The document discusses different types of planimeters, which are instruments used to measure the area of arbitrary two-dimensional shapes. It describes the main types as polar, linear, and Prytz planimeters. Polar planimeters work by having a pointer trace the shape's perimeter, causing a measuring wheel to rotate and indicate the area. Linear planimeters can measure long areas without restriction. The document provides details on the construction, working principles, and mathematical justification of different planimeter designs.

1. DYP
CONSTRUCTION MATERIAL PROJECT
CERTIFICATE
Name Roll no.
Hrishikesh Tarange Y-1171
Omkar Kumbarde Y-1139
Avinash sakhere Y-1162
Yogesh Yelavi Y-1177
Atharva Patil Y-1154
Aditya Gothmare Y-1121
This is to certify that, has satisfactorily carried out and complete the project work entitled
types of stones
This Work Is Being Submitted for the Award of Diploma in Civil Engineering Partial
Fulfillment of Prescribed Syllabus of M.S.B.T.E Mumbai for Academic For year 2019-2020
Mr. Amit Handguide Mr. Abhijeet Patil
GUIDE HOD OF CIVIL DEPARTMENT
Prof. A.S. Kondekar
PRINCIPAL

2. types of Stones
Aim/Benefits of Micro-Project: -
To know the types of cements, characteristic, properties, and their specifications.
Objective: -
This project helps us to know types of cement and their properties.
Proposed Methodology: -
● First this process requires the land which will be types of stone and proper instruments.
● Then it will require the method by which the stones would be carried out.
● When the stones are extracted then they are crushed to produce aggregate, which is then
screened into the sizes required for immediate use.
● If required, they will be coated with bitumen to make bituminous macadam (bitmac) or
asphalt.
Action Plan: -
S
no.
Details of activity Planned
Start date
Planned
Finish date
Name of responsible
team members
1) Getting information of the
topic
14/01/2020 16/01/2020 Yogesh, hrishikesh,
omkar
2) Completing and getting
project ready for submission
16/01/2020 19/01/2020 Avinash, Aditya,
atharva
Resources Required: -
S
no.
Name of resource/material Specifications Quantity
1) Notebook and textbook Providing
information
1
2) Laptop Presentation 1
Names of Team Members with Roll No: -

3. Name Roll no.
Hrishikesh Tarange Y-1171
Omkar Kumbarde Y-1139
Avinash sakhere Y-1162
Yogesh Yelavi Y-1177
Atharva Patil Y-1154
Aditya Gothmare Y-1121
Name Of Programme : Micro project on the topic types of cement
Semester : ll nd
Course Title : Construction Material
Title : Type of cement
Sr. no. Characteristics to be
assessed
Poor
(marks 1-3 )
Average
(marks4-5)
Good
(marks 6-8)
Excellent
(marks 9-10)
Sub
Total
(A) Process and product Assessment (convert above total marks out of 6 marks)
1.
Relevance to the
Course
2. Literature
Review/information
3.C Completion of the
target as per project
proposal
4. Analysis of data and
representation
5. Quality of
Prototype/Model

4. 6. Report preparation
(B) Individual Presentation/viva (convert above marks out of 4 marks)
7. Presentation
8. Viva
Name Roll no. (A)
Process And
Product
Assessment
(6 marks)
(B)
Individual
Presentation/Viva
(4 marks)
Total
Marks
Hrishikesh Tarange Y-1171
Omkar Kumbarde Y-1139
Avinash sakhere Y-1162
Yogesh Yelavi Y-1177
Atharva Patil Y-1154
Aditya Gothmare Y-1121
Comment/Suggestions about teamwork/leadership/interpersonal communication
Name and designation of the teacher_____MrAmit
Ghongade.______________________________

5. Dated Signature____________________________________
Planimeter
A planimeter, also known as a platometer, is a measuring instrument used to
determine the area of arbitrary two-dimensional shape.

6. Construction
There are several kinds of planimeters, but all operate in a similar way. The precise
way in which they are constructed varies, with the main types of mechanical planimeter
being polar, linear and Prytz or "hatchet" planimeters. The Swiss mathematician Jakob
Amsler-Laffon built the first modern planimeter in 1854, the concept having been
pioneered by Johann Martin Hermann in 1814. Many developments followed Amsler's
famous planimeter, including electronic versions.
various types of planimeters
●
Polar planimeter
●
A planimeter (1908) measuring the indicated area by tracing its perimeter
●
Amsler polar planimeter
●
A linear planimeter. Wheels permit measurement of long areas without
restriction.

7. ●
Three planimeters: digital, Prytz's (hatchet) and Amsler's (polar)
●
Prytz planimeter with a wheel at the left
The Amsler (polar) type consists of a two-bar linkage. At the end of one link is a pointer,
used to trace around the boundary of the shape to be measured. The other end of the
linkage pivots freely on a weight that keeps it from moving. Near the junction of the
two links is a measuring wheel of calibrated diameter, with a scale to show fine
rotation, and worm gearing for an auxiliary turns counter scale. As the area outline is
traced, this wheel rolls on the surface of the drawing. The operator sets the wheel,
turns the counter to zero, and then traces the pointer around the perimeter of the
shape. When the tracing is complete, the scales at the measuring wheel show the
shape's area.
When the planimeter's measuring wheel moves perpendicular to its axis, it rolls, and
this movement is recorded. When the measuring wheel moves parallel to its axis, the
wheel skids without rolling, so this movement is ignored. That means the planimeter
measures the distance that its measuring wheel travels, projected perpendicularly to
the measuring wheel's axis of rotation. The area of the shape is proportional to the
number of turns through which the measuring wheel rotates.
The polar planimeter is restricted by design to measuring areas within limits
determined by its size and geometry. However, the linear type has no restriction in one
dimension, because it can roll. Its wheels must not slip, because the movement must
be constrained to a straight line.
Developments of the planimeter can establish the position of the first moment of area
(center of mass), and even the second moment of area.
Various types of planimeters

8. ●
Linear planimeter
●
Polar planimeter
The images show the principles of a linear and a polar planimeter. The pointer M at
one end of the planimeter follows the contour C of the surface S to be measured. For
the linear planimeter, the movement of the "elbow" E is restricted to the y-axis. For the
polar planimeter, the "elbow" is connected to an arm with its other endpoint O at a
fixed position. Connected to the arm ME is the measuring wheel with its axis of rotation
parallel to ME. A movement of the arm ME can be decomposed into a movement
perpendicular to ME, causing the wheel to rotate, and a movement parallel to ME,
causing the wheel to skid, with no contribution to its reading.
Principle
The working of the linear planimeter may be explained by measuring the area of a
rectangle ABCD (see image). Moving with the pointer from A to B the arm EM moves
through the yellow parallelogram, with area equal to PQ×EM. This area is also equal
to the area of the parallelogram A"ABB". The measuring wheel measures the distance
PQ (perpendicular to EM). Moving from C to D the arm EM moves through the green
parallelogram, with area equal to the area of the rectangle D"DCC". The measuring
wheel now moves in the opposite direction, subtracting this reading from the former.
The movements along BC and DA are the same but opposite, so they cancel each
other with no net effect on the reading of the wheel. The net result is the measuring of
the difference between the yellow and green areas, which is the area of ABCD.

9. Mathematical derivation
The operation of a linear planimeter can be justified by applying Green's theorem onto
the components of the vector field N, given by:
{displaystyle !,N(x,y)=(b-y,x),}
where b is the y-coordinate of the elbow E.
This vector field is perpendicular to the measuring arm EM:
{displaystyle {overrightarrow {EM}}cdot N=xN_{x}+(y-b)N_{y}=0}
and has a constant size, equal to the length m of the measuring arm:
{displaystyle !,|N|={sqrt {(b-y)^{2}+x^{2}}}=m}
Then:
{displaystyle {begin{aligned}&oint _{C}(N_{x},dx+N_{y},dy)=iint _{S}left({frac
{partial N_{y}}{partial x}}-{frac {partial N_{x}}{partial
y}}right),dx,dy[8pt]={}&iint _{S}left({frac {partial x}{partial x}}-{frac {partial (b-
y)}{partial y}}right),dx,dy=iint _{S},dx,dy=A,end{aligned}}}
because:
{displaystyle {frac {partial }{partial y}}(y-b)={frac {partial }{partial y}}{sqrt
{m^{2}-x^{2}}}=0,}
The left hand side of the above equation, which is equal to the area A enclosed by the
contour, is proportional to the distance measured by the measuring wheel, with
proportionality factor m, the length of the measuring arm.
The justification for the above derivation lies in noting that the linear planimeter only
records movement perpendicular to its measuring arm, or when
{displaystyle Ncdot (dx,dy)=N_{x}dx+N_{y}dy}

10. is non-zero. When this quantity is integrated over the
closed curve C, Green's theorem and the area follow.
Polar coordinates
The connection with Green's theorem can be understood in terms of integration in
polar coordinates: in polar coordinates, the area is computed by the integral
{displaystyle scriptstyle int _{theta }{tfrac {1}{2}}(r(theta ))^{2},dtheta ,}
where the form being integrated is quadratic in r, meaning that the rate at
which area changes with respect to change in angle varies quadratically with the
radius.
For a parametric equation in polar coordinates, where both r and θ vary as a function
of time, this becomes
{displaystyle int _{t}{tfrac {1}{2}}(r(t))^{2},d(theta (t))=int _{t}{tfrac
{1}{2}}(r(t))^{2},{dot {theta }}(t),dt.}
For a polar planimeter the total rotation of the wheel is proportional to
{displaystyle scriptstyle int _{t}r(t),{dot {theta }}(t),dt,}
as the rotation is proportional to the distance traveled, which at any point in
time is proportional to radius and to change in angle, as in the circumference of a circle
(
{displaystyle scriptstyle int r,dtheta =2pi r}
).
This last integrand
{displaystyle scriptstyle r(t),{dot {theta }}(t)}
can be recognized as the derivative of the earlier integrand
{displaystyle scriptstyle {tfrac {1}{2}}(r(t))^{2}{dot {theta }}(t)}
(with respect to r), and shows that a polar planimeter computes the area
integral in terms of the derivative, which is reflected in Green's theorem, which equates
a line integral of a function on a (1-dimensional) contour to the (2-dimensional) integral
of the derivative.