Centroid for System of Particles Mechanical Engineering Notes | EduRev

Engineering Mechanics - Notes, Videos, MCQs & PPTs

Mechanical Engineering : Centroid for System of Particles Mechanical Engineering Notes | EduRev

The document Centroid for System of Particles Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.
All you need of Mechanical Engineering at this link: Mechanical Engineering

Centroid: point defines the geometric center

If the material composing a body is uniform or homogeneous, the density or specific weight will be constant throughout the body, then the centroid is the same as the center of gravity or center of mass

Geometric Properties of Line and Area Elements

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Find Centroid of area?

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroids: Using Single Integration

1) DRAW a differential element on the graph.

2) Label the centroid Centroid for System of Particles Mechanical Engineering Notes | EduRev of the differential element.

3) Label the point where the element intersects the curve (x, y)

4) Write down the appropriate general equation to use.

5) Express each term in the general equation using the coordinates describing the curve or function.

6) Determine the limits of integration

7) Integrate

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Centroids: Using Single Integration

1) DRAW a differential element on the graph.

2) Label the centroid Centroid for System of Particles Mechanical Engineering Notes | EduRev of the differential element.

3) Label the point where the element intersects the curve (x, y)

4) Write down the appropriate general equation to use.

5) Express each term in the general equation using the coordinates describing the curve or function.

6) Determine the limits of integration

7) Integrate

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Determine the center of gravity of a thin homogeneous wire

Centroid for System of Particles Mechanical Engineering Notes | EduRev

segmentL (mm)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
AB600300018x104 
BC65030012519.5x1048.125x104
CA250012503.125x104
 1500  37.5x10411.25x104

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Locate the centroid Centroid for System of Particles Mechanical Engineering Notes | EduRev of the uniform wire bent in the shape shown. 

Centroid for System of Particles Mechanical Engineering Notes | EduRev

The given composite line can be divided intofollowing three parts having simpler shapes: 

Centroid for System of Particles Mechanical Engineering Notes | EduRev

 

SegmentL (mm)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
1150075011250
210050150500015000
35075130 37506500
4130506565008450
55025012500
Σ480  1650041200

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Locate the distanceyto the centroid of the member’scross-sectional area. 

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

 

Particle #A (in2)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
17.54.7535.625
21.8751.52.8125
31.8751.52.8125
460.53.0
ΣΣA = 17.25 Centroid for System of Particles Mechanical Engineering Notes | EduRev = 44.25

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

The given composite line can be divided into following three parts having simpler shapes: 

Centroid for System of Particles Mechanical Engineering Notes | EduRev

SegmentL (mm)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
1π(60)=188.560-38.2011310-72000
240020008000
320040-100800200
ΣΣL= 248.5   ΣxL=11310ΣyL=
-5600
ΣzL=
-200

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Locate the centroid of the plate area shown in Fig

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

SegmentA (m2)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
10.5 * 3 * 3 = 4.5114.54.5
23 * 3 = 9-1.5-13.513.513.5
3−2 * 1 = −2-2.525-4
ΣΣA = 11.5  Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

                                     Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

SegmentA (cm2)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
S.circleπ/2*22=6.2826.8512.5643.02
Rectangle6*4=24234872
Triangle1/2*3*6=9-12-918
Q. circle-π/4*22 = −3.143.150.85-9.892.67
Σ36.14  41.67130.35

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Find: The centroid of the part

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Solution: 1. This body can be divided into the following pieces:

rectangle (a) + triangle (b) + quarter circular (c) – semicircular area (d). (Note the negative sign on the hole!)

Steps 2 & 3: Make up and fill the table using parts a, b, c, and d.

SegmentA (m2)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
Rectangle1831.55427
Triangle 4.57131.54.5
Q. Circle 9π⁄4−4 * 3⁄3π4 * 3⁄3π-99
Semi-Circle−π⁄20−4 * 1⁄3π0-2/3
Σ28.0  Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid.

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Solution:

Divide the area into a triangle, rectangle, and semicircle with a circular cutout. Calculate the first moments of each area with respect to the axes. Find the total area and first moments of the triangle, rectangle, and semicircle.  Subtract the area and first moment of the circular cutout. Compute the coordinates of the area centroid by dividing the first moments by the total area.

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

SegmentA, mm2Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
Rectangle120 * 80 = 9.6 * 1036040576 * 103384 * 103
Triangle1/2*120 * 60 = 3.6 * 10340-20144 × 103−72 × 103
Semicircle1/2 *π * 602 = 5.655 * 10360105.46 339.3 × 103596.4 × 103
Circle−π * 402 = −5.027* 1036080−301.6* 103−402.2 * 103
Σ∑A= 13.828 * 103  Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev

SegmentA (mm2)Centroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRevCentroid for System of Particles Mechanical Engineering Notes | EduRev
120*60=1200103012,00036,000
21/2* 30 * 36 = 540303616,20019,440
Σ1740  28,200 55,440

Centroid for System of Particles Mechanical Engineering Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Dynamic Test

Content Category

Related Searches

Viva Questions

,

ppt

,

practice quizzes

,

Important questions

,

Extra Questions

,

Summary

,

Centroid for System of Particles Mechanical Engineering Notes | EduRev

,

shortcuts and tricks

,

video lectures

,

Free

,

Semester Notes

,

MCQs

,

Exam

,

Previous Year Questions with Solutions

,

Objective type Questions

,

pdf

,

study material

,

Centroid for System of Particles Mechanical Engineering Notes | EduRev

,

Sample Paper

,

Centroid for System of Particles Mechanical Engineering Notes | EduRev

,

past year papers

,

mock tests for examination

;