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2. Mass (Weight) Measurement
3. Force Measurement
4. Torque Measurement
5. Calibration of Mass, Force, and Torque Measuring Sensors
Mass, force, and torque are covered together within this section because they are closely related quantities. Mass describes the quantity of matter that a body contains. Force is the product of mass times acceleration, according to Newton's second law of motion:
Force= Mass _ acceleration:
Forces can be applied in either a horizontal or a vertical direction. A force applied in a downward, vertical direction gives rise to the term weight, which is defined as the downward force exerted by a mass subject to a gravitational force:
Weight = Mass x acceleration due to gravity:
The final quantity covered in this section, torque, can be regarded as a rotational force. When applied to a body, torque causes the body to rotate about its axis of rotation. This is analogous to the horizontal motion of a body when a horizontal force is applied to it.
2. Mass (Weight) Measurement
The mass of a body is always quantified in terms of a measurement of the weight of the body, this being the downward force exerted by the body when it’s subject to gravity. Three methods are used to measure this force.
The first method of measuring the downward force exerted by amass subject to gravity involves the use of a load cell. The load cell measures the downward force F, and then the mass M is calculated from the equation:
M = F/g,
…where g is acceleration due to gravity.
Because the value of g varies by small amounts at different points around the earth's surface, the value of M can only be calculated exactly if the value of g is known exactly. Nevertheless, load cells are, in fact, the most common instrument used to measure mass, especially in industrial applications.
Several different forms of load cells are available. Most load cells are now electronic, although pneumatic and hydraulic types also exist. These types vary in features and accuracy, but all are easy to use as they are deflection-type instruments that give an output reading without operator intervention.
The second method of measuring mass is to use a spring balance. This also measures the downward force when the measured mass is subject to gravity. Hence, as in the case of load cells, the mass value can only be calculated exactly if the value of g is known exactly. Like a load cell, the spring balance is also a deflection-type instrument and so is easy to use.
The final method of measuring mass is to use some form of mass balance instrument. These provide an absolute measurement, as they compare the gravitational force on the mass being measured with the gravitational force on a standard mass. Because the same gravitational force is applied to both masses, the exact value of g is immaterial. However, being a null-type instrument, any form of balance is tedious to use.
The following paragraphs consider these various forms of mass-measuring instruments in more detail.
Electronic Load Cell (Electronic Balance):
The electronic load cell is now the preferred type of load cell in most applications. Within an electronic load cell, the gravitational force on the body being measured is applied to an elastic element. This deflects according to the magnitude of the body mass. Mass measurement is thereby translated into a displacement measurement task.
The elastic elements used are specially shaped and designed, some examples of which are shown in --- +++1. The design aims are to obtain a linear output relationship between the applied force and the measured deflection and to make the instrument insensitive to forces that are not applied directly along the sensing axis. Load cells exist in both compression and tension forms. In the compression type, the measured mass is placed on top of a platform resting on the load cell, which therefore compresses the cell. In the alternative tension type, the mass is hung from the load cell, thereby putting the cell into tension.
Cylindrical block Proving frame Parallelogram-cut proving frame Octagonal-cut proving frame Rectangular block Proof ring
-- Various types of displacement transducers are used to measure the deflection of the elastic elements. Of these, the strain gauge is used most commonly, as this gives the best measurement accuracy, with an inaccuracy figure less than _0.05% of full-scale reading being obtainable.
Load cells, including strain gauges, are used to measure masses over a very wide range between 0 and 3000 ton. The measurement capability of an individual instrument designed to measure masses at the bottom end of this range would typically be 0.1-5 kg, whereas instruments designed for the top of the range would have a typical measurement span of 10-3000 ton.
Elastic force transducers based on differential transformers (LVDT) to measure defections are used to measure masses up to 25 ton. Apart from having a lower maximum measuring capability, they are also inferior to strain gauge-based instruments in terms of their _0.2% inaccuracy value. Their major advantages are their longevity and almost total lack of maintenance requirements.
The final type of displacement transducer used in this class of instrument is the piezoelectric device. Such instruments are used to measure masses in the range of 0 to 1000 ton. Piezoelectric crystals replace the specially designed elastic member used normally in this class of instrument, allowing the device to be physically small. As discussed previously, such devices can only measure dynamically changing forces because the output reading results from an induced electrical charge whose magnitude leaks away with time. The fact that the elastic element consists of a piezoelectric crystal means that it’s very difficult to design such instruments to be insensitive to forces applied at an angle to the sensing axis. Therefore, special precautions have to be taken in applying these devices. Although such instruments are relatively inexpensive, their lowest inaccuracy is _1% of full-scale reading and they also have a high temperature coefficient.
Electronic load cells have significant advantages over most other forms of mass-measuring instruments in terms of their relatively low cost, wide measurement range, tolerance of dusty and corrosive environments, remote measurement capability, tolerance of shock loading, and ease of installation. However, one particular problem that can affect their performance is the phenomenon of creep. Creep describes the permanent deformation that an elastic element undergoes after it has been under load for a period of time. This can lead to significant measurement errors in the form of a bias on all readings if the instrument is not recalibrated from time to time. However, careful design and choice of materials can largely eliminate the problem.
Several compression-type load cells are often used together in a form of instrument known as an electronic balance. Commonly, either three or four load cells are used in the balance, with the output mass measurement being formed from the sum of the outputs of each cell. Where appropriate, the upper platform can be replaced by a tank for weighing liquids, powders, and so on.
Pneumatic and Hydraulic Load Cells:
Pneumatic and hydraulic load cells translate mass measurement into a pressure measurement task, although they are now less common than the electronic load cell. A pneumatic load cell is shown. Application of a mass to the cell causes deflection of a diaphragm acting as a variable restriction in a nozzle-flapper mechanism. The output pressure measured in the cell is approximately proportional to the magnitude of the gravitational force on the applied mass. The instrument requires a flow of air at its input of around 0.25 m^3/h at a pressure of 4 bar. Standard cells are available to measure a wide range of masses. For measuring small masses, instruments are available with a full-scale reading of 25 kg, while instruments with a full-scale reading of 25 ton are obtainable at the top of the range. Inaccuracy is typically _0.5% of full scale in pneumatic load cells.
The alternative, hydraulic load cell. In this, the gravitational force due to the unknown mass is applied, via a diaphragm, to oil contained within an enclosed chamber. The corresponding increase in oil pressure is measured by a suitable pressure transducer. These instruments are designed for measuring much larger masses than pneumatic cells, with a load capacity of 500 ton being common. Special units can be obtained to measure masses as large as 50,000 ton. In addition to their much greater measuring range, hydraulic load cells are much more accurate than pneumatic cells, with an inaccuracy figure of _0.05% of full scale being typical. However, in order to obtain such a level of accuracy, correction for the local value of g (acceleration due to gravity) is necessary. A measurement resolution of 0.02% is attainable.
Intelligent Load Cells:
Intelligent load cells are formed by adding a microprocessor to a standard cell. This brings no improvement in accuracy because the load cell is already a very accurate device. What it does produce is an intelligent weighing system that can compute total cost from the measured weight, using stored cost per unit weight information, and provide an output in the form of a digital display. Cost per weight values can be pre-stored for a large number of substances, making such instruments very flexible in their operation.
In applications where the mass of an object is measured by several load cells used together (e.g., load cells located at the corners of a platform in an electronic balance), the total mass can be computed more readily if the individual cells have a microprocessor providing digital output. In addition, it’s also possible to use significant differences in the relative readings between different load cells as a fault detection mechanism in the system.
Mass Balance (Weighing) Instruments
Mass balance instruments are based on comparing the gravitational force on the measured mass with the gravitational force on another body of known mass. This principle of mass measurement is known commonly as weighing and is used in instruments such as the beam balance, weigh beam, pendulum scale, and electromagnetic balance. Various forms of mass balance instruments are available, as discussed next.
--- Known standard mass; Pointer and scale; Unknown force
Beam balance (equal arm balance)
In the beam balance standard masses are added to a pan on one side of a pivoted beam until the magnitude of the gravity force on them balances the magnitude of the gravitational force on the unknown mass acting at the other end of the beam. This equilibrium position is indicated by a pointer that moves against a calibrated scale.
Instruments of this type are capable of measuring a wide span of masses. Those at the top end of the range can typically measure masses up to 1000 grams, whereas those at the bottom end of the range can measure masses of less than 0.01 gram. Measurement resolution can be as good as 1 part in 10^7 of the full-scale reading if the instrument is designed and manufactured very the knife edges, leading to deterioration in measurement accuracy and measurement resolution.
A further problem affecting their use in industrial applications is that it takes a relatively long time to make each measurement. For these reasons, the beam balance is normally reserved as a calibration standard and is not used in day-to-day production environments.
Graduated bar; Movable standard mass; Movable standard masses; Null marker, Null marker, Fixed standard mass, Unknown mass, Unknown mass, Notched bars
The weigh beam, sketched in two alternative forms, operates on similar principles to the beam balance but is much more rugged. In the first form, standard masses are added to balance the unknown mass and fine adjustment is provided by a known mass that is moved along a notched, graduated bar until the pointer is brought to the null, balance point. The alternative form has two or more graduated bars (three bars shown). Each bar carefully. The lowest measurement inaccuracy value attainable is _0.002%.
One serious disadvantage of this type of instrument is its lack of ruggedness. Continuous use and the inevitable shock loading that will occur from time to time both cause damage to carries a different standard mass, which is moved to appropriate positions on the notched bar to balance the unknown mass. Versions of these instruments are used to measure masses up to 50 ton.
Counterweights Steel tape Unknown force Pointer and scale
The pendulum scale is another instrument that works on the mass-balance principle. In one arrangement, the unknown mass is put on a platform that is attached by steel tapes to a pair of cams. Downward motion of the platform, and hence rotation of the cams, under the influence of the gravitational force on the mass, is opposed by the gravitational force acting on two pendulum-type masses attached to the cams. The amount of rotation of the cams when the equilibrium position is reached is determined by the deflection of a pointer against a scale. The shape of the cams is such that this output deflection is linearly proportional to the applied mass. Other mechanical arrangements also exist that have the same effect of producing an output deflection of a pointer moving against a scale. It’s also possible to replace the pointer and scale system by a rotational displacement transducer that gives an electrical output. Various versions of the instrument can measure masses in the range between 1 kg and 500 ton, with a typical measurement inaccuracy of _0.1%.
Recently, the instrument has become much less common because of its inferior performance compared with instruments based on newer technology such as electronic balances. One potential source of difficulty with the instrument is oscillation of the weigh platform when mass is applied. Where necessary, in instruments measuring larger masses, dashpots are incorporated into the cam system to damp out such oscillations. A further possible problem can arise, mainly when measuring large masses, if the mass is not placed centrally on the platform. This can be avoided by designing a second platform to hold the mass, which is hung from the first platform by knife edges. This lessens the criticality of mass placement.
-- Unknown force Known mass Null detector Amplifier Light detector Window Voltage source Light source Ammeter Permanent magnet Bearings Coil
The electromagnetic balance uses the torque developed by a current-carrying coil suspended in a permanent magnetic field to balance the unknown mass against the known gravitational force produced on a standard mass. A light source and detector system is used to determine the null-balance point. The voltage output from the light detector is amplified and applied to the coil, thus creating a servo system where deflection of the coil in equilibrium is proportional to the applied force. Its advantages over beam balances, weigh beams, and pendulum scales include its smaller size, its insensitivity to environmental changes (modifying inputs), and its electrical form of output. Despite these apparent advantages, it’s no longer in common use because of the development of other instruments, particularly electronic balances.
Spring balances provide a method of mass measurement that is both simple and inexpensive.
The mass is hung on the end of a spring and deflection of the spring due to the downward gravitational force on the mass is measured against a scale. Because the characteristics of the spring are very susceptible to environmental changes, measurement accuracy is usually relatively poor. However, if compensation is made for changes in spring characteristics, then a measurement inaccuracy less than _0.2% is achievable. According to the design of the instrument, masses between 0.5 kg and 10 ton can be measured.
This section is concerned with the measurement of horizontal forces that either stretch or compress the body that they are applied to according to the direction of the force with respect to the body. If a force of magnitude, F, is applied to a body of mass, M, the body will accelerate at a rate, A, according to the equation:
F = MA:
The standard unit of force is the Newton, this being the force that will produce an acceleration of 1meter per second squared in the direction of the force when applied to amass of 1 kilogram.
One way of measuring an unknown force is therefore to measure acceleration when it’s applied to a body of known mass. An alternative technique is to measure the variation in the resonant frequency of a vibrating wire as it’s tensioned by an applied force. Finally, forms of load cells that deform in the horizontal direction when horizontal forces are applied can also be used as force sensors. These techniques are discussed next.
Use of Accelerometers
The technique of applying a force to a known mass and measuring the acceleration produced can be carried out using any type of accelerometer. Unfortunately, the method is of very limited practical value because, inmost cases, forces are not free entities but are part of a system (from which they cannot be decoupled) in which they are acting on some body that is not free to accelerate. However, the technique can be of use in measuring some transient forces and also for calibrating forces produced by thrust motors in space vehicles.
Vibrating Wire Sensor:
This instrument consists of a wire that is kept vibrating at its resonant frequency by a variable-frequency oscillator. The resonant frequency of a wire under tension is given by ...
...where M is the mass per unit length of the wire, L is the length of the wire, and T is the tension due to the applied force, F. Thus, measurement of the output frequency of the oscillator allows the force applied to the wire to be calculated.
Use of Load Cells
Special forms of electronic load cells designed to deflect in the horizontal direction are used to measure horizontal forces applied to them.
Measurement of applied torques is of fundamental importance in all rotating bodies to ensure that the design of the rotating element is adequate to prevent failure under shear stresses.
Torque measurement is also a necessary part of measuring the power transmitted by rotating shafts. The four methods of measuring torque consist of (i) measuring the strain produced in a rotating body due to an applied torque, (ii) an optical method, (iii) measuring the reaction force in cradled shaft bearings, and (iv) using equipment known as the Prony brake. Of these, the first two should be regarded as "normal" ways of measuring torque at the present time as the latter two are no longer in common use.
Measurement of Induced Strain
Measuring the strain induced in a shaft due to an applied torque has been the most common method used for torque measurement in recent years. The method involves bonding four strain gauges onto a shaft, where the strain gauges are arranged in a d.c. bridge circuit. The output from the bridge circuit is a function of the strain in the shaft and hence of the torque applied. It’s very important that positioning of the strain gauges on the shaft is precise, and the difficulty in achieving this makes the instrument relatively expensive.
This technique is ideal for measuring the stalled torque in a shaft before rotation commences.
However, a problem is encountered in the case of rotating shafts because a suitable method then has to be found for making the electrical connections to the strain gauges. One solution to this problem found in many commercial instruments is to use a system of slip rings and brushes for this, although this increases the cost of the instrument still further.
Optical Torque Measurement
Optical techniques for torque measurement have become available recently with the development of laser diodes and fiber-optic light transmission systems.
Two black-and-white striped wheels are mounted at either end of the rotating shaft and are in alignment when no torque is applied to the shaft. Light from a laser diode light source is directed by a pair of fiber-optic cables onto the wheels. The rotation of the wheels causes pulses of reflected light, which are transmitted back to a receiver by a second pair of fiber-optic cables. Under zero torque conditions, the two pulse trains of reflected light are in phase with each other. If torque is now applied to the shaft, the reflected light is modulated. Measurement by the receiver of the phase difference between the reflected pulse trains therefore allows the magnitude of torque in the shaft to be calculated. The cost of such instruments is relatively low, and an additional advantage in many applications is their small physical size.
Reaction Forces in Shaft Bearings:
Any system involving torque transmission through a shaft contains both a power source and a power absorber where the power is dissipated. The magnitude of the transmitted torque can be measured by cradling either the power source or the power absorber end of the shaft in bearings, and then measuring the reaction force, F, and the arm length, L.
The torque is then calculated as the simple product, FL. Pendulum scales are used very commonly for measuring the reaction force. Inherent errors in the method are bearing friction and windage torques. This technique is no longer in common use.
The Prony brake is another torque-measuring system that is now uncommon. It’s used to measure the torque in a rotating shaft and consists of a rope wound round the shaft. One end of the rope is attached to a spring balance and the other end carries a load in the form of a standard mass, m. If the measured force in the spring balance is F s, then the effective force, F e, exerted by the rope on the shaft is given by
Fe = mg _ Fs:
Spring balance; Dead load; Rotating brake drum
If the radius of the shaft is Rs and that of the rope is Rr, then the effective radius, Re, of the rope and drum with respect to the axis of rotation of the shaft is given by:
Re = Rs + Rr:
The torque in the shaft, T, can then be calculated as:
T = F eRe:
While this is a well-known method of measuring shaft torque, a lot of heat is generated because of friction between the rope and shaft, and water cooling is usually necessary.
Calibration of Mass, Force, and Torque Measuring Sensors
One particular difficulty that arises in the calibration of mass, force, and torque measuring instruments is variability in the value of g (acceleration due to gravity). Apart from instruments such as the beam balance and pendulum scale, which directly compare two masses, all other instruments have an output reading that depends on the value of g.
The value of g is given by Helmert's formula:
g = 980:6 _ 2:6 cos f _ 0:000309h, where f is the latitude and h is the altitude in meters.
It can be seen from this formula that g varies with both latitude and altitude. At the equator (cosf = 0_), g = 978.0, whereas at the poles (cosf = 90_), g = 983.2. In Britain, a working value of 980.7 is normally used for g, and very little error can normally be expected when using this value. Where necessary, the exact value of g can be established by measuring the period and length of a pendulum.
Another difficulty that arises in calibrating mass, force, and torque sensors is the presence of an upward force generated by the air medium in which the instruments are tested and used.
According to Archimedes' principle, when a body is immersed in a fluid (air in this case), there is an upward force proportional to the volume of fluid displaced. Even in pure mass-balance instruments, an error is introduced because of this unless both the body of unknown mass and the standard mass have the same density. This error can be quantified as where SGa is the specific gravity of air, SGu is the specific gravity of the substance being measured, and SGm is the specific gravity of the standard mass.
Fortunately, maximum error due to this upward force (which has the largest magnitude when weighing low-density liquids such as petrol) won’t exceed 0.2%. Therefore, in most circumstances, the error due to air buoyancy can be neglected. However, for calibrations at the top of the calibration tree, where the highest levels of accuracy are demanded, either correction must be made for this factor or it must be avoided by carrying out the calibration in vacuum conditions.
The primary requirement in mass calibration is maintenance of a set of standard masses applied to the mass sensor being calibrated. Provided that this set of standard masses is protected from damage, there is little reason for the value of the masses to change.
Despite this, values of the masses must be checked at prescribed intervals, typically annually, in order to maintain the traceability of the calibration to reference standards.
The instrument used to provide this calibration check on standard masses is a beam balance, a weigh beam, a pendulum scale, an electromagnetic balance, or a proof ring-based load cell.
A beam balance is used for calibrating masses in the range between 10 mg and 1 kg. The measurement resolution and accuracy achieved depend on the quality and sharpness of the knife edge that the pivot is formed from. For high measurement resolution, friction at the pivot must be as close to zero as possible, and hence a very sharp and clean knife edge pivot is demanded.
The two halves of the beam on either side of the pivot are normally of equal length and are measured from the knife edge. Any bluntness, dirt, or corrosion in the pivot can cause these two lengths to become unequal, causing consequent measurement errors. Similar comments apply about the knife edges on the beam that the two pans are hung from. It’s also important that all knife edges are parallel, as otherwise displacement of the point of application of the force over the line of the knife edge can cause further measurement errors. This last form of error also occurs if the mass is not placed centrally on the pan.
Great care is therefore required in the use of such an instrument, but, provided that it’s kept in good condition, particularly with regard to keeping the knife edges sharp and clean, high measurement accuracy is achievable. Such a good condition can be confirmed by applying calibrated masses to each side of the balance. If the instrument is then balanced exactly, all is well.
In order to use it as a calibration standard, a weigh beam has to be manufactured and maintained to a high standard. However, providing these conditions are met, it can be used as a standard for calibrating masses up to 50 ton.
Like the weigh beam, the pendulum scale can only be used for calibration if it’s manufactured to a high standard and maintained properly, with special attention to the cleanliness and lubrication of moving parts. Provided that these conditions are met, it can be used as a calibration standard for masses between 1 kg and 500 ton.
Various forms of electromagnetic balance exist as alternatives to the three instruments just described for calibration duties. A particular advantage of the electromagnetic balance is its use of an optical system to magnify motion around the null point, leading to higher measurement accuracy. Consequently, this type of instrument is often preferred for calibration duties, particularly for higher measurement ranges. The actual degree of accuracy achievable depends on the magnitude of the mass being measured. In the range between 100 g and 10 kg, an inaccuracy of _0.0001% is achievable. Above and below this range, inaccuracy is worse, increasing to _ 0.002% measuring 5 ton and _0.03% measuring 10 mg.
Proof ring-based load cell:
The proof ring-based load cell is used for calibration in the range between 150 kg and 2000 ton. When used for calibration, displacement of the proof ring in the instrument is measured by either an LVDT or a micrometer. As the relationship between the applied mass/ force and the displacement is not a straight-line one, a force/deflection graph has to be used to interpret the output. The lowest measurement inaccuracy achievable is _0.1%.
Force Sensor Calibration:
Force sensors are calibrated using special machines that apply a set of known force values to the sensor. The machines involved are very large and expensive. For this reason, force sensor calibration is normally devolved to either specialist calibration companies or manufacturers of the measurement devices being calibrated, who will give advice about the frequency of calibration necessary to maintain the traceability of measurements to national reference standards.
Calibration of Torque Measuring Systems:
As for the case of force sensor calibration, special machines are required for torque measurement system calibration that can apply accurately known torque values to the system being calibrated.
Such machines are very expensive. It’s therefore normal to use the services of specialist calibration companies or to use similar services provided by the manufacturer of the torque measurement system. Again, the company to which the calibration task is assigned will give advice on the required frequency of calibration.
We have covered the measurement of all three quantities-mass, force, and torque-in this section as the three quantities are closely related. We also learned that weight was another related quantity as this describes the force exerted on a mass subject to gravity.
Mass is measured in one of three distinct ways, using load cell, using a spring balance, or using one of several instruments working on the mass-balance principle. Of these, load cells and spring balances are deflection-type instruments, whereas the mass balance is a null-type instrument. So, a balance is somewhat tedious to use compared with other forms of mass-measuring instruments.
In respect of load cells, we looked first at the electronic load cell, as this is now the type of load cell preferred in most applications where masses between 0.1 kg and 3000 ton in magnitude are measured. We learned that pneumatic and hydraulic load cells represent somewhat older technology that is used much less frequently nowadays. However, special types of hydraulic load cells still find a significant number of applications in measuring large masses, where the maximum capability is 50,000 ton. We noted that variations in the local value of g (the acceleration due to gravity) have some effect on the accuracy of load cells but observed that the magnitude of this error was usually small. Before leaving the subject of load cells, we also made some mention of intelligent load cells.
Looking next at mass balance instruments, we saw that a particular advantage that they had was their immunity to variations in the value of g. We studied the various types of balance available in the form of the beam balance, weigh beam, pendulum scale, and electromagnetic balance.
We then ended the review of mass-measuring instruments by looking at the spring balance. Our conclusion about this was that, while simple and inexpensive, its measurement accuracy is usually relatively poor.
Moving on to force measurement, we noted that transient forces could be measured by an accelerometer. However, static forces were measured either by a vibrating wire sensor or by a special form of load cell.
Looking next at torque measurement, we saw that the main two current methods for measuring torque were to measure the induced strain in a rotating shaft or measure the torque optically.
Brief mention of two older techniques was made, in the form of measuring the reaction forces in the bearings supporting a rotating shaft and in using a device called the Prony brake.
However, we noted that neither of these is now in common use.
We then concluded the section by examining the techniques used for calibrating the measuring devices covered in the section. We noted that calibration of mass-measuring sensors involved the use of a set of standard masses. As regarding the calibration of force and torque sensors, we saw that both of these required the use of special machines that generate a set of known force or torque values. Because such machines are very expensive, we noted that it was normal to use the services of either specialist calibration companies or the manufacturers of the measurement devices being calibrated.
+++1. What is the difference between mass and weight? Discuss briefly the three main methods of measuring the mass of a body.
+++2. Explain, using a sketch as appropriate, how each of the following forms of load cells work: (a) electronic, (b) pneumatic, (c) hydraulic, and (d) intelligent.
+++3. Discuss the main characteristics of the four kinds of load cells mentioned in Problem 2. Which form is most common, and why?
+++4. Discuss briefly the working characteristics of each of the following: (a) beam balance, (b) weigh beam, and (c) pendulum scale.
+++5. How does a spring balance work? What are its advantages and disadvantages compared with other forms of mass-measuring instruments?
+++6. What are the available techniques for measuring force acting in a horizontal direction?
+++7. Discuss briefly the four main methods used to measure torque.
+++8. Discuss the general principles of calibrating mass-measuring instruments.
+++9. Which instruments are used as a reference standard in mass calibration? What special precautions have to be taken in manufacturing and using such reference instruments?
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Updated: Friday, 2014-03-28 5:43 PST