Find out the centroid of an area of a semicircle, Mechanical Engineering

Assignment Help:

Find out the centroid of an area of a semicircle:

Find out the centroid of an area of a semicircle illustrated in Figure.

1262_Find out the centroid of an area of a semicircle.gif

Solution

You shall note that it is not possible to divide a semicircle in a few appropriate standard-areas, for each of which magnitude with location of its centroid is known. Therefore, in such a condition, we should let a very large number of extremely thin strips of width dx and suppose that its centroid is at the centre of the strip.

Let the semicircle with the centre of the full circle at O, and its straight boundary EOD along with y-axis. Therefore, the axis OX is the axis of symmetry of the area. It means that the quadrant OEF is the mirror-image of quadrant OFD.

Let an elementary area such as a strip AB at a distance x from C with its width dx tending to zero. If a is the radius of semicircle

AB = 2 a sin θ

Elemental Area, where, dA = 2 a sin θ × dx

which gives,   x = a cos θ

dx = - a sin θ dθ

1620_Find out its centre of gravity1.png                  (neglecting the -ve sign of dx)

Since      cos 2θ = 1 - 2 sin 2  θ

2 sin 2  θ = [1 - cos 2θ]

2284_Find out its centre of gravity2.png

Since value of θ =  π/2 ,

∴ A = π a 2 / 2

Which is already a known value of area of semicircle as area of full circle is (π a2); so one could have as well supposed it and proceeded as follows.

Letting the first moment of elemental areas about y-axis, we have

                                         ∴ (π a2/2 )x¯ =  2 a3 / 3

                                  ∴ x =  4 a /3 π ≈ 0.424 a = OG

725_Find out its centre of gravity3.png


Related Discussions:- Find out the centroid of an area of a semicircle

Specific steam consumption in turbine, how degree of super heat reduces spe...

how degree of super heat reduces specific steam consumption

Machin of dynamics, Ask question Prove maximum fluctuation of energy ?E=2EC...

Ask question Prove maximum fluctuation of energy ?E=2ECS.

Stitch welding, Stitch welding Here, a series of overlapping spot welds...

Stitch welding Here, a series of overlapping spot welds are made with the spot welding machine using the conventional electrode tip. Both water tight and gas tight joints can b

Definition of screw conveyor, Definition of screw conveyor Screw convey...

Definition of screw conveyor Screw conveyors belong to the conveyer with traction component. It depends on the shaft with spiral blades in hematic chute to make material move o

Balancing of several masses rotating in different planes, The crank shaft o...

The crank shaft of a high speed pump has a parallel shaft of 2.5 cm diameter and 48 cm in length. It carries three eccentrics each of 6 cm in diameter and uniform thickness of 1.8

Define the adjustable auger mechanisms, Define the Adjustable auger mechani...

Define the Adjustable auger mechanisms They improved mounting arrangement for the auger and auger drive mechanism of a floating screed type asphalt paver. And they provide actu

Efficiency test on sand blasting, Efficiency Test on Sand Blasting:   ...

Efficiency Test on Sand Blasting:             After the study on sand blasting, the result was acceptable and the next step is planning to make order for new machine and searc

Determine the workdone by the fluid on the piston, Explain Zeroth law and 1...

Explain Zeroth law and 1 st law of thermodynamics. Describe Internal energy and Enthalpy. What is a PMMFK. Give the expression for reversible workdone in a thermodynamic system

Piping material specifications, Q. Show Piping Material Specifications? ...

Q. Show Piping Material Specifications? Piping Material Specifications include the development and / or review of material specifications, and the review of data sheets, materi

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd