Tip:
Highlight text to annotate it
X
Now we go to the next chapter force measurement. Until now we have seen displacement, velocity
and acceleration all the three put together is called motion measurement. So now it is
a new chapter on force measurement.
What is force? Force equal to m into a where m is the mass and a is the acceleration. So
that is by Newton’s law of motion, force is given by mass into acceleration. So if
you want to have unit for mass then you analyze whatever units it is made up of. It is made
up of mass which we have not yet seen the unit for mass whereas acceleration, we know
acceleration made up of length that is meter per Second Square. So we know the standard
for the standard unit for meter and we also know this unit for the second. So now we have
to see what is the unit for mass under accepted by all countries that is 1 kilogram which
is the unit for mass.
What is that one kilogram? It is some randomly selected body say platinum iridium cylinder
kept in a Paris such that amount of mass is assumed. They might have selected some other
mass also but once it is accepted that is one kilogram, now it is always referred there
as one kilogram mass. So one kilogram mass is nothing but the unit, it's a mass of the
unit or the product cylinder which is kept there near Paris that mass is always taken
as one kilogram. So now it is made up of all the three units mass, length and time. All
the three units made up of the force. Now the SI unit for force is one Newton. What
is Newton? Newton is the force required to accelerate a mass of one kilogram through
an acceleration of one meter per square. So one Newton is equal to one kilogram mass through
one meter per second square acceleration. When it accelerates then that force is called
one Newton or one Newton force will accelerate a mass of one kilogram through an acceleration
of one meter per Second Square that is our unit Newton but for calibration purposes we
have got the weight. What is weight? Weight is the force of attraction of a mass by the
earth, that earth attraction is called a weight on any mass. So that is g, mass multiplied
by g, instead of acceleration here we have got g, g is what we have taken the nominal
value is 9.81 meter per second square. That is the g value but it is a variable one at
different places in the earth this value g varies but all countries have agreed to use
this average value for all computation purposes.
So whenever there is a mass on the earth, it is always acted upon by g so we always
realize a weight. So weight was there, for example this chock piece mass cannot be separated
from its weight because it is acted on by the weight always, mass to separate it from
weight it is not possible. Then you have to take it into space where there is no earth
attraction then you have the mass there. Since we have the weight here and this nominal value
also is there, the weight is used. The unit for weight is one kilogram force or one kilogram
weight that is the unit for the calibration purposes. By having the one kilogram weight
we can calibrate. What is it in terms of Newton? This is equal to one kilogram force that is
one kilogram mass into 9.81 meter per Second Square. That is now one kilogram through one
meter per Second Square is one Newton, so here 9.81 so one kilogram force equal to 9.81
Newton. That is the relation between one kilogram force and the Newton.
So now people under different places they can go for the one kilogram weight itself
as standard for the force, so going for Newton because kilogram force and Newton can be related
by this relation. So kilogram weight is taken as the standard for calibration purposes because
one Newton to realize it, you should create a force that is some force is available already
in nature. That is force due to attraction of earth so that is what we use it for all
calibration purposes. Now under this topic force measurement we are going to see how
the force or weight or different objects are weighed.
So far we have given importance to the conversion from the mechanical quantity into electrical
quantity immediately but under this one we have got some mechanical devices, very often
used or it is still used. Suppose a mechanical engineer is asked to make one such mechanical
device, he should know what are the different types and principles behind it. Hence we are
not going immediately to conversion of force into a voltage signal, before we do it we
would like to see few of the mechanical devices. All the mechanical devices now we have got
can be brought under the title lever balances.
Under lever balances we have got many devices and they can be grouped under 4 category equal
arm balance and b, unequal arm balance and bent lever balance and then compound lever
balance. These are the 4 main types of the mechanical devices. You will see briefly what
are the constructions. The common balance still used in grocery shops, very old one.
Whatever the accuracy that is available in this common balance is more than sufficient.
So we can weigh things of say plus or minus 10 or 20 grams, when we measure few kilogram
of weight we can have the plus or minus 10 or 20 grams weight error that is acceptable
and in such one you will find, only at the middle of the lever. This is the lever, equal
arm balance. This is the pivot; pivot is made up of knife edge bearing. It is one of the
precision bearings available and very cheap also. This is made up of some hardened steel
and the pan or the support is little bit harder. Support should be sometime synthetic crystals
or something like that synthetic diamond are made use of or more hardened metal is taken,
otherwise the knife will flow into this support. To avoid it the pan or the support is harder
than the knife. So the knife diameter is of the order of 10 microns radius that is knife
edge is not a sharp end but it is always rounded, the knife end is always rounded to the radius
of 10 microns.
So you find the friction radius is 10 microns or is equal to 0.01 millimeter. Hence you
will find the friction torque in such knife is very small hence we got such accuracies.
So here we have only one knife edge bearing and the two ends equal, this is l1 this is
also l1 from the pivot. That is why it is called equal arm and from two hooks the pan
will be hung and standard weight is put on one side. We all see whenever we go to shop
and our commodity is put on the other one, until the pointer comes to stationary mark,
until the mark comes and stands here we have to add the commodity and when it is there.
Otherwise it will be tilting, until it comes to the standard mark we add weight and then
we measure it like that. So that is accuracy say few 10 of grams may be the accuracy whereas
we have got analytical balance.
People call it chemical balance or the physical balance and where the accuracy can be of the
order of 0.05 gram up to that it is possible, 0.005 gram it is possible, up to that 5 milligram
it is possible. We have got standard weight boxes and we have 3 knife edge bearings, at
the middle one and at end 2, 3 knife edge bearings are there, two pans from this point
it is equal distance the other two knife edges and cg of this lever below the middle knife
edge. Hence you find any little difference, this is the commodity, any small difference
this will be oscillating in a scale, the reference point will be marked. The pointer will be
oscillating. By taking for 3 oscillations readings the correct measurement can be this
is w1 known, this is w2 unknown weight can easily be found out by making 3 oscillations.
This we have learnt in physics that is analytical balance. To maintain the precision, sometimes
the manipulators are used to load this pan or to put the weight because if we stand near
like this, this one may be expanded by the body of the heat whereas the other arm may
be shorter. So in order to avoid this body heat, people stand at a distance and use manipulators
to put the weight or put the standard weight and so on. Such operations are also available
with analytical balance. A very precise measurements are made and more precise measurement can
be made by calculating the different volumes it occupy and buoyancy force of air acting
on this bodies of standard weight as well as this because standard weight will be normally
occupying smaller volume and commodity may occupy larger volume. So here buoyancy force
will be more than this. That can be accounted and the precision of one part in 10 to the
power of 8 also can be found out by this analytical balance methods.
So for precision measurement we can go for it but nowadays we have got electronic balance
which gives up to 0.001, that is 0.1milligram accuracy, you have got in electronic balances.
So they were used but in some places there may be found this is analytical balances.
Next one is unequal arm balance, here one is drawn.
Now the advantage is with a smaller weight, this is standard weight. By having a smaller
weight we can measure the higher unknown weight that is advantage of this standard weight
that is moved over a scale at a longer lever arm than the loading point. This is loading
point where unknown load is connected. So by having small weight we can measure a larger
weight but before that we should see one more thing, even though we compare the mass here
but in essence we are comparing two moments. The moment we put a mass immediately g come
into picture attraction due to earth and it becomes a weight so weight w1. When weight
w1 acts at a distance from a pivot, it is w1 into l1. So w into l1 is the movement of
this mass about this pivot. Similarly when add weight, this weight w2 is obtained until
w2 into l1 is equal to w1 into l1. So in essence the process of measuring weight is the process
of comparing two moments. Even though mass we are measuring but actually the instrument
measures to moments, until the two moments come equal we are adding the weight that is
when they are equal we find the pointer stands vertical.
So in all measurements we are comparing only moment here, more or less moment because mass
cannot be separated from the force and force acts from a distance, it is a moment. Here
because longer arm, the movement with a smaller weight we get this larger moment and here
larger load with a smaller moment and we can equalize until this is moved to and fro, until
the end tip of lever stands against the reference mark, after adding the weight here. Then we
make a reading about what is the weight and that is how it is made. One advantage is here
in the equal arm balance, the unknown weight and the standard weight should be equal. So
we should have standard weight until we could reach the range full range but here by having
a smaller weight we can have the larger weight we can compare.
The problem here is we have to move this and then find the weight, so sometimes we may
not be able to or we may not like to do that motion. When the moment we put the load we
want to have the reading, such a thing is there in the bent lever balance. This is a
bent lever balance that is third type bent lever balance. Say this is the lever bent
in this form and at the one end we have got this knife edge and the pan. The other end
we have got fixed weight w1 acting at a distance b from the pivot, from the knife edge. This
is the knife edge supported and above knife edge we have got, as part of the lever a pointer
is attached.
So the moment we put the unknown weight here then unknown weight if you call w2, w2 into
a is the moment because essentially all these lever principles we are comparing moments
w2 and a. So it will deform until the distance b is such that w1 into b1. That is w2 into
a will be equal to w1 into b1. So the distance b, now as it tilts this b is increasing, it
will increase until this relation is realized. At that time wherever the pointer stands,
so this may be zero and this may be say 200 or say 2 kilogram whatever it is 2 kilogram
force until then. Even though we call 2 kilogram or 2 kilogram force it represents force only,
2 kilogram means 2 kilogram force. So the moment we put the unknown immediately it deflects
and we find the reading. A modified version of this bent lever is letter weighing machine.
Still in some of the post offices this letter weighing machine is used this actually is
your bent lever, up to here it is bent lever.
So loading is done by having another linkage, so this is the coupler and that is called
input link, they are called as guiding link that actually you find here you have got 4
pin joints, so within this we have got a four bar linkage mechanism. We realize within these
four linkages, this is a frame which is fixed to the ground, it is a fixed link whereas
we can call it as one of them, this is the input link this is the coupler, this you can
call it output link. You find input link and output link when they are equal, you find
the coupler always makes motion parallel to the fixed link. That is what is happening
here. When you put letters here, it moves up and down vertically, vertical motion only
will be there but during its motion it may shift towards or away from the fixed link
and always it is parallel.
That is what is made use of here as a coupler. From coupler that property is made is also
when you put letter it doesn’t tilt, it always moving vertically up and down that
is the advantage. So when you put weight here it acts at the end of the bent lever, so bent
lever tilts and at the other end we have got the weight w1 and as where it stands that
itself is a pointer and it moves over the scale. So from zero probably it is 200 gram
up to that we can measure in this letter weighting machines.
The last type is compound lever balance that is what it is written here, compound lever
balance. One of the designs is shown here in the diagram. It is mainly made up of a
long lever. It is also called platform scale for compound lever mechanism. It is normally
used for very large weight say for example in lorry weighing machines, lorry loaded with
some commodities, if the police want to measure the lorry weight they will ask the lorry to
be stopped over this platform. That is why it is called platform scale and then they
will read the weight of the lorry whether it is permitted weight or overloaded that
they will decide. The main advantage of this platform scale is you can put this weight
w in this platform; actually this is your platform. You can put this weight anywhere
in this platform; the reading will not be affected that is the advantage of this platform
scale. Anywhere you can put here or there but the measurement will not be affected by
the position of the weight. That is advantage we are proving that now, how the position
of w doesn’t affect the measurement.
For that this is the say here you have a platform that is mounted on another platform here,
one is on the lever, this lever pivoted at O2. This main lever pivoted at O1 and the
lever edges are like this a and b, this is b lever edge and this is a. Up to the middle
it is a, that is the knife edge there. So if you are taking movement at O1 you can write
the equation T into b, T is the tension in the… this is called tie rod due to the weight
here, you will have the tension in the tie rod. So T into b is equal to ws into a where
ws is called the pan weight, by having different sets of pan weight and moving this, this is
the poise weight.
Suppose a lorry stands here then add number of pan weight and move this poise weight such
that this end of the lever stands against the reference mark. Then you have measured,
just add this pan weight and this reading here will give the total weigh of the lorry
whatever it is that has placed over the platform that is the method of measurement.So the principle
is T into b is equal to ws into a and now taking moment about O2, we can write T into
c, c is the total distance that I have not marked there, total distance from this pivot
to the other pivot. This is your c, that is T into c that is it is trying to tilt the
lever in this direction clockwise. The other reactions tend to rotate the lever in the
anticlockwise directions. That is two places two forces; w1 part is w1 into f by d. That
is w1 into f is this distance into whole lever this is simply supported beam, the reaction
here will be f by d and that into e will be the movement about O2 that is from w1 from
w2 into h, it is simple w2 into h, this is the motion.
Now taking h is a common factor then w1 into f by d into e by h plus w2. Now the lever
edge of this platform scale is taken in such a way that you have f by d is equal to h by
e that ratio we can easily select while you design. This way if it is selected then this
cancels out, this equal to T into c is equal to h into w1 plus w2. W1 plus w2 is always
w so h into w, so you find this instrument functioning is not dependent on the w and
w2, it depends only on the total weight. Hence we say this instrumentation is insensitive
for position of the load. Now what we have seen is irrespective of the position of the
load we get the correct measurement, the measurement doesn’t get affected. Now what is this magnification
or multiplication factor? For measuring a few tonnes of weight we use a few kilograms
only. So what is that multiplication factor, we obtain by eliminating those from those
two equations.
T into b is ws into a, we derived first that is the first lever and in second lever over
O2 we have got this equation. Eliminating T we have got w is equal to that weight, what
measure is equal to a by b into c by h into ws. So all these things we call it as R so
R into ws, so R is our multiplication factor. So that is if you measure 110 suppose 1000
kilogram force, if multiplication factor is 100 then by having 10 kilogram weight here
we can measure 100. That is the advantage, by having a smaller weight we measure a very
large weight. So that is pan weight but how the poise weight is decided? How it is made
use of? Normally pan weight is added for every 10 gram or every 100 kilogram, suppose it
is 1000 every hundred kilogram of weight we have to add one pan weight and in between
say 100 to 200 we have to move this poise. That is how the measurement is made use of
by using this compound balance between the pan weight, if the weight is there however
to that extent we are supposed to move the poise weight. This is poise weight. How the
poise weigh is decided and pan weight is decided? This is an example here. Suppose a is 78 mm,
b is equal to 13 millimeter, c is equal to 80 millimeter, i is equal to 6 millimeter,
wmax 100 kilogram. That is the maximum load here, I assume it as 100 kilogram we want
to measure and in this example so R is equal to multiplication factor is 80 that means
the total weight what we have to put here is 100 by 80 that is 1.25 kilogram that pan
weight we should have total.
Suppose we want to measure here for every 10 kilogram weight we want to put one pan
weight then that each pan weight should measure 125 that is 1.25 by 10 pan weights of equal
weight, 125gram. If that is the pan weight then in between measurement we should have
a poise weight which should be little larger than this one, so that it does not move over
this from its movement. So when it is zero it will be standing here, at that position
this is the counter weight. It is fixed in such a way when there is no weight here and
put this at zero and no pan weight, only the holder will be there and we move it until
the lever stands against reference mark and fix it there that is zero.
We have got pan weight for every 10 kilogram, between 10 kilogram say it 1, 2, 3, 4 or 11,
12 we do not measure. We should have for one kilogram force we should have the reading
here. For that purpose let us assume here weight of one kilogram force here. That is
w minimum assume as one kilogram force. What should be the poise weight here that we have
to find out. So for that assuming a distance of x is moved with poise weight, poise weight
being wp we move by distance of x for the next graduations. If x is distance for that
say for one kilogram let us see later. If we moved like that then additional movement
is wp into x that is being balanced in this side of the pivot. The other side comes from
the one kilogram weight here that is we call it w minimum. So w minimum into h by c that
is tension on the tie rod will be h w minimum, h by T is equal to T equal to see this side,
into b will give the moment about O so that is why into b, w minimum h by c into b will
be the moment in the other side.
We equate these two motions. Due to the w minimum we have got the movement here and
w p we have got this moment w into x. They are being compared and we find x is given
by this equation w minimum w p h into b into b by c. Now for this pan weight of 125 gram
we should have a little larger weight of poise weight 200 gram and for this w minimum 1 kilogram
and x comes about 4.9 millimeter. So every 4.9 millimeter distance, you write 1, 2, 3
up to 10 so 10 kilogram means it will come here. That is how the poise weight and its
graduations are finalized.