Bangladesh University of Engineering and Technology Department of Industrial and Production Engineering
IPE – 302 Report should contain
1.
Exp. number and name of the experiment
2.
Objectives
3.
Figure and deduction of equation used
4.
Equipments used
5.
Data and sample calculation
6.
Results and graphs (if necessary)
7.
Discussions.
8.
Answer to the given questions.
Note for the Students
1.
The student should read the supplied sheet thoroughly and try to understand conceptually
2.
Reports must be competent with the r equired contents
3.
The student should be familiar with t he measuring instruments beyond these experiments.
4.
The student should deduce the formula to be used.
5.
The procedure for each experiment mentioned in the sheet is just a guideline. For details, books are referred.
References
1.
Grumman, E.V: Metrology in Alignment of M/c. tools.
2.
Sharp K.W.B: Practical Engineering Metrology
3.
ASTME & Handbook of Industrial Metrology
4.
Kenedy, C.W: Inspection and Gaging.
IPE – 302 Expt. No. 1 i) ii)
Expt. No. 2 i) ii) iii) iv)
Check the normality by X 2 (Chi-square) test. Measurement of angle of Template by, a) Bevel protractor b) Angle gauge.
Measurement of Splines Determination of radius of Convex are. Determination of radius of Concave are Measurement of Screw thread by wire
Expt. No. 3 i) ii) iii) Expt. No. 4 i)
ii) iii)
Design and Implementation of a Lot- By- Lot Acceptance Sampling Plan Calibration of Slip Gauges using Supermicrometer Measurement of propeller by using Pitchometer.
Measurement of angles by a) 3 rollers and b) 2 rollers. Measurement of gear teeth by gear teeth Vernier Calipers Measurement of Bore by, a) Two balls. b) Pin gauge.
Expt. No. 2 Measurement of Tapers: i) ii) iii)
By rollers, Slip gauges and Micrometer. Ring gauge measurement by unequal Balls. Taper Plug measurement by Sine Bar and Dial Gauges General Information
Factors affection the accuracy of measurement 1. Least count of subdivisions 2. Line matching 3. Parallax in reading 4. Elastic deformation of the instrument and work-piece 5. Temperature effect
2
Measurement Instruments: A.
Linear i) Direct-reading: Machinist scale, combination set, Rule-depth gauge, Vernier calipers, Vernier height gauge, Micrometer. ii) Indirect-reading: There are various methods.
B.
Angle: Bevel protractor, m attachment on the combination set, Universal bevel, Sine bar, Angle gauge clock.
C.
Taper : Taper plug gauge, Ring gauge.
Gauge: A comparison type measuring instrument. Common gauges are: Plug gauges, plain plug gauges, Taper Plug gauges, Thread plug gauges, Spline gauges, Ring gauges, Ring thread gauges, Snap gauges, Slip gauges, Form gauges, Radius gauges, Pin Hole gauge, telescopic gauges, Screw thread pitch gauges, dial indicators.
Experiment No. 1 1(i)
Check the normality by
2
(chi-square) test
1.
Measure the diameter of 150 rollers.
2.
Calculate mean (μ) and standard deviation ( σ)
3.
Make ten equal intervals (between the lowest and highest measurement)
4.
Count the number of rollers in each interval
5.
Draw the histogram (frequency distribution)
6.
Apply χ test to check the conformity with normal distribution (assume 2
α – 0.05)
A goodness of fit test between observed and expected frequencies is based on the quantity r (oi − ei ) 2 = X cal ei i =1
∑
χ2 =
is the value of a random variable χ approximately very close to chi-square distribution oi = observed frequency ei = Expected frequency
whose
sampling
distribution
is
Expected frequency (e i): Expected frequency for each interval is obtained from a normal curve having the same mean and standard deviation as our sample. For ith interval, ei = (Area under the interval from normal curve)
× Total no. of sample.
No. of degrees of freedom = No of interval or cell – No of quantities needed from the sample to calculate the expected frequency.
3
Test for good fit of the Normal distribution: 1. Normally distributed 2. H Non-normally distributed 3. Level of significance a (given) 2 4. χ cri (critical), degrees of freedom (from table) 5. 1(ii)
2 cal (From data) χ cal
6. Conclusion: Accept or reject : Measurement of angle of template by a) Bevel protractor b) Angle gauge a) Bevel protractor consists of a round body with a fixed blade on which a graduated turret rotates. The turret is slotted to accommodate a non-graduated blade. Through locking mechanism any desired angle up to 360 can be measured. b) Angle gauges consist of pieces of hardened and stabilized steel. There are thirteen pieces in the set one being square plate and with these few places it is possible to set up any angle to the nearest 3 sec.
The set consists of three serves of values of the angle, namely: 1, 2, 9, 27 and 41 degrees 1, 3, 9 and 27 minutes 0.1, 0.3 and 0.5 minutes
Exp. No. 2 2(i)
Measurement of Splines
To measure the angle between splines, it is first necessary to measure the root diameter of the shaft and the width of each spline.
Fig : 2.1
4
Referring to the figure, Sin α Sin β
+ d ) ( W + d ) = 1/2 ( D + d ) ( D + d ) 1/2(S + d) S + d = = 1/2(D + d) D + d =
1/2 ( W
2(ii) Determination of radius of a Convex Arc with the help of fixed roller instrument Referring to figure the distance ‘a’ from the arc to the line joining the roller centers is measured by means of the micrometer. From this dimension together with the center distance of the rollers and their diameter, the radius is calculated. The instrument is first placed o a true straightedge and a reading of the micrometer taken with the spindle end in contact with the straightedge. It is then placed on the arc and a reading taken on this. The difference between the two readings is the height of the arc above the tangent to the rollers (dimension h).
Fig : 2.2 a=(
d
− h)
2 d ( R + ) 2 2
∴ R =
=(
C
) 2 + (R + a)2
2 C d = ( )2 + ( R + − h) 2 2 2
C 2
−
(d − h)
8h 2 Assignment: Find the possible error in the radius,
5
δc δd δh
Where,
δ R =
2(iii)
± 0.002 mm ± 0.001 mm ± 0.005 mm
= = = δ R
.δ h +
δ h Radius of a Concave Arc R =
T 2 g
6π 2
δ R
δ R .δ C + .δ d δ C δ d
+ r
Figure: 2.3
2(iv)
ii)
R g T t
= = = =
r
=
Radius of arc in inch 2 386.4 inch/sec Time of one oscillation in sec = t/10 The time for 10 oscillation repeat this 3 three timer then take average. Radius of roller
Measurement of Screw Thread by Wires
i) Find the number of thread per inch (tpi) ii) Find the flank angle iii) Determine the effective diameter iv) The flank angle can be measured from the following formula: θ d 2 − d 1 sin = 2 (D 2 − D1 ) − (d 2 − d 1 ) where θ/2 is the flank angle and d 1, d2, D1, D2 are dimensions as shown in the Fig. 2. 4
(iii) Effective Diameter: This element cannot be measured directly, but must be calculated from the measurements which are made. To measure this two small wires are inserted in the vees of the thread. The diameter must be the same for both wires and must be known very accurately. It may, however, have any value within the range which permits the wires to contact the straight flank of the thread and to protrude above the crests. This applies only on the assumption that the flank angles are correct: if there is any error in the angler, a correct value of the effective
6
diameter will be obtained only if the wires make contact at the pitch line. Wires of this sizes are referred to as “best size” wires. The diameter of the wires used should be such that they contact the flanks at a distance from the pitch line not exceeding one-twentieth of the length of the straight flank. The effective diameter E is given by: E
=
T+P
T is measured as shown in the Fig. 2.4
Fig. 2.4 Here, T = D-2d P is a constant whose value depends on the diameter of the wires P=
f
2
cot θ / 2 − d (cos ec θ / 2 - 1)
where f =1/TPI Experiment No. 3 (i): Design and Implementation of a Lot- By- Lot Acceptance Sampling Plan Sampling is a quality assurance tool. It is based on the idea that a random selection of a sample from a homogenous lot represents the quality of the lot. It is obvious that sampling involves risks but apart from the necessity of acceptance sampling as in the case of destructive testing, advantages of sampling are significant. It is also to be remembered that the purpose of sampling is to guide the course of action, not to estimate lot quality. Further, acceptance sampling is not an attempt to ‘control’ the quality but to ‘assure’ the quality. A typical application of acceptance sampling is as follows: a company receives a shipment of product from a vendor. This product is often a component or raw material used in the company’s manufacturing process. A sample is taken from the lot, and some quality characteristics of the units in the sample is inspected. On the basis of the information in this sample, a decision is made regarding lot disposition. Most generally applied sampling scheme is single sampling plan, where other sampling plans like double sapling plans, multiple sampling plan and sequential sampling plans are other advanced varieties. A single sampling plan is a lot sentencing procedure in which one sample
7
of n units is selected at random from the lot and the disposition of the lot is determined based on the information contained in that sample. The real problem in most acceptance sampling is to design a satisfactory acceptance sampling system or more commonly, to select such a system from a number of possible systems already developed. To judge the suitability of any proposed acceptance sampling system in a particular case, it is desirable to have an understanding of the strategy and tactics built into the various available types of systems. Based on manufacturer’s product quality, data-type e.g. variables and attributes, can also classify sampling plan. Variables, of course, are quality characteristics that are measured on a numerical scale. Attributes are quality characteristics that are expressed on a “go, no-go” basis. Present day industrial practice of acceptance sampling is by attributes and will doubtless continue to be so. But, the growth of knowledge of statistical quality control techniques has led to a considerable increase in the industrial use of acceptance sampling by variables. It seems likely that this tendency will continue. Acceptance sampling by variables is often preferable, though costly, to acceptance sampling by attributes, particularly for those quality characteristics that are source of troubles. The great advantage of the use acceptance sampling by variables is that more information is obtained about the quality characteristics in question. In this experiment, an acceptance sampling plan for variables will be designed and implemented. Nomenclatures and Equations: α = Producer’s risks β = Consumer risk P1 = Lot fraction defective for which probability (Acceptance) ≥ 1- α P2 = Lot fraction defective for which probability (Acceptance) ≤ β LSL = Lower Specification Limit USL = Upper Specification Limit n = Sample Size k = Critical Value X = Sample mean Pˆ LSL = Fraction defective estimates corresponding to LSL
Pˆ USL = Fraction defective estimates corresponding to USL M = Allowable fraction defective
S
=
ZLSL=
∑ ( X − X )2 n −1 X − LSL
ZUSL=
s USL − X s
8
Procedure: The nomograph shown in figure 3.1 is used to find the required sample size n and the critical value k to meet a set of given conditions P1 ,P2, 1- α, β for both the σ known and σ unknown cases. The greater uncertainty in the case where the standard deviation is unknown requires a larger sample size than does σ known case but the same value of k is used. After determining n and k from the nomograph, the diameters of the balls supplied is measured and X & s is calculated. From given LSL and USL values Z LSL and ZUSL is calculated. From figure 3.2 the value of M is determined. Pˆ LSL and Pˆ USL can be determined from figure 3.3. If Pˆ LSL+ Pˆ US L ≤ M , the will be accepted, otherwise it will be rejected.
Figure 3.1 : Nomograph for Designing Variables Sampling Plan
9
Figure 3.2 Chart for determining maximum allowable fraction defective M
ˆ from Z Figure 3.3 Chart for determining P
10
Assignment: The density of a plastic part used ina packet calculator is required to be at least 3 0 .70 gm/cm .The parts are supplied in large lots and a variable sampling plan is to be used to sentence the lots. It is desired to have α =0.1, β= 0.05, p1=0.02 & p2=0 .1. The variability of the manufacturing process is unknown but will be estimated by the sample standard deviation.
a) Find an appropriate variables sampling plan. b) Suppose that a sample of appropriate size was taken, and X =0.73 and S = 1.05 x 10 . Should the lot be accepted or rejected. -2
Experiment No. 3 (ii) : Calibration of Slip Gauges using Supermicrometer Slip gauges are rectangular block of steel having a cross-section of about 30 by 10 mm. these may be used as reference standards for transferring dimensions of the unit of length from the primary standard to gauge blocks of lower accuracy and for verification and graduation of measuring apparatus and length measures for the regulation and adjustment of indicating measuring apparatus for direct measurement of linear dimensions of industrial component. Slip gauges are first hardened to resist w ear and carefully stabilized so that they are independent of any subsequent variation in size or shape. After being hardened , blocks are carefully finished on the measuring faces to such a fine degree of finish, flatness and accuracy that any two such faces when perfectly clean may be wrung together. The phenomenon of wringing occurs due to molecular adhesion between a l iquid film and the mating surfaces. When two gauges are wrung together and the overall dimension of a pile made of two or more blocks so joint is exactly the sum of the constituent gauges. Due to handling in the laboratory or inspection room, slip gauges are liable to show signs of wear after appreciable period of use and , therefore, they should be checked and recalibrated at regular intervals. Experiment No. 3 (iii): Measurement of propeller by using Pitchometer With the help of a pitchometer , the horizontal distance of any point on the helicoidal surface of the propeller blade from its center can be measured. The vertical distance of any two points can also be determined with its vertical scale.
Figure 3.4: Propeller
For a propeller , the pitch diameter = 0.75 x Propeller diameter Referring to figure: 3.4 2 2 2 AB =AC +BC – 2AC.BC. cosθ θ in degrees From which θ can be calculated.
11
Again, let the vertical distance traveled by the screw fitted on any point of pitch circle be for θ angle of rotation. Then the pitch, P = 360
h
θ
Experiment No. 4
4(i):
3 rollers and 2 rollers
The standard units of angular measurement is the degree, m Radian for mathematical investigation. The selection of an angle measuring instrument depends upon the component and the accuracy of measurement required. Part I (a)
In this method 3 rollers of equal diameter are used referring to Fig..4.1 the distance H is measured by a depth micrometer or a straight edge and slip gauges and hers,
AC = H
Cos
θ 2
=
AC AB
=
H D
where D is the diameter of the roller. This arrangement is used for angles between 180 and about 60. But for angles below 90 it may be preferable to measure the gap between the two outer rollers as shown in F ig (b) in this case. θ S + D sin = 2 2 D
Fig. 4.1(a) Angle measurement by equal discs
12
Fig. 4.1(b) Angle measurement by equal discs
Fig. 4.1(c) Angle measurement by equal discs
(b)
For angles less than 60 only two rollers are used as shown in fig. (C). The distance S being measured by slip gauges with this arrangement, BC = S and BC S Sinθ = = AB D
Questions:
(a)
What is the Indirect measuring instrument used in the experiment?
(b)
Which angle measuring instrument used in the laboratory is comparatively more accurate?
(c)
Sing θ/2 = (S + D)/2D. Assuming S to be 5
± 0.01 mm, find error
4(ii) Measurement of gear teeth by Gear Teeth Venire Caliper 1.
Calculate the chordal thickness (w) of the teeth.
13
2. 3. 4. 5.
Calculate the chordal addendum (H) of the teeth Measure the height (h) with width (w) Measure the width (w) with height (h) Find the percentage errors.
Here the chordal thickness (w) is, W = DSin
90 T
=
T P
Sin
90 T
Where, P = Diametrical pitch P
=
T + 2 O. D.
Pitch Circle radius, R =
Nm
2
=
T
2P
Chordal addendum ( H ) = R (1 − Cos
90 T
)+
1 P
Experiment No. 4 (iii)
a)
Two ball method:
This method can be applied to obtain the dia meter of a recessed hole. Higher accuracy cannot be attained with this method. Referring to fig.4. 2. C2
=(
d1
D =C+
+ d2 2 d1
)2
− (H 2 − H1 +
d2
− d1 2
)2
+ d2 2
Fig. 4.2: Measurement of Diameter of Hole by Two Balls
14
D
(b)
= C +
(d 1
+ d2 ) 2
Pin gauge method:
Internal diameter is measure by a pin with spherical ends, the length of which is slightly less than the diameter to be measured. The pin is placed in the bore with its ends in contact with the bore surface, keeping one end in contact with the bore surface, the other end is brought across to make contact at the other side of a diameter. The distance between the two points of contact is measured and from this dimension and the length of the pin gauge, the diameter of the bore is determined. Referring to the Fig. 4.3 AE.EB = DE.EC = W2 2 2 2 AE = AC – EC
Fig. 4.3: Measurement of bore by pin gauge D = AE + EB
=
W2
− W2 +
L2
L2
− W2
L2
=
− W2
L2
It may be simplified (applying binomial theorem) as, D = L (L 2
2
−W
2
)
−
1 2
= L(1 −
W2 L2
)
−
1 2
= (L +
W2 2L2
)
Question:
i) ii) iii)
What is the two ball method more advantageous? What is the difference between a measuring tool and a gauge How do the error in measurement can be eliminated?
15
iv) v) vi)
What is the contact surfaces of two slip gauges? How do we find a venire constant? What happens when the diameter of both the balls are equal for 2 ball method?
Experiment No. 5
5(i)
Measurement of Tapers
Measurement of a Taper Plug gauge by Rollers, Step Gauge and Slide Calipers
In this method, measurements are made over two equal rollers standing on slip gauges at each side of the taper plug. Two measurements are needed, one near each and of the plug. From these, together with the difference in heights of the stacks of slip gauges. The angle of the taper is easily calculated. The diameters at the ends of the plug are derived from additional measurements of the axial distances from the ends to the planes through the roller centers.
The half angle of the taper is given by tan θ/2 =
− M1 2(h 2 − h 1 ) M2
Fig. 5.1
Measurement of Taper Plug Gauge by Means of Rollers
The diameters in the planes through the roller centers are obtained from D1 = M1 – d(1 + sec θ/2) D2 = M2 – d(1 + sec θ/2) The diameter at any other position is then easily found. Thus at the ends Ds = D1-2 (h1 – H1) tan θ/2 DL= D2 + 2(H2 – h2) tan θ/2 Equipments used: 1. Rollers of equal diameters
16
2. Slide calipers 3. Depth gauge 4. Slip gauge In order to obtain satisfactory results a number of practical details require attention. The plug gauge should be set up on its small end, so that the force exerted by the micrometer used for the micrometer used for the diametrical measurements tends to wedge he rollers against the cone and the slip gauges. If set up on its large end, as may sometimes be unavoidable, care must be taken to prevent the rollers riding up the taper. The force applied by the slide calipers must be kept small by using the instrument with a light touch, to avoid errors caused by elastic indentation. The roller and gauge make contract at a point and the stress here may be very high. An error of 0.0025 mm can quite easily occur. Generally, it will be possible to stand the gauge directly on the surface pate, thereby dispensing with the slip gauges on which it is shown supported in the figure. This reduces the number of dimension to be observed and results in slightly increased accuracy. With a small gauge, especially one which is long in relation to its diameter, some difficulty may be experienced in keeping it on its base. In this case, a small vertical force may be applied to the bop of the gauge by means of a spring-loaded plunger with a rounded and which is supported from a vertical column with a heavy base resting on the surface plate. The rollers should preferably be of about the same diameter as the slide calipers anvils, which can then be rested on the slip gauges to ensure that the measurement is made at right angles to the gauge axis. This is particularly important on gauges of small diameter. The uniformity of taper is checked by taking a third diametrical measurement about mid way along the gauge and comparing this with the value as this position calculated from the two other readings. An additional test is done for the straightness of the contact against an illuminated background. To test for any out-of-roundness of the cross section, three sets of readings should be taken along different diameters at each axial position. If these readings do not vary by more than the variation in the readings of any one diameter, the out-of-roundness is not measurable with the equipment and skill employed. The possible error in the angle of taper may be calculated in the following manner. Let
M2 – M1 = M
and tan Then, since
h2 – h1 = h M 2 − M1 θ 2
δθ =
=
2(h 2
δθ δ M
= 2.Cos 2
− h1 )
.δ M +
=
δθ δ h
θ 1 . .δδ 2 2h
M 2h
.δ h
− 2.Cos 2
θ M . .δδ 2 2h 2
17
Now,
Cos
θ 2
1 (
M 2
)
(
M 2
+ h2
)2
2h 2 1 M [ .δ M − 2 ] M 2h 2h δ h. ( )2 +h 2 2
∴ δθ =
=
h
=
[h.δ M − M .δ h]
+h
2
2
Usually, the error is very small, being that in the slip gauges used. If this error is neglected, the expression reduces to h
δθ = (
M 2
)
2
.δ M
+h
2
from which it is clear that, for any given value of h, the error the angle increases as the value of M decreases. This means that the error in the measured value of the angle of a slow taper is larger than that of a quick taper. It is, however, only slightly larger, but, when expressed as a percentage of the angle, given it larger figure. As examples, suppose that, m in measuring two this gauges, the dimension h is 100 mm in both cases and the values of h are 75 mm for one gauge and 6 mm for the other, with a possible error on 0.005 mm. The angle of the taper is approximately 41 degree for the first gauge and 2.5 degree for the second.
5(ii)
Ring Gauge Measurement by Unequal Balls
Many internal tapers are either too long or too small in diameter to permit of using the method described in the preceding experiment. In these cases, two balls are used of such sizes that one will rest in the cone near the large and one near the small end. Their distances from a datum surface, such as one end of the gauge, are measured by a micrometer depth gauge. Referring to Fig. 5.2 the semi-angle of the taper is given by 1 sin
θ 2
=
CD BC
2
= h1
(d 1
+ h2 −
− d 2 )
1 2
(d 1
− d 2 )
18
cos ec
or
θ 2
=
+ h2 ) −1 ( d 1 − d 2 )
2 ( h1
The diameter in the places of the ball centres are θ θ D1 = d 1sec D2 = d 2 sec 2 2
Fig. 5.2: Internal Taper Measurement by Unequal Balls. Then the diameters at the ends of the taper are
= D1 + 2(
1
− h1 )
tan
= d 1 sec + (d 1 − 2h1 )
tan
D L
2
d 1
θ 2
and
DS
θ
2 θ 2
θ
= d 2 sec − 2( H − 2h2 − d 2 ) 2
tan
θ 2
Equipments used: 1. Two unequal balls 2. Micrometer depth gauge 3. Slide calipers
The uniformity of the taper can be checked by taking a measurement to a third ball seated midway along the taper.
19
The errors which may occur in the results obtained from these measurements are calculated from the following expressions. Let the distance between ball centres = c and the difference between ball radii = r Then
sin
θ
=
r
2 c The error in the angle is δθ ≈
δθ
.δ r +
δθ
.δ c δ c 1 r ≈ 2 sec(θ / 2) ⎡⎢ .δ r − 2 .δ c⎤⎥ c ⎣c ⎦ δ r
2
≈ c c
5(iii)
2
− r 2
(c.δ r − r .δ c )
Taper Plug Measurement by Sine Bar and Dial Gauge
The arrangement for this method is shown Fig. 5.iii.1 The sine bar is set up on a surface place to the nominal angle of the taper plug which is then placed in position on the bar, being prevented from sliding down by the stop plate at the end. Care must be taken to ensure that the axis of the plug gauge is aligned with the sine movement. The dial gauge, supported in a stand on the surface place, is then passed over the plug gauge near each end and at one or two positions between the ends. If there is any variation in the readings two alternatives are available for finding the true angle of the cone. Either the variation over a measured distance along the surface of the plug gauge can be used to obtain the difference between the true angle and the angle set up or the height of the slip gauge pile can be adjusted until no variation occurs in the readings of the dial gauge. If only the angle of the gauge is to be found. The former method is quicker, but adjustment of the slip gauge pile is preferable if the diameter of the small end is also to be determined by the method about to be described.
Fig. 5.3(a): Measurement of Taper Plug Gauge by Sine Bar and Dial Gauge
20
If the length of the sine bar is L and the height of the slip gauge pile is H, the angle of taper, H θ = sin −1 L To determine the small end diameter, a roller, having a diameter slightly less than the diameter at the end of the plug gauge, is place across the sine bar in the manner shown in Fig. 5.3(a) The slip gauges between the roller and the end of the plug gauge are varied until the dial gauge reading is the same over the roller as it is over the plug gauge. Referring to Fig. 5.3(b), the line CD is drawn parallel to the surface of the sine bar. Then,
AC = S + d/2 AD = (s + d/2) tan θ/2 Ds = 2AE = 2(AD + DE) = (2s + d) tan θ/2 + d1sec θ/2
Fig. 5.3(b) Determination of small end diameter of Taper Plug Gauge by Roller and Dial Gauge. Instead of adjusting the pile of slip gauges between the roller and the plug gauge, the roller can be placed directly in contact with the end of the plug gauge and the height h, shown in Fig, found by means of the dial gauge.
21
DF DE Ds
= = = =
BG sec θ/2 (h + d) sec θ/2 2.d/2 tan θ/2 (h + d) sec θ/2 + d tan θ/2
The diameter at the large end of the plug gauge can then be found from the equation DL = Ds + 2H. tan θ/2 Where, H is the axial length of the gauge. Too much reliance should not be placed upon the accuracy of the diameter obtained in this way. It is considerably less than the accuracy of the small end diameter, because of the effect of errors in the angle.
22
