# Chap 3

```Vernier calliper
How To Read A Vernier Caliper
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A quick guide on how to read a vernier caliper. A vernier caliper outputs measurement
readings in centimetres (cm) and it is precise up to 2 decimal places (E.g. 1.23 cm).
Note: The measurement-reading technique described in this post will be similar for
vernier calipers which output measurement readings in inches.
Measurement Reading Technique For Vernier Caliper
In order to read the measurement readings from vernier caliper properly, you need to
remember two things before we start. For example, if a vernier caliper output a
measurement reading of 2.13 cm, this means that:


The main scale contributes the main number(s) and one decimal place to the
reading (E.g. 2.1 cm, whereby 2 is the main number and 0.1 is the one decimal
place number)
The vernier scale contributes the second decimal place to the reading (E.g. 0.03
cm)
Let’s examine the image of the vernier caliper readings above. We will just use a two
steps method to get the measurement reading from this:


To obtain the main scale reading: Look at the image above, 2.1 cm is to
the immediate left of the zero on the vernier scale. Hence, the main scale
To obtain the vernier scale reading: Look at the image above and look closely
for an alignment of the scale lines of the main scale and vernier scale. In the
image above, the aligned line correspond to 3. Hence, the vernier scale reading
is 0.03 cm.
and vernier scale reading together. This will give 2.1 cm + 0.03 cm = 2.13 cm.
Let’s go through another example to ensure that you understand the above steps:
Main scale reading: 10.0 cm (Immediate left of zero)
Vernier scale reading: 0.02 cm (Alignment of scale lines)
Compensating For Zero Error
Now, we shall try with zero error. If you are not familiar on how to handle zero error
for vernier calipers, I suggest that you read up on Measurement of Length.
The reading on the top is the measurement obtained and the reading at the bottom is
the zero error. Find the actual measurement. (Meaning: get rid of the zero error in the
measurement or take into account the zero error)
Measurement with zero error: 3.34 cm
Zero error: – 0.04 cm (negative because the vernier scale is to the left)
Measurement without zero error: 3.34–(–0.04)=3.383.34–(–0.04)=3.38 cm
If you do not understand the subtraction of the negative zero error from the
measurement, please read on. Since the zero error is -0.04 cm, this means that all
measurements taken by the vernier calipers will be SMALLER by 0.04 cm. Hence, you
will have to ADD 0.04 cm to ALL measurements in order to get
the TRUE measurement. The subtraction is done in the above case is to have an
elegant way of obtaining a resultant addition: 3.34+0.04=3.383.34+0.04=3.38 and
to make it COMPATIBLEwith positive zero error. This means that once you have
determined the nature of the zero error (positive or negative), you can just subtract the
Consider a zero error of +0.04 cm. With my method, 3.34–(+0.04)=3.303.34–
(+0.04)=3.30 cm.
Normal method: Since the zero error is +0.04 cm, this means that all measurements
taken by the vernier calipers will be larger by 0.04 cm. Hence, you will have
to SUBTRACT 0.04 cm from ALLmeasurements in order to get the true measurement.
The final calculation will be 3.34–0.04=3.303.34–0.04=3.30 cm, which is the same
as my method.
Vernier caliper
Main article: Vernier scale
Diagram of Vernier calipers. The labeled parts are
1.
Outside large jaws: used to measure external diameter or width of an object
2.
Inside small jaws: used to measure internal diameter of an object
3.
Depth probe/rod: used to measure depths of an object or a hole
4.
Main scale (Metric): scale marked every mm
5.
Main scale (Imperial): scale marked in inches and fractions
6.
Vernier scale (Metric) gives interpolated measurements to 0.1 mm or better
7.
Vernier scale (Imperial) gives interpolated measurements in fractions of an inch
8.
Retainer: used to block movable part to allow the easy transferring of a measurement
The calipers in the diagram show a primary reading on the metric scale of about 2.475 cm (2.4 cm read
from the main scale plus about 0.075 cm from the Vernier scale). Calipers often have a "zero point error",
which is when the calipers do not read 0.000 cm when the jaws are closed. The zero point error must
always be subtracted from the primary reading. Let us assume these calipers have a zero point error of
0.013 cm. This would give us a length reading of 2.462 cm. For any measurement, reporting the error on
the measurement is also important. Both the primary reading and the zero point reading are bounded by
plus/minus half the length corresponding to the width of the smallest interval on the Vernier scale
(0.0025 cm). These are "absolute" errors and absolute errors add, so the length reading is then bounded
by plus/minus the length corresponding to the full width of the smallest interval on the Vernier scale
(0.005 cm). Assuming no systematics affect the measurement (the instrument works perfectly), a complete
measurement would then read 2.462 cm ± 0.005 cm.
The vernier, dial, and digital calipers give a direct reading of the distance measured with
high accuracy and precision. They are functionally identical, with different ways of reading the result.
These calipers comprise a calibrated scale with a fixed jaw, and another jaw, with a pointer, that
slides along the scale. The distance between the jaws is then read in different ways for the three
types.
The simplest method is to read the position of the pointer directly on the scale. When the pointer is
between two markings, the user can mentally interpolate to improve the precision of the reading.
This would be a simple calibrated caliper; but the addition of a vernier scale allows more accurate
interpolation, and is the universal practice; this is the vernier caliper.
Vernier, dial, and digital calipers can measure internal dimensions (using the uppermost jaws in the
picture at right), external dimensions using the pictured lower jaws, and in many cases depth by the
use of a probe that is attached to the movable head and slides along the centre of the body. This
probe is slender and can get into deep grooves that may prove difficult for other measuring tools.
The vernier scales may include metric measurements on the lower part of the scale
and inch measurements on the upper, or vice versa, in countries that use inches. Vernier calipers
commonly used in industry provide a precision to 0.01 mm (10 micrometres), or one thousandth of
an inch. They are available in sizes that can measure up to 1,829 mm (72 in
Pendulum
"Simple gravity pendulum" model assumes no friction or air resistance.
A pendulum is a weight suspended from a pivot so that it can swing freely.[1] When a pendulum is
displaced sideways from its resting, equilibrium position, it is subject to a restoring force due
to gravity that will accelerate it back toward the equilibrium position. When released, the restoring
force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging
back and forth. The time for one complete cycle, a left swing and a right swing, is called the period.
The period depends on the length of the pendulum and also to a slight degree on the amplitude, the
width of the pendulum's swing.
The period of a pendulum gets longer as the amplitude θ0(width of swing) increases.
The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity,
and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called
the amplitude.[8] It is independent of the mass of the bob. If the amplitude is limited to small
swings,[Note 1] the period T of a simple pendulum, the time taken for a complete cycle, is:[9]
where is the length of the pendulum and is the local acceleration of gravity.
For small swings the period of swing is approximately the same for different size swings: that
is, the period is independent of amplitude. This property, called isochronism, is the reason
pendulums are so useful for timekeeping.[10] Successive swings of the pendulum, even if
changing in amplitude, take the same amount of time.
For larger amplitudes, the period increases gradually with amplitude [12]
where is in radians.The difference between this true period and the period for small
swingsabove is called the circular error.
For real pendulums, the period varies slightly with factors such as the buoyancy and viscous
resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is
attached to the string, and flexibility and stretching of the string.[11][13] In precision applications,
corrections for these factors may need to be applied to eq. (1) to give the period accurately.
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