper has a precision of 0.1 mm (or 0.01 cm), so it is less precise than a micrometer. The micrometer has a precision of 0.01 mm.

⁠

1.4.1 Parts/Components of the micrometer

⁠

As visible in the figure, the micrometer screw gauge cannot accommodate long objects. For instance, the one shown in the figure can measure a maximum length of 25 mm (2.5 cm).

It is important to know and remember the names of the main parts/features of a micrometer: anvil, spindle, lock, datum line, sleeve, thimble, and ratchet.

Anvil: the short cylindrical part, labelled as 1. It is a fixed component and cannot be adjusted.

Spindle: the long cylindrical part, labelled as 2. The spindle rotates and either moves toward the anvil or away from it. The object to be measured is held between the anvil and the spindle, and the spindle is then moved forward until the object is firmly gripped.

Lock nut: the lock, labelled as 3. It is used to “lock” the spindle into position so that it does not move while a measurement is being taken. If the spindle moves, the measurement will change.

Sleeve: it is the cylindrical component, labelled as 4. It shows the main scale of the micrometer. It is fixed and cannot be adjusted.

Thimble: the outer cylindrical part, labelled at 5. It is the part that rotates and controls the movement of the spindle. When the thimble is rotated, the spindle also rotates, and moves toward or away from the anvil.

Ratchet: the small stop at the end of the instrument, labelled as 6. It allows uniform pressure to be applied to the object and prevents the object from being held “too loosely” or “too tightly” between the anvil and the spindle.

1.4.2 Using a micrometer

The object is placed between the anvil and spindle. The anvil is fixed and the spindle moves. The thimble is then rotated so that the spindle moves forward.

When the object is almost secured between the anvil and the spindle, the ratchet is rotated to ensure that the object is firmly gripped. When the ratchet is rotated, a “clicking” sound is heard once the object is firmly in place.The clicking sound indicates that the object is firmly held in place and the thimble (or ratchet) should not be rotated any further.

Of course, you should note that the thimble rotates with the ratchet. So, when you rotate the ratchet, the thimble will automatically rotate with it.

Finally, once the object is held firmly in place, the lock nut is turned so that the spindle does not move.

The micrometer measurement is then recorded.

1.4.3 Reading a micrometer

The reading of a micrometer is made up of two parts: the reading shown by the sleeve and the reading shown by the thimble.

You need to note the reading shown by the sleeve and the reading shown by the thimble, and then add the two readings to obtain the overall measurement.

Let’s use figure 1.6, shown below, to demonstrate how a measurement should be made, using a micrometer.

⁠

Part 1: note the sleeve reading

Note the last mark that is visible before the sleeve touches the thimble. In figure 5, the last mark that can be observed is shown by the blue dot. It represents a reading of 6.5 mm.

Therefore, the sleeve measurement is 6.5 mm.

Part 2: note the thimble reading

Observe the mark on the thimble that coincides with the datum line — in this case, it is 13. This means that the thimble measurement is given by the 13th mark, as shown by the purple dot in figure 5.

The smallest division on the thimble represents 0.01 mm, so 13 divisions represent 0.13 mm (0.01 mm x 13).

Therefore, the thimble measurement is 0.13 mm.

Part 3: add the two readings to obtain the micrometer measurement

Micrometer measurement = sleeve reading + thimble reading

Micrometer measurement = 6.5 mm + 0.13 mm

Micrometer measurement = 6.63 mm

1.4.4 Uses of a micrometer

Micrometers give very precise measurements of short lengths; for instance, the diameter of a marble, the thickness of a few sheets of paper, the thickness of a mobile phone, etc.

The micrometer has a precision of 0.01 mm, which means that it gives measurements to the nearest 0.01 mm. For instance, one possible measurement is 6.53 mm.

In today’s time, it is very common to see digital micrometers. Unlike analog micrometers, they display the measurement directly on a small screen. However, the syllabus requires you to know how to read analog micrometers.

⁠

2. Measurement of Time

Any periodic motion can be used to keep track of time. In fact, all devices that measure time make use of repeating oscillations to calculate the passage of time.

Periodic motion is any motion that repeats itself at regular intervals. For example, the following two examples are typical cases of periodic motion:

The earth rotates about its own axis once every 24 hours, and the same rotation repeats itself.

The earth completes one revolution about the sun every 365 days (approximately), and the same motion repeats itself.

Time can be measured using a pendulum, clock, or stopwatch.

Not all these devices are equally useful to measure time. We need to consider the motion of the object and the required accuracy (or precision) before selecting the most appropriate device. In simple words, we need to consider how many decimal places we are interested in.

For example, if we want to measure the time it takes for a runner to complete a 100 m sprint, it would not be too useful to use a standard clock. That is because we would only be able to record the time to the nearest second.

Let’s suppose there are two runners: the first one takes 11.6 s and the second one takes 11.8 s to complete. In such a case, we would think that both runners finished the 100 m race in 12 s and we would declare a tie. However, that would be wrong.

To address that issue, we should use a device that can measure time more precisely (one that can provide 1 to 2 decimal places). Therefore, we would use a stopwatch to measure the sprinters’ times.

However, if you’re performing deep breathing exercises, you don’t really need a stopwatch. A simple clock would suffice because you would just be interested in counting the number of whole seconds. Also, it’s not quite possible to hold your breath for exactly 5.23 seconds, so recording time to 2 decimal places in such a case would be practically useless.

⁠

3. Measurement of Volume

3.1 Volume of a liquid

A measuring cylinder is used to measure the volume of a liquid. A measuring cylinder is shown below in figure 1.7

⁠

Recall that parallax error must be avoided while recording such readings. The figure shows the line of sight that you should maintain while reading a measuring cylinder.

The procedure for avoiding parallax error here is exactly the same as the one mentioned for taking a reading on a ruler: the line of sight should be perpendicular to the cylinder.

In this case, there is one additional precaution that must be taken — the mark should be read at the bottom of the meniscus. The curved surface of the liquid is called the meniscus. If you take a reading at the top of the curve (meniscus), you will end up with the wrong reading.

This is a basic question that you can expect to come across.

3.2 Volume of an irregular solid

Irregular solids do not have a standard, well-defined shape that can be categorized easily. There are no standard names either, unlike those for regular shapes, e.g. sphere, cube, cylinder, etc.

For example, if you stumble across a large number of rocks and you inspect 10-15 of those, you would realize that no two rocks are identical. All irregular shapes are like that — we cannot generalize them, which means that we cannot use a standard formula or equation to calculate the volume of an irregular solid.

The formulas given in the equations section can only be used for regular shapes.

To find the volume of an irregular shape, we can use the displacement method. It is a simple method in which we take a known quantity of a liquid (usually water) and note the difference in volume that occurs after we add an object to it. The difference in volume gives the volume of the object.

The following procedure is used to calculate the volume of an irregular solid:

Add some water to a measuring cylinder and record its volume as V_1

Gradually lower the irregular solid into the measuring cylinder.

Note down the new volume of the liquid and record it as V_2

The volume of the object, V, will be given by: V = V_2 - V_1

Like any other practical procedure, it is very important to follow good practices to avoid experimental errors. In this case, the following precautions should be taken:

The solid should be lowered into the measuring cylinder without causing spillage (no liquid should leave the cylinder).

Parallax error should be avoided while taking readings on the measuring cylinder. That can be done by ensuring that the line of sight is at the same level as the mark that is being read; in other words, the line of sight must be perpendicular to the measuring cylinder.

This is the simplest case but there are other cases that are trickier to deal with.