9.1 Centroids by Integration

Transcription

1 9.1 Centroids b Integration

2 9.1 Centroids b Integration Procedures and Strategies, page 1 of 2 Procedures and Strategies for Solving Problems Involving Calculating Centroids b Integration = f () (, ) 1. Determine the coordinates of the centroid b evaluating integrals such as ( el, el ) c = el da da el = For a planar area, the differential area da is usuall a rectangular strip of finite length and differential width d (for a vertical strip) or d (for a horizontal strip). Use a vertical strip if the curve bounding the planar region is given as a function of, = f(). Use a horizontal strip if the bounding curve is given as a function of, = g(). The integrand el is the coordinate of the centroid of the strip. It must be epressed as a function of for a vertical strip and as a function of for a horizontal strip. d ( el, el ) el = /2 = g()/2 (, ) = g() d

3 9.1 Centroids b Integration Procedures and Strategies, page 2 of 2 2. For a line (a wire), the area element da is replaced b dl = (d) 2 + (d) 2 ) = 1+ (d/d) 2 d if the line is given as a function of : = f(). Use dl d d dl = (d/d) d if the line is given as function of : = g(). 3. For volumes with some degree of smmetr (for eample, a solid of revolution), da can be replaced b a circular disk of finite radius and differential thickness. 4. Using the integral function on a scientific graphing calculator simplifies the work and helps avoid errors. z Radius = d z = f()

4 9.1 Centroids b Integration Problem Statement for Eample 1 1. Locate the centroid of the plane area shown. Use a differential element of thickness d. = ft 2 ft

5 9.1 Centroids b Integration Problem Statement for Eample 2 2. Locate the centroid of the plane area shown, if a = 3 m and b = 1 m. Use a differential element of thickness d. a = a sin( ) 2b b

6 9.1 Centroids b Integration Problem Statement for Eample 3 3. Locate the centroid of the plane area shown. 1 in = in. 1 in.

7 9.1 Centroids b Integration Problem Statement for Eample 4 4. Locate the centroid of the plane area shown. 0.5 m = 1 2 m 0.5 m 2 m

8 9.1 Centroids b Integration Problem Statement for Eample 5 5. Locate the centroid of the plane area shown. = (13 ) 6 6 m 2 m = m 4 m

9 9.1 Centroids b Integration Problem Statement for Eample 6 6. Locate the centroid of the plane area shown. 1 m = 3 = m 1 m

10 9.1 Centroids b Integration Problem Statement for Eample 7 7. Locate the centroid of the plane area shown. Use a differential element of thickness d. = h b h b

11 9.1 Centroids b Integration Problem Statement for Eample 8 8. Locate the centroid of the plane area shown. Use a differential element of thickness d. b = a[1 ( ) 2 ] b a

12 9.1 Centroids b Integration Problem Statement for Eample 9 9. A sign is made of 0.5 in. thick steel plate in the shape shown. Determine the reactions at supports B and C. B = 50 + (10) sin in. Specific weight = 490 lb ft 3 C 50 in.

13 9.1 Centroids b Integration Problem Statement for Eample Locate the centroid of the wire shown. 3 m = m

14 9.1 Centroids b Integration Problem Statement for Eample Locate the centroid of the wire shown. = 300[1 ( ) 4 ] mm 300 mm

15 9.1 Centroids b Integration Problem Statement for Eample The rod is bent into the shape of a circular arc. Determine the reactions at the support A. 0.2 lb/ft A 3 ft 20

16 9.1 Centroids b Integration Problem Statement for Eample a) Locate the centroid of the Gatewa Arch in St. Louis, Missouri, USA. b) During the pre-dawn hours of September 14, 1992, John C. Vincent of New Orleans, Louisiana, USA, climbed up the outside of the Arch to the top b using suction cups and then parachuted to the ground. Estimate the length of his climb. Approimate equation of centerline: = ft (68.78 ft) cosh[( ft -1 )] 625 ft 299 ft 299 ft

17 9.1 Centroids b Integration Problem Statement for Eample Locate the centroid of the cone shown. Radius = 2 m 3 m O z

18 9.1 Centroids b Integration Problem Statement for Eample Locate the centroid of the volume shown. One-eighth of a sphere of radius "a" a z

19 9.1 Centroids b Integration Problem Statement for Eample Determine the coordinate of the centroid of the solid shown. The solid consists of the portion of the solid of revolution bounded b the z and z planes. a b z = a[1 ( z b )2 ] (This curve is rotated about the -ais to generate the solid.)

20 9.1 Centroids b Integration Problem Statement for Eample Locate the centroid of the pramid shown. h b b a z a

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