Find the centre of gravity of T-section with the
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Answers
Answer:
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Explanation:
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Centre of Gravity
1. 78 A Textbook of Engineering Mechanics 78 Centre of Gravity 6 C H A P T E RC H A P T E RC H A P T E RC H A P T E RC H A P T E R Contents 1. Introduction. 2. Centroid. 3. Methods for Centre of Gravity. 4. Centre of Gravity by Geometrical Considerations. 5. Centre of Gravity by Moments. 6. Axis of Reference. 7. Centre of Gravity of Plane Figures. 8. Centre of Gravity of Symmetrical Sections. 9. Centre of Gravity of Unsymmetrical Sections. 10. Centre of Gravity of Solid Bodies. 11. Centre of Gravity of Sections with Cut out Holes.6.1. INTRODUCTION It has been established, since long, that every particle of a body is attracted by the earth towards its centre. The force of attraction, which is proportional to the mass of the particle, acts vertically downwards and is known as weight of the body.As the *distance between the different particles of a body and the centre of the earth is the same, therefore these forces may be taken to act along parallel lines. We have already discussed inArt. 4.6 that a point may be found out in a body, through which the resultant of all such parallel forces act. This point, * Strictly speaking, this distance is not the same. But it is taken to the same, because of the very small size of the body as compared to the earth. Contents
2. Chapter 6 : Centre of Gravity 79 through which the whole weight of the body acts, irrespect of its position, is known as centre of gravity (briefly written as C.G.). It may be noted that every body has one and only one centre of gravity. 6.2. CENTROID The plane figures (like triangle, quadrilateral, circle etc.) have only areas, but no mass. The centre of area of such figures is known as centroid. The method of finding out the centroid of a figure is the same as that of finding out the centre of gravity of a body. In many books, the authors also write centre of gravity for centroid and vice versa. 6.3. METHODS FOR CENTRE OF GRAVITY The centre of gravity (or centroid) may be found out by any one of the following two methods: 1. By geometrical considerations 2. By moments 3. By graphical method As a matter of fact, the graphical method is a tedious and cumbersome method for finding out the centre of gravity of simple figures. That is why, it has academic value only. But in this book, we shall discuss the procedure for finding out the centre of gravity of simple figures by geometrical considerations and by moments one by ones. 6.4. CENTRE OF GRAVITY BY GEOMETRICAL CONSIDERATIONS The centre of gravity of simple figures may be found out from the geometry of the figure as given below. 1. The centre of gravity of uniform rod is at its middle point. Fig. 6.1. Rectangle Fig. 6.2. Triangle 2. The centre of gravity of a rectangle (or a parallelogram) is at the point, where its diagonals meet each other. It is also a middle point of the length as well as the breadth of the rect- angle as shown in Fig. 6.1. 3. The centre of gravity of a triangle is at the point, where the three medians (a median is a line connecting the vertex and middle point of the opposite side) of the triangle meet as shown in Fig. 6.2. 4. The centre of gravity of a trapezium with parallel sides a and b is at a distance of 2 3 h b a b a ⎛ ⎞+ × ⎜ ⎟ +⎝ ⎠ measured form the side b as shown in Fig. 6.3. Contents
3. 80 A Textbook of Engineering Mechanics 5. The centre of gravity of a semicircle is at a distance of 4 3 r π from its base measured along the vertical radius as shown in Fig. 6.4. Fig. 6.3. Trapezium Fig. 6.4. Semicircle 6. The centre of gravity of a circular sector making semi-vertical angle α is at a distance of 2 sin 3 r α α from the centre of the sector measured along the central axis as shown in Fig. 6.5. Fig. 6.5. Circular sector Fig. 6.6. Hemisphere 7. The centre of gravity of a cube is at a distance of 2 l from every face (where l is the length of each side). 8. The centre of gravity of a sphere is at a distance of 2 d from every point (where d is the diameter of the sphere). 9. The centre of gravity of a hemisphere is at a distance of 3 8 r from its base, measured along the vertical radius as shown in Fig. 6.6. Fig. 6.7. Right circular solid cone Fig.6.8. Segment of a sphere 10. The centre of gravity of right circular solid cone is at a distance of 4 h from its base, measured along the vertical axis as shown in Fig. 6.7. Contents