Figure 1: Sine bar.

A sine bar is a precision measuring instrument usedto measure or set angles accurately. It operates based on the principles of trigonometry, specifically the sine function. It is as shown in Fig. 1, which consists of two rollers of same diameters fixed at a distance (L) between them, called as length of sine bar.

Working Principle of Sine Bar

The sine bar works on the principle of trigonometry. It is primarily used to measure or set angles. The angle of inclination (θ) is calculated using the formula:

\[ \sin \theta = \frac{\text{Height of gauge block (h)}}{\text{Length of sine bar (L)}}\]

To measure an angle:

1. One roller is placed on a flat surface, while the other is raised using gauge blocks.
2. The height of the gauge blocks corresponds to the sine of the angle to be measured.

Figure 2: Principle of Sine bar.

Sine bar works on basic principle of trigonometric sine function as explained in Figure 2. If one of the roller is lifted and the appropriate value of slip gauge are placed (total height, h), then,

\[ \sin \theta =\frac{h}{L}\]

\[ \theta ={{\sin }^{-1}}(h/L)\]

To calculate θ, sine function is used in above Equation, hence the instrument is called as sine bar.

Construction of Sine Bar

A sine bar consists of:

• Body: A precisely machined rectangular block.
• Rollers: Two accurately machined cylindrical rollers fixed at the ends of the bar. The distance between the roller centers is standardized (e.g., 100 mm or 200 mm).
• Relief Holes: These reduce the weight and ensure easy handling.
• Upper Surface: Precisely flat and used as a reference for placing the workpiece.
• Lower Surface: Parallel to the upper surface, forming the base.
• End Faces: Flat and perpendicular to the length of the sine bar.

Types of Sine Bars

1. Plain Sine Bar: Standard sine bar used for general angular measurements.
2. Compound Sine Bar: Used for measuring compound angles.
3. Sine Center: Designed to hold cylindrical workpieces.
4. Sine Table: A sine bar integrated into a table for easier adjustment.
5. Special Sine Bars: Custom-designed for specific applications.

Advantages of Sine Bar

1. Highly accurate for angle measurements.
2. Simple to use and reliable.
3. Requires minimal maintenance.
4. Suitable for precise machining and inspection tasks.

Limitations of Sine Bar

1. Used for angle measurement below 45º only.
2. Time consuming process of measurement.
3. Accuracy/grade of sine bar matters on accuracy of angle measured.

Errors in Sine Bar

Following could be errors in sine bar, by removing which accuracy can be improved.

1. Roller diameters
2. True roller cylinders
3. Error in center distance of rollers
4. Setting of rollers
5. Flatness of upper face of sine bar

Care of Sine Bar

1. Proper length of sine bar is to be selected.
2. Sine bar should be free from dust, dirt etc.
3. Clamping of job should be done properly on sine bar while measuring.

Uses of Sine Bar

1. Setting workpieces at specific angles in machining operations.
2. Checking angles of machine parts and tools.
3. Calibrating and inspecting angular gauges.
4. Measuring taper angles of components.

Use for Small Job Measurement / Taper Plug Gauge :

Figure 3: Small job measurement using Sine bar.

First, calculate the approximate value of angle of given job using bevel protractor. (e.g. 5º). Calculate the height h for the same angle using,

\[\sin \theta =h/L\]

\[\sin 5=h/100\text{ }\left( \text{Assuming L = 100 mm} \right)\]

\[h=8.7155\text{ mm}\]

Arrange slip gauges of size 8.715 mm by selecting from a slip gauge box by using proper wringing method (e.g. 8.715 = 1.005 + 1.41 + 1.3 + 5). Put the assembled slip gauges under one roller of sine bar (Fig. 3) and set the dial gauge with stand. By moving the dial gauge throughout the length of job, observe the deviation of dial gauge pointer. Rearrange the slip sizes by adding or subtracting slips if required and repeat till zero deviation is observed. For example, at height of 8.710 mm, zero deviation is observed. Calculate, new angle θ using the same Equation.

\[\sin \theta =\frac{h}{L}=\frac{8.710}{100}\]

\[\theta =4{}^\circ 5{9}’48.4{6}”\]

Use for Big Job Measurement :

Figure 4: Big job measurement using Sine bar.

Sine bar can also be used for measurement of angles of heavy jobs. Basic principle is same, but the arrangement is only different i.e. sine bar is to be kept on the job. (Fig. 4). Place proper height of slip gauges by making arrangement as shown in Fig. 4 and get θ.

Why is Sine Bar Limited Below 45°?

Figure 5.

We know that sinθ = h/l is the basic principle of working of sine bar. Differentiating equation,

\[\cos \theta .d\theta =\frac{l.dh-h.dl}{{{l}^{2}}}=\frac{dh}{l}-\frac{h.dl}{{{l}^{2}}}\]

\[\cos \theta .d\theta =\frac{dh}{l}\sin \theta -\frac{dl}{l}\sin \theta \]

\[d\theta =\tan \theta \left( \frac{dh}{h}-\frac{dl}{l} \right)\]

This indicates that error is a function of tan θ and below 45º error is smaller which suddenly increases above 45º. Because of this sine bar is preferred for measuring angle below 45º. Refer Fig. 5.

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Figure 1: Angle Gauge.

Working Principle of Angle Gauges

Angle gauges are used as a standard for angle measurement/angle setting in workshops. They are made up of hardened steel blocks of about 75 mm long and 15 mm wide. The two surfaces, which make the angle are granted and lapped to high degree of accuracy. A set of angle gauges consists of 14 gauges in total including a 90º square block.

Sizes of gauges are :

1º, 3º, 9º, 27º, 41º     Qty:5

1′, 3′, 9′, 27″               Qty:4

3″, 6″, 18″, 30″           Qty:4

Square block (90º)   Qty:1

Thus, the total of 14 gauges.

The angle gauge is shown in Fig. 1. It has markings of angle size and direction of angle.

Addition of Angle Gauges and Subtraction of Angle Gauges

Figure 2: Addition of Angle Gauges.

Figure 3: Subtraction of Angle Gauges.

Angles can be added or subtracted by keeping them on one another as shown in Fig. 2 and Fig. 3. Note the position of marks (<) for addition and subtraction.

Use of Angle Gauges

Figure 4: Use of Angle Gauges (Less than 90º).

Figure 5: Use of Angle Gauges (Greater than 90º).

Angle gauges can be used for the measurement of angles of given jobs for less than 90º as shown in Fig. 4. Angle greater than 90º can be measured by using a square block as shown in Fig. 5.  It is possible to set any angle to the nearest of 3″. These are similar to slip gauges, i.e., we can built up angle in combinations. Direct combination of all gauges given an angle upto 81º 40.9′ and more than this can be developed by using 90º square block.

Applications of Angle Gauge

Angle gauges can be used for :

1. Built up of given angle (as a standard).
2. Measuring angle between two surfaces by inserting in the surfaces (If less than 90º angle).
3. Measuring of angle greater than 90º by using square black and other angle gauges.

Angle gauges can be calibrated using interferometry or slip gauges and sine bar combinations.

Difference between Angle Gauge and Slip Gauge

The post What is Angle Gauge? Working Principle, Construction, Diagram & Reading Procedure appeared first on ElectricalWorkbook.

Figure 1: Spirit Level.

A spirit level is a tool used to measure and verify the alignment of a surface, ensuring it is either perfectly horizontal (level) or vertical (plumb). It works on the principle of gravity and uses a sealed vial filled with liquid and an air bubble. The bubble moves within the vial and aligns with the center markings when the surface is correctly positioned.

Construction of Spirit Level

A glass tube, whose surfaces are grounded, is fixed in a frame as shown in Fig. 1. A glass tube has got radius ‘R’ which is very large and filled with ether liquid, as well as ether bubble (vapor). Its construction includes:

• Body: Typically made of wood, metal, or plastic, providing durability and a straight edge.
• Glass Tube (Vial): A sealed, curved, and transparent tube filled with liquid (alcohol or oil) and a bubble of air.
• Ether Vapor Bubble: The bubble in the vial indicates the level when it is centered between the marked lines.
• Cross Spirit Level (90°): A secondary vial positioned at 90° to check for vertical alignment.
• Markings: Lines on the vial help determine precise leveling.

Working Principle of Spirit Level

The spirit level operates based on gravity:

1. The glass vial is slightly curved, and the bubble naturally moves to the highest point.
2. When the surface is level or plumb, the bubble aligns perfectly between the marked lines on the vial.

Working of Spirit Level

Figure 2: Spirit Level.

The position of the bubble in the tube will be always at the topmost level which is the principle of working of the spirit level. Fig. 2 shows the principle sketch of spirit level. In this, R-radius of curved glass tube A – is the topmost position of the bubble at horizontal position of spirit level. If bubble moves one division, i.e. from B to B’, the angle of tilt is θ. The level sensitivity per division i.e. the tilt causing the displacement of bubble by one division of the scale is expressed in mm per meter, e.g. 0.02 mm/meter.

For measuring very small angle in terms of seconds, the movement of bubble in terms of divisions is measured and multiplied by least count/level sensitivity. If R is the radius of curvature of the inside surface of the glass tube and α is angle of tilt in radians,

Displacement of bubble, l = R · α

The sensitivity of a sprit level depends on R As R increases, sensitivity increases. If α is expressed in seconds,

\[l=\frac{R{\alpha }”}{206265}\]

[1 radians = 206265 seconds of arc]

If the space between two lines is 2 mm and for a radius of R = 206 m,

\[{\alpha }”=\frac{2\times 206265}{206\times 1000}={2}”\]

This indicates inclination of 2″ of a spirit level will cause a bubble to move by 1 division or 2 mm. Spirit levels are to be handled very carefully and they are sensitive to temperature.

Advantages of Spirit Level

1. Easy to use and provides quick results.
2. Lightweight and portable.
3. Requires no power or complex setup.
4. Durable and reliable for long-term use.
5. High accuracy for both professional and DIY applications.

Uses of Spirit Level

• Checking the horizontal and vertical alignment of surfaces.
• Ensuring the proper leveling of furniture, tiles, and construction work.
• Plumbing and carpentry for installing shelves, cabinets, and frames.
• Aligning machinery and industrial equipment.
• Calibration tasks in surveying and engineering.

Care of Spirit Level

The following care is to be taken when spirit level is used.

1. Spirit level should be used properly for avoiding errors.
2. Temperature variation may effect on the accuracy of spirit level.
3. Calibration should be carried out time to time.
4. Care to be taken to avoid deformation of glass tube to avoid error.
5. It should not made vertical during handling.
6. As it contains glass tube, should be handled carefully.

Types of Spirit Levels

1. Standard Spirit Level: Used for general leveling tasks.
2. Torpedo Level: A compact version for tight spaces.
3. I-Beam Level: A durable and lightweight option, often used in construction.
4. Box-Beam Level: Known for high accuracy, suitable for professionals.
5. Cross-Check Level: Features vials for both horizontal and vertical leveling simultaneously.
6. Digital Spirit Level: Includes a digital display for precise angle readings.

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Figure 1: Micrometer.

Micrometer screw gauge works on the principle of nut and bolt. It consists of an accurate screw with about 10 or 20 threads per cm and revolves in a fixed nut. The rotation of thread is measured, which is already calibrated. In metric micrometers, the pitch of the screw is 0.5 mm so that one rotation of screw moves it axially by 0.5 mm. If main scale on barrel has least count of 0.5 mm.

\[\text{Least count = }\frac{\text{Smallest division on main scale}}{\text{Total number of division on circular scale}}\]

i.e.,

\[\text{LC = }\frac{0.5}{50}=0.01\text{ mm}\]

Micrometers can have least count of 0.01 mm, 0.001 mm, etc as per requirement and manufacturer.

Reading Micrometer (Least Count)

Four different cases are explained below to know reading micrometer.

Total Reading = (Main scale reading) + (Circular scale division) × (Least count)

Figure 2.

Case (a):

Find MSR i.e. number of divisions visible on the main scale. Find number of divisions crossed on circular scale by the horizontal line on main scale (Reference line), that is CSD. Refer to Fig. 2(a),

TR = MSR + CSD × LC

= 7 + 15 × 0.01 = 7.15 mm

Case (b):

Refer to Fig. 2(b)

TR = MSR + CSD ×  LC

= 2.5 + 0 × 0.01 = 2.5 mm

Case (c):

Refer to Fig. 2(c)

TR = MSR + CSD × LC

= 4.5 + 1 × 0.01 = 4.51 mm

Case (d):

Refer to Fig. 2(d)

TR = MSR + CSD × LC

= 7 + 47 × 0.01 = 7.47 mm

Construction and Working of Simple micrometer

Simple micrometer consists of a screw with fixed pitch and nut arrangement. The end of screw acts as a spindle / anvil in the base of the frame. Spindle can be advanced or retracted by using thimble connected to spindle. The barrel has engraved lines of main scale (LC = 0.5 mm) which is true scale. Thimble has circular scale with number of divisions engraved on it. A lock nut is provided for locking a dimension. (Fig. 1). The Ratchet screw provides a stop, if excessive pressure is applied by the anvils, which is one of the most important part for avoiding errors. When two jaws touches each other, zero reading is expected. If it is not observed, error can be adjusted by rotating barrel itself. Fig. 3 shows a digital micrometer.

Figure 4: Digital Micrometer.

Depth Micrometer

Figure 4: Depth Micrometer.

Depth micrometer is used for measuring depths of holes, slots, etc. Its principle is similar to micrometer. It consists of one shoulder that acts as the reference while measurement is carried out. Extension rods can be used for large ranges of measurement e.g.

Rod-1: Range 0 – 25

Rod-2: Range 25 – 50

Rod-3: Range 50 – 75

Rod-4: Range 75 – 100

If extension rods are used, the reading is to be calculated as :

Total Reading = [MSR + (CSD × LC)] + [Lower range of extension rod.]

A very important point to be noted is that its scale is calibrated in the reverse direction. Refer to Fig. 4.

Inside Micrometer

Figure 5: Inside Micrometer.

Fig. 5 shows inside micrometer with micrometer unit, extension rods, spacing collars, etc. The basic micrometer consists of a nut bolt principle and range starting from 50 onwards or like that. For various inner dimensions, extension rods can be attached as per the requirements. Number of extension rods can be attached at a time and larger distance can be measured e.g. Minimum 175 i.e. [Basic unit (50) + Extension rod (25) + Extension rod (100)] to maximum 188 (i.e. 175 + 13).

Collar is provided for avoiding errors in the readings provided on extension rods. Zero error can be adjusted by using master unit provided with instrument.

Precautions While Using Micrometer

1. Instruments must be used by proper process.
2. It must be calibrated.
3. It is to be used with standard environmental conditions.
4. Vibrations or such errors are to be avoided.
5. Proper knowledge is needed about method of reading an instrument.
6. Proper least count range is to be selected.

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Figure 1: Parkinson’s Gear Tester.

Instead of measuring individual errors in gears, composite errors are checked. The composite error indicates the combination of errors in a gear. In Parkinson’s Gear Tester, the gear to be tested is rolled in mesh with a master gear and the errors will be indicated by dial gauge or recording unit.

Construction and Working of Parkinson’s Gear Tester

One fixed spindle and other movable spindle is mounted on a flat base. The movable spindle moves along with base by rolling action on the main base plate as shown in Fig. 1.

A master gear is mounted on the fixed spindle whereas, a gear to be tested is to be mounted on a movable spindle. The dial gauge is set to note the errors as shown in Fig. 1 whose pointer touches the floating body.

Procedure for Measurement using Parkinson’s Gear Tester

Figure 2: Profiles of gears.

When master gear is rotated slowly, a gear to be tested will also get rotation movement because of their meshing. Errors in the manufactured gear cause the gear to move away from the center line of spindle. When gear (on test) moves, the floating body, also move, by the same distance. Because of displacement of floating body, dial gauge gives displacement. The variations in the readings can be observed and plotted in the graphical format. Fig. 2 shows a typical graphs for unsatisfactory, moderate and satisfactory gears.

Limitations of Parkinson’s Gear Tester

1. Maximum 300 mm diameter gears can be tested.
2. The floating body is very sensitive and hence readings are to be taken very carefully.
3. Accuracy upto 1 µm can be possible while measurement
4. Only composite errors in the gears can be checked, not individual one.
5. Measurement depends upon the master gear.

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Figure 1: Tool Maker’s Microscope.

Tool Maker’s Microscope is a direct reading measuring instrument used extensively in the shop for inspecting gauges, tools, screw threads, jigs, dies, and fixtures. The essential features of the microscope are shown in Fig. 1.

Construction and Working of Tool Maker’s Microscope

The workpiece to be inspected is mounted on a micrometer stage under a microscope. The stage is operated by two accurate micrometer screws on compact slides moving on precision ball bearings.

The movement of the slide can be measured to 0.0002 mm in the horizontal plane in two mutually perpendicular directions. The microscope provides a magnification of 10 to 100 times. The dimension is measured by first aligning the microscope hairlines at one end of the dimension to be measured and then moving the table till the other end of the dimension reaches the hairline.

The movement of the table as read by the micrometer is the length of the dimension. A protractor ocular with two angular scales is provided with the tool maker’s microscope for angular measurements. This can be used for measurement of angles with a least count of 1 minute. To make an angle measurement the eye piece hair line is first made parallel to one side of the angle being measured and the reading on the protractor scale noted. The hairline is then rotated to touch on the other side of the angle and the angle is read again. The difference in the two readings is the angular measurement desired. When making measurements on screw threads, worms, hobs, and other similar tools it is necessary that the microscope be tilted at an angle from the vertical. This adjustment is provided on the microscope with the help of two tangent screws.

A common tool maker’s microscope has a longitudinal range of 100 mm and a cross range of 50 mm. 50 mm of the longitudinal movement are obtained by the screw while the remaining 50 mm are obtained by insertion of gauge blocks.

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Figure 1: Gear Tooth Vernier Caliper.

The gear tooth thickness measuring vernier is as shown in Fig. 1. It measures the thickness of the tooth on pitch circle i.e. chordal thickness.

Construction of Gear Tooth Vernier Caliper

1. Main Scale: A fixed linear scale used to measure lengths and provide a reference for the vernier scale.
2. Vernier Scale: A secondary scale that provides precise measurements by dividing the main scale’s least division into smaller increments.
3. Fixed Jaw: A stationary jaw used as a reference point during measurements.
4. Movable Jaws:
   • Movable Jaw 1: Measures the chordal thickness of the gear tooth.
   • Movable Jaw 2: Measures the depth of the gear tooth.
5. Lock Nuts: Used to fix the movable parts of the caliper in position during measurement to avoid errors.
6. Fine Adjustment Screw: Enables precise movement of the movable jaws for accurate measurements.
7. Bevel Edge: The edges of the jaws are beveled to fit the contours of gear teeth.

Working of Gear Tooth Vernier Caliper

Gear Tooth Vernier Caliper consists of two beams which are in square with each other. There are two sliding vernier scales which moves along the beams of main scale. The tooth thickness on the pitch circle is measured as the distance between a fixed jaw and a movable jaw by fixed adjustable jaw of vertical vernier beam. In other words it is a combination of two vernier with a common jaw.

Working Principle of Gear Tooth Vernier Caliper

Figure 2: Working Principle of Gear Tooth Vernier Caliper

Gear tooth thickness is measured at pitch circle, and it is also called as pitch line thickness (See Figure 2).  In the above sketch ‘d’ is chordal addendum, which can be calculated as,

\[\text{d = }\frac{\text{Nm}}{\text{2}}\left[ \text{1 + }\frac{\text{2}}{\text{N}}-\text{cos}\left( \frac{\text{90}}{\text{N}} \right) \right]\]

Where,

N – Number of teeth

m – Module

Tooth thickness can be calculated using gear tooth vernier. By setting ‘d’ in vertical vernier, horizontal vernier gives ‘W. ‘W’ can be verified by,

\[\text{W = N}\text{.m sin}\left( \frac{\text{90}}{\text{N}} \right)\]

Procedure for Measurement using Gear Tooth Vernier Caliper

• Find chordal addendum by using by using Equation (Fig. 3).

\[\text{h = }\left( \frac{\text{mN}}{\text{2}} \right)\left[ \frac{\text{2}}{\text{N}}+1-\text{cos}\frac{\text{90}}{\text{N}} \right]\]

• Set height ‘h’ on vertical vernier by using fine adjustment screw.
• Apply vernier to gear tooth as shown in Fig. 3, so that the fix jaw can touch the flank of tooth.
• Push movable jaw of horizontal vernier and lock it.
• This gives the gear tooth thickness ‘W’.

Applications of Gear Tooth Vernier Caliper

1. Gear Manufacturing: To inspect the accuracy of gear dimensions during production.
2. Maintenance: To measure wear and tear on gear teeth and determine if replacements are needed.
3. Quality Control: Used in industries to ensure gears meet design specifications.

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Construction & Working Dead Weight Tester

A typical dead weight or piston gauge is schematically shown in the following figure 1,

Figure 1: Dead Weight Tester.

It consists of a chamber in which fluid is filled. The calibrated gauge is connected to the chamber. A pump and bleed valve are attached to the chamber so as to adjust the pressure of the fluid present in the chamber. A vertical piston cylinder is also connected to the chamber and different standard weights are applied to this cylinder. The third pressure is applied to the other side of the piston by operating the plunger. This pressure is applied until the piston and weights are seen to float. At this point the third gauge pressure is given by.

\[P=\frac{W}{a}\]

Where,

P = Fluid gauge pressure

W = Dead weight supported by the piston

a = Area of the piston.

Thus the pressure caused due to the weight placed on the vertical piston is calculated by using the above equation. To achieve high accurate results, frictional force between the cylinder and piston must be reduced which is generally accomplished by rotating the piston while the reading is taken.

Calibration of Pressure Measuring Instrument Using Dead Weight Tester

Figure 2: Dead Weight Tester for Pressure Calibration.

Static calibration of pressure transducers is carried by dead weight tester. It is the basic standard pressure calibration device. The dead weight tester can be operated by hydraulic means or pneumatic means. The schematic of dead weight tester is shown in the figure 2.

The dead weight tester consists of a chamber filled with fluid. Two accurately machined cylinders of known cross-section area are attached to the chamber. An accurately machined cylinder is closely fitted into one of the cylinders. This piston has a platform on its top so that the piston can be loaded with weights. A screwed plunger is incorporated into the chamber from one of its sides as shown in figure 2.

Dead Weight Tester Calibration Procedure 

The pressure transducer to be calibrated is connected to the other cylinder of dead weight tester and the piston is loaded with standard weights of high accuracy. Then, due to the gravitational force acting on these standard weights, a force is exerted on the known area i.e., an equivalent force of piston-weight combination acts on the fluid and develops a pressure on the fluid in the chamber.

Now, pressure is applied on the fluid from the other end by operating the plunger inwards until the combination of piston and standard weights float freely. At this condition the force of the piston-weight combination is equal to the force of the fluid pressure.

i.e.,

\[F\text{ }=\text{ }PA\]

Where,

F = Equivalent force of the piston-weight combination

= Mg (M = Total mass, g = Acceleration due to gravity)

P = Fluid pressure

A = Equivalent area of piston-cylinder combination

\[P=\frac{F}{A}\]

\[P=\frac{Mg}{A}…(1)\]

At equilibrium condition, the pressure developed on the fluid in the chamber is transmitted to the pressure gauge under calibration. The reading then indicated by the pressure gauge is equal to the value of P calculated from equation (1) above.

1. The static calibration of low pressure transducers is carried out by standard mercury or water manometers because these manometers provide accurate measurements of low pressure.
2. Pressure transducers can also be calibrated using secondary standards of pressure such as helical type bourdon gauges made up of quartz and force balance type pressure transducer.

Advantages of Dead Weight Tester

1. Its construction is simple and is very easy to operate.
2. It is used as a standard for calibration of a wide range of pressure measuring devices.
3. Fluid pressure can be varied easily either by adding pistons or by changing the piston cylinder.

Disadvantages of Dead Weight Tester

1. Friction between the piston and cylinder affects the accuracy of the gauge.
2. The gravitational force also affects the accuracy of the gauge.

Applications of Dead Weight Tester

It is used to measure pressure and also to calibrate all kinds of pressure gauges.

Factors Affecting the Accuracy of Dead Weight Tester

The following are the factors affecting the accuracy of dead weight tester,

1. The presence of friction between the piston and the cylinder.
2. Uncertainties in the value of gravitational constant.
3. Ambiguous value of effective area of piston cylinder.
4. Gravity
5. Mass, height and air buoyancy
6. Head of transmitting fluid
7. Elastic deformation of piston and cylinder
8. Thermal expansion
9. Weight of the fluid buoyancy
10. Levelness and cleanliness

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Figure 1: Inclined Tube Manometer.

Working & Construction of Inclined Tube Manometer

In an inclined tube manometer, the limb having a large cross-sectional area is known as well and the limb having small cross-sectional area is known as column. Therefore, it is considered as a single-column manometer. The column of this manometer is inclined at an angle θ with respect to the horizontal. The tube is filled with the manometer liquid.

Derivation & Formula of Inclined Tube Manometer

When no pressures are applied or when equal pressures are applied to the limbs of the manometer, the liquid in both the limbs (well and column) will be at same level (i.e. 0-0 level). When two different pressures are applied to the limbs, the liquid level decreases in the well, while the liquid level increases in the inclined column.

This leads to a difference between the liquid levels of the two limbs. The difference in the liquid levels is given as,

\[\Delta h\text{ }=\text{ }{{h}_{1}}+\text{ }{{h}_{2}}\]

Where,

h₁ = Level of liquid from 0-0  level in well

h₂ = Level of liquid from 0-0 level in column

Δh corresponds to the difference of the two pressures applied to the manometer and the relationship between Δh and pressures is given by tile following equation.

\[\Delta h=\frac{{{P}_{1}}-{{P}_{2}}}{\rho g}\]

also

\[{{h}_{1}}+{{h}_{2}}=\frac{{{P}_{1}}-{{P}_{2}}}{\rho g}\]

Where.

P₁ = Pressure applied to well

P₂ = Pressure applied to column

Due to increase and decrease in liquid level of column and well respectively, the displacement in the volume of the two limbs is equal.

i.e.

\[{{V}_{1}}={{V}_{2}}\]

\[{{A}_{1}}{{V}_{1}}={{A}_{2}}{{V}_{2}}\]

\[{{h}_{1}}=\frac{{{A}_{2}}}{{{A}_{1}}}{{h}_{2}}\]

But,

\[{{h}_{2}}=l.\sin \theta ….(1)\]

Where,

l = Slant height of the liquid in inclined column

A₁ = Cross-sectional area of well

A₂ = Cross-sectional area of column

From equation (1), we have.

\[\left( {{P}_{1}}-{{P}_{2}} \right)=\rho g\left( {{h}_{1}}+{{h}_{2}} \right)\]

\[=\rho g\left( \frac{{{A}_{2}}}{{{A}_{1}}}{{h}_{2}}+{{h}_{2}} \right)\]

\[=\rho g{{h}_{2}}\left( \frac{{{A}_{2}}}{{{A}_{1}}}+1 \right)\]

Also

\[\left( {{P}_{1}}-{{P}_{2}} \right)=\rho gl.\sin \theta \left( \frac{{{A}_{2}}}{{{A}_{1}}}+1 \right)\]

If

\[{{A}_{1}}\gg {{A}_{2}}\]

\[\frac{{{A}_{2}}}{{{A}_{1}}}\approx 0\]

\[\left( {{P}_{1}}-{{P}_{2}} \right)=\rho gl.\sin \theta ..(2)\]

\[\left( {{P}_{1}}-{{P}_{2}} \right)=\rho g{{h}_{2}}….(3)\]

Equations (2) and (3) represents the equations for differential pressure based on the movement of the liquid in the inclined only i.e., the differential pressure can be determined by measuring either h₂ or l.

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Figure 1: Infrared Pyrometer.

Every material with temperature above absolute zero emits infrared radiations proportional to the temperature. Infrared radiations are invisible to the human eye and can be sensed as heat. An infrared pyrometer is a non-contact-type sensor used to detect infrared radiation from a hot body. The radiation emitted by hot body has different wavelengths. Radiations having longer wavelengths than visible light are known as infrared radiations which possess less energy and are less harmful. A part of the infrared energy radiated by target is detected by an infrared detector to measure. The surface of the target begins to radiate when it attains a temperature of around 500 – 600°C. The infrared energy increases with increase in temperature.

Working Principle of Infrared Pyrometer

The benefit of infrared sensor is formed in Wien’s displacement I law which states that the maximum radiant intensity of low temperature bodies occurs in the infrared for example, at 25°C (298K) the body radiates at the maximum wavelength of 9.7 µm as per Wien’s law.

λ_max is the wavelength at which the maximum value of monochromatic emissive power occurs. Wien’s law states that the product of λ_max and T is constant i.e.

   λ_max T = 2.898 x 10^-3

Thus,

          λ_max for 298 K = 9.7 x 10^-6 m

Maximum wavelength = 9.7 µm    (λ_max for 298 K )

Normally radiations with the wavelength range of 0.7 – 14 µm are employed for measurement.

Construction & Working of Infrared Pyrometer

An infrared pyrometer consists of which four the infrared wave detector (temperature sensor). The infrared radiation is absorbed and converted into an electrical signal by the temperature detector. The amount of radiation striking the detector influences the electrical output signal. The amplified electrical output signal is then displayed on the display device with temperature units. Variation due to ambient temperature is properly compensated before display.

Advantages of Infrared Pyrometer

The infrared pyrometer provides better accuracy if emissivity of the material is known. Higher the emissivity of the material higher the accuracy of measurement.

Applications of Infrared Pyrometer

An infrared pyrometer is also useful for the measurement of temperatures of moving objects and in situations where objects are placed in vacuum or in a controlled atmosphere, or the distance between the source of the temperature and the instrument of measurement is large and when the objects are in inaccessible areas and hazardous conditions.

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