Vernier Calliper
The meter scale enables us to measure the length to the nearest millimeter only. Engineers and scientists need to measure much smaller distances accurately. For this a special type of scale called Vernier scale is used.
The Vernier scale consists of a main scale graduated in centimeters and millimeters. On the Vernier scale 0.9 cm is divided into ten equal parts. The least count or the smallest reading which you can get with the instrument can be calculated as under:
Least count = one main scale (MS) division - one vernier scale (VS) division.
= 1 mm - 0.09 mm
= 0.1 mm
= 0.01 cm
The least count of the vernier
= 0.01 cm
The Vernier calliper consists of a main scale fitted with a jaw at one end. Another jaw, containing the vernier scale, moves over the main scale. When the two jaws are in contact, the zero of the main scale and the zero of the vernier scale should coincide. If both the zeros do not coincide, there will be a positive or negative zero error.
After calculating the least count place the object between the two jaws.
Record the position of zero of the vernier scale on the main scale (3.2 cm in figure below).
You will notice that one of the vernier scale divisions coincides with one of the main scale divisions. (In the illustration, 3rd division on the vernier coincides with a MS division).
Reading of the instrument = MS div + (coinciding VS div x L.C.)
= 3.2 + (3 x 0.01)
= 3.2 + 0.03
= 3.23 cm
To measure the inner and outer diameter of a hollow cylinder or ring, inner and outer callipers are used. Take measurements by the two methods as shown in figure below.
Micrometer Screw-Gauge
Micrometer screw-gauge is another instrument used for measuring accurately the diameter of a thin wire or the thickness of a sheet of metal.
It consists of a U-shaped frame fitted with a screwed spindle which is attached to a thimble.
The screw has a known pitch such as 0.5 mm. Pitch of the screw is the distance moved by the spindle per revolution. Hence in this case, for one revolution of the screw the spindle moves forward or backward 0.5 mm. This movement of the spindle is shown on an engraved linear millimeter scale on the sleeve. On the thimble there is a circular scale which is divided into 50 or 100 equal parts.
When the anvil and spindle end are brought in contact, the edge of the circular scale should be at the zero of the sleeve (linear scale) and the zero of the circular scale should be opposite to the datum line of the sleeve. If the zero is not coinciding with the datum line, there will be a positive or negative zero error as shown in figure below.
While taking a reading, the thimble is turned until the wire is held firmly between the anvil and the spindle.
The least count of the micrometer screw can be calculated using the formula given below:
Least count
= 0.01 mm
The wire whose thickness is to be determined is placed between the anvil and spindle end, the thimble is rotated till the wire is firmly held between the anvil and the spindle. The rachet is provided to avoid excessive pressure on the wire. It prevents the spindle from further movement. The thickness of the wire could be determined from the reading as shown in figure below.
= (2.5 + 0.46) mm = 2.96 mm
Relationship in the Metric system of length 1 kilometer (km) = 103 m
1 centimeter (cm) = 10-2 m 1 millimeter (mm) = 10-3 m
Mass
Moving objects have inertia too. A moving object needs force to make it stop. A moving car has more inertia than a moving book. It needs more force to make it stop.
Measurement of Mass
This balance consists of a beam and two scale pans (shown in figure below), the beam being balanced at its mid point on a knife-edge. The scale pans also hang on knife edges and rest on the base board. When the balance is not in use the beam rests on the beam support.
- Use the leveling screws, attached beneath the base board to make sure that the beam is horizontal. It can be verified with the help of the plumb- line provided shown in the diagram.
- Use the arrestment knob to raise the beam and the adjusting screw at the two ends of the beam, to bring the pointer to the middle or zero mark on the scale.
- Lower the beam using the arrestment knob again.
- Place the body to be weighed on the left scale pan and put weights on the
Mass
If you push a book, it moves faster than if you push a car with the same force. This is because the car has more mass than the book. If you had two identical boxes, one containing iron and the other containing cotton we could identify them by pushing the boxes. We can say that the car and iron box are more reluctant to move than the book and the cotton box. We call this reluctance to move "inertia". The larger the mass of an object, the larger its inertia. Hence mass of a body is a measure of its inertia.
Moving objects have inertia too. A moving object needs force to make it stop. A moving car has more inertia than a moving book. It needs more force to make it stop.
Measurement of Mass
This balance consists of a beam and two scale pans (shown in figure below), the beam being balanced at its mid point on a knife-edge. The scale pans also hang on knife edges and rest on the base board. When the balance is not in use the beam rests on the beam support.
- Use the levelling screws, attached beneath the base board to make sure that the beam is horizontal. It can be verified with the help of the plumb- line provided shown in the diagram.
- Use the arrestment knob to raise the beam and the adjusting screw at the two ends of the beam, to bring the pointer to the middle or zero mark on the scale.
- Lower the beam using the arrestment knob again.
- Place the body to be weighed on the left scale pan and put weights on the right hand scale pan to balance the beam (when pointer is at zero).
State of Matter's Volume Measurements
Volume of some of the regular solids can be calculated using the formulae given in the following table.
| Object | Volume(formula) |
|---|---|
| Cube | (length)3 = l3 |
| Cuboid | length X breadth X height = l x b x h |
| Sphere | 4/3 x π x (radius)3 = 4/3 π r3 |
| Cylinder | π x radius2 x height = π r2 h |
| Cone | 1/3 x π x (radius)2 x height = 1/3 π r h |
= 100 cm x 100 cm x 100 cm
= 1000000 cm3
= 106cm3
Volume of irregularly shaped objects can be found out by measuring the volume of displaced liquid by an object. We can find the volume of these solids with the help of a measuring cylinder.
Pour water into a clean measuring cylinder (nearly three fourth of its volume) and note the level of water. To avoid parallax error, reading must be taken at the lowest level of meniscus or curved surface of the liquid as shown in the diagram below. Attach a string to the solid and lower it into the water and note the new level of water. The difference in the above two readings will give you the volume of the solid.
If the object is lighter, it will float on water. In that case its volume can be found out with the help of a sinker. Pour some water in a measuring cylinder and lower the sinker (any body which sinks) into it. Note the level of water.
Tie the object to the sinker and lower them into the measuring cylinder and note the reading of water level. The difference in water level will give you the volume of the object.
Volume of an irregular solid can also be found by displacement method using an overflow vessel.
Fill the overflow vessel with water until the water starts overflowing. When water stops overflowing place a measuring cylinder below the overflow tube. Now gently lower the solid into the water with the help of a string. When water stops overflowing, read the volume of water collected in the measuring cylinder.
Volume of a liquid can be measured with the help of a pipette, burette, measuring flask, or a measuring cylinder.
The burette is a long narrow cylinder ending with a tap and a jet at the bottom. It is graduated from the top downwards.
The pipette is used to transfer a fixed volume of liquid very accurately from one container to another. It consists of a long tube blown into a bulb in the middle.
Volume of a liquid is usually measured in a unit called 'liter' 1 liter = 1000 cm3 (cc)
1 m3 = 1000 liters 1 liter = 1000 mL (milliliter)
1 mL = 1 cm3 (1 cc)
Burette or Buret
The burette is a long narrow cylinder ending with a tap and a jet at the bottom. It is graduated from the top downwards.
It is calibrated in milliliters (mL) and is graduated from 0 to 50 mL. With its help exact volume can be measured.
Pipette
The pipette is used to transfer a fixed volume of liquid very accurately from one container to another. It consists of a long tube blown into a bulb in the middle.
The lower end is drawn out into a jet, and a mark is made on the upper part of the tube to show up to what level the liquid should be drawn. To use the pipette the lower end A is dipped into the liquid and air is drawn out from the pipette for the liquid to come up.
Volume of a liquid is usually measured in a unit called 'liter'
1 liter = 1000 cm3 (cc)
1 m3 = 1000 liters
1 liter = 1000 mL (milliliter)
1 mL = 1 cm3 (1 cc)
Density
"Which is heavier, a kg of iron or a kg of wood?" The answer is of course, "Both are equally heavy."If, however, you are asked "Which is heavier, iron or wood?" You would say, "Iron".
To be precise, when we compare the heaviness of two different materials, we must refer to the same volume of each material. This leads to the concept of density.
The density of a substance is defined as its mass per unit volume
Density
The SI unit of density is kg m-3 and in the cgs system, the unit of density is g cm-3.
The densities of different substances vary widely. Densities of most of the solids are greater than that of liquids, though there are exceptions. Densities of wood, wax and cork are less than that of water and the density of mercury is greater than most of the solids.
Densities of some of the common substances are listed in the following table:
| Substance | Density in kg m - 3 | Substance | Density in kg m - 3 |
|---|---|---|---|
| Solids | Liquids | ||
| Aluminium | 2700 | Alcohol | 800 |
| Copper | 8920 | Glycerine | 1260 |
| Cork | 220 to 260 | kerosene | 800 |
| Glass | 2250 to 2600 | Mercury | 13600 |
| Gold | 19300 | Pure water at 40C | 1000 |
| Ice | 920 | Sea water | 1020 to 1140 |
| Iron | 7860 | Turpentine | 870 |
| lead | 11340 | ||
| Silver | 10500 | ||
| Gases at 00C and 760 mm Pressure | |||
| Air(Dry) | 1.29 | Carbon-di-oxide | 1.98 |
| Hydrogen | 0.09 | Nitrogen | 1.25 |
| Oxygen | 1.43 | ||
The following figure shows the masses of six metals each having the same volume (unit volume 1 cm3). Aluminium is the lightest metal and gold is the heaviest.
The density of aluminium is about 2.7 g cm-3 (2700 kg m-3). The density of gold is 19.3 g cm-3 (19300 kg m-3).
If the mass and volume of a substance can be measured, its density can be calculated using the formula
Let us take a piece of wood which has a volume of 1000 cm3 and a mass of 600 g.Then its density
= 0.6 g cm-3 (600 kg m-3)
A piece of aluminium of volume 100 cm3 has a much greater mass, about 270 g, than the same volume of wood.
Density of aluminium
= 2.7 g cm-3 (2700 kg m-3)
Example 1:
A concrete slab 1.0 m by 0.5 m by 0.1 m has a mass of 120 kg. What is the density of concrete slab?
Suggested Answer :
Volume of the concrete slab = l x b x h
= 1.0 m x 0.5 m x 0.1 m
= 0.05 m3
= 2400 kg m-3
Example 2 :
An alloy is made by mixing 360 g of copper of density 9 g cm-3, with 80 g of iron of density 8 g cm-3. Find the density of the alloy, assuming the volume of each metal used does not change during melting.
Suggested answer :
Volume of copper
= 40 cm3
Volume of iron
= 10 cm3
Total volume = (40 + 10)
= 50 cm3
Total mass = (360 + 80)
= 440 g
Density of alloy
= 8.8 g cm-3
Water is the most common liquid. Density of water at 4oC is 1 g cm-3 (1000 kg m-3).
Example :
28 kg of fuel completely fills a 40 liter petrol tank of a vehicle. What is the density of the fuel? (1 liter = 10-3 m3)
Suggested Answer :
Mass of fuel = 28 kg
Volume of fuel = 40 liter
= 40 x 10-3 m3
= 700 kg/m3
Measurement of Time
Thus just as length has units like kilometer, meter, centimeter, millimeter etc., time also has units like year, month, day, hour etc. You must have been hearing quite a lot about the ending of the millennium and starting of the third millennium in the year 2001. Here we have larger units measuring more time - thousand years. A millennium consists of 10 centuries. Each century consists of 100 (years) and in the range we finally come to the "second". One second is the smallest unit of time that we can see in clocks and watches by the clicking of the second hand.
In the past when there were no clocks people used water clocks, the hour glass, candle clocks and the sun dials.
We must thank Galileo Galilei for replacing all these devices by clocks and watches. Some of them are so accurate that they are beyond our imagination. The pendulum that you see in wall clocks started this replacement and Galileo was responsible for this.
Galileo is considered to be the father of modern science. In 1583, he discovered that the time taken for a single oscillation of a pendulum of a given length is always constant. This paved the way for the use of pendulum for controlling the time element in the clocks.
In the past when there were no clocks people used water clocks, the hour glass, candle clocks and the sun dials.
Galileo is considered to be the father of modern science. In 1583, he discovered that the time taken for a single oscillation of a pendulum of a given length is always constant. This paved the way for the use of pendulum for controlling the time element in the clocks.
Simple Pendulum
The length of the pendulum is the distance from the point of suspension to the center of gravity of the bob. The resting position of a simple pendulum is known as the mean position.
The time taken for one oscillation is known as the time period (T).
The number of oscillations made by the pendulum in one second is called its frequency (symbol n or f). Frequency is measured in hertz (Hz).
- The period of a simple pendulum of constant length is independent of its mass, size, shape or material.
- The period of a simple pendulum is independent of the amplitude of oscillation, provided it is small.
- The period of a simple pendulum is directly proportional to the square root of length of the pendulum.
- The period of a simple pendulum is inversely proportional to the square root of the acceleration due to gravity.
where l = length of the pendulumg = gravitational constant. A seconds pendulum is a pendulum which takes 2 seconds for one oscillation.
Experiment :
Simple Pendulum Experiment
Aim :
To prove that T
l.Procedure :
Tie the hook of the bob on one end of a thread (more than 1 meter). Clamp the other end firmly between the gap of a split cork which is fixed to the clamp of the retort stand as shown in the diagram.
Measure the length 'l' from the middle of the bob to the lower edge of the split cork.
Pull the bob to one side (making an angle of 10o with the vertical line) and allow it to oscillate in one plane. Using a stopwatch record the time (t) taken for 20 complete oscillations. Repeat the experiment for different lengths (l) and record the corresponding time (t) in the tabular form as shown below:
Observations :
| No. of Trails | Lengths 'l' of pendulum (cm) | Time for 20 Oscillations 't' (s) | Time for one oscillation 'T' (s) | T2(S)2 | T2/l (s2 cm-1) |
|---|---|---|---|---|---|
| 1 | 20 | ||||
| 2 | 40 | ||||
| 3 | 60 | ||||
| 4 | 80 | ||||
| 5 | 100 |
(ii) Draw a graph of l against T
Graphs are very helpful in comparing measurements. The general rule in plotting the graph of a given data is to plot the independent variable on the horizontal axis (X-axis). The data, which changes, is plotted on the vertical axis (Y-axis). It is possible to use the graph to predict the reading of measurements that lie between those actually made. This is known as interpolation. If the pattern of the graph is extended beyond the observed data in either direction, a prediction may also be made of readings lying outside the observed data. Predicting readings by this method is called extrapolation.
Every graph should have the following:
- a title (e.g., Load against extension)
- both axes labeled with units
- scales being used on each axis
- points plotted with crosses or with dot and circle.
The plotted points must be joined with a single straight line or a continuous curve. The graph should cover as much of the area of the graph paper as possible. This requires a sensible choice of scale for each axis. It is unlikely that points plotted from real experimental results will all lie on a straight line or smooth curve due to errors. Hence try to produce a straight line or smooth curve which passes through as many of the plotted points as possible or which leaves an equal distribution of points on either side. Refer the graph shown in figure (a) above which shows a best fitting line.
Solved Examples :
Example 1:
The diagram shown is a section of Vernier Calliper. Find the least count of the instrument and also the final reading, which is the thickness of a metal sheet.
Suggested answer :
Least count of the Vernier
= 0.01 cm
Reading of the instrument = Main scale reading
+ (coinciding v.s. div x least count)
= 4.3 + (8 x 0.01)
= 4.3 + 0.08
= 4.38 cm
Example 2:
In a Vernier calliper 1 cm of the main scale is divided into 20 equal parts. 19 divisions of the main scale coincide with 20 divisions on the vernier scale. Find the least count of the instrument.
Suggested answer :
The value of one main scale division
Number of divisions on vernier scale = 20
Least count of the vernier scale
= 0.025 cm
Example 3:
The circular scale (head scale) of a screw gauge is divided into 100 equal parts and it moves 0.5 mm ahead in one revolution. Find the pitch and the least count.
Suggested answer :
Pitch = distance moved in one revolution
= 0.5 mm
Least count
= 0.005 mm
Example 4:
The accompanying diagram represents a screw gauge. The circular scale is divided in to 50 divisions and the linear scale is divided into millimeters. If the screw advances by 1 mm when the circular scale makes 2 complete revolutions, find the least count of the instrument and the reading of the instrument in figure below.
Suggested answer :
Pitch of the screw
= 0.5 mm
Least count
= 0.01 mm
Reading = L.S. reading + (coinciding circular scale x least count)
= 3.5 mm + (32 x 0.01)
= 3.5 + 0.32
= 3.82 mm
Example 5:
Two simple pendulums are of lengths 40 cm and 1.6 m respectively. What will be the ratio of their time periods?
Suggested answer :
Since
therefore,
or T1 : T2 = 1 : 2
Example 6 :
The diagram below shows part of the main scale and vernier of a calliper, which is used to measure the diameter of a metal ball. Find the radius of the ball (sphere).
Suggested answer :
Least count
= 0.01 cm
Reading (diameter) = M.S. reading + (coinciding V.S. reading x Least count)
= 4.3 cm + (7 x 0.01)
= 4.3 + 0.07
= 4.37 cm
Diameter = 4.37 cm
Radius 
= 2.185 cm
= 2.18 cm
To the required number of significant figures.
Example 7:
The diagram shown below shows a part of the linear scale and head scale (circular scale) of a micrometer screw which is used to measure the thickness of a glass plate. Calculate the thickness (pitch=0.5 mm), Total number of divisions on head scale = 50.
Suggested answer :
Pitch = 0.5 mm
Least count 
= 0.01 mm
Thickness = Linear S.D + (Circular S.D. x Least count)
= 3.5 mm + (11 x 0.01) mm
= 3.61 mm.
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