Solved Example & Practice Problems: Centre of Gravity, Centre of Mass and Centroid

Solved Examples for Understanding

Example 5.2.1 

Determine the x and y co-ordinates of centroid of the rectangle shown in Fig. 5.2.4 from first principles.

Solution :

Consider the differential element to be a vertical strip of width dx as shown in Fig. 5.2.4 (a). The area of rectangle is

A = a b

The area of differential element is

dA = b dx

⸫ A = ab

The distance of centroid of differential element from Y-axis is x.

The distance of centroid of differential element from X-axis is b/2.

Example 5.2.2 

Find the centroid of the area enclosed by a right angled triangle from first principles.

Solution: 

Consider a right-angle triangle of base b and height h as shown in Fig. 5.2.5. 

Consider an elementary strip of width dx and height y parallel to Y-axis as shown.. As dx is very small, it is approximately a rectangular area. Its area is

dA = y dx

By similarity of triangles,

where y is the centroidal distance of area dA from x-axis. 

Example 5.2.3 

Determine the Y-co-ordinate of centroid of the triangular area of base b and height h shown in Fig. 5.2.6 from first principles.

Solution : 

Consider a horizontal strip of length x and width dy as shown in Fig. 5.2.6 (a) 

The area of differential element which is approximately a rectangle, is

dA = x dy

Using similarity of triangles,

Example 5.2.4 

Obtain centroidal distance for a sector of angle 2a from first principles. Use it to locate centroid of semicircular and quarter circular plane laminae. 

Solution: 

Consider a sector as shown in Fig. 5.2.7 (a) with inscribed angle 2 α.

Let the X-axis be the axis of symmetry. Then,

Consider a triangular element as shown in Fig. 5.2.7 (b). The area of the triangular element will be

The centroid of a sector lies on the line of symmetry at a distance of from the centre.

For a semicircular area, the inscribed angle is 2α = π, i.е., α = π/2 radians.

Thus, for a semicircular area, the centroid is at a distance of 4r / 3π from the centre on the axis of symmetry as shown in Fig. 5.2.7 (c).

For a quarter circular area the inscribed angle is 2α = π/2, i.e., α = π/4 radians

Thus the centroid of a quarter circular plane lamina can be obtained by traversing a distance 4r / 3π along one of the radius and then again a distance 4r / 3π in a perpendicular direction as shown in Fig. 5.2.7 (d).

Example 5.2.5 

Determine the centroid of a semi-circular area of radius r from rst principles.

Solution : 

Consider a semicircle as shown in Fig. 5.2.8. As the x-axis is the axis of symmetry.

Consider an elementary area of inscribed angle dθ as shown in Fig. 5.2.8.

As dθ is very small, the area is a triangle having its centroid at distance 2r /3 from O.

The area of the triangular element is

The x- co-ordinate of centroid of the triangular elementary area is 2/3 r cos θ.

The centroid of a semicircle lies on the line of symmetry at a distance of 4r/3π from the centre.

Example 5.2.6 

Determine the centroid of a quarter-circular area of radius r from first principles.

Solution: 

Consider a quarter circle as shown in Fig. 5.2.9.

Consider an elementary area of inscribed angle de as shown. As de is very small, the area is a triangle having its centroid at a distance 2r/3 from O.

The area of triangular element is

The x- co-ordinate of centroid of the triangular elementary area is 2r/3 cos θ.

The line of symmetry is a line making angle 45° with X-axis. This line divides the quarter circle into two equal areas. As centroid lies on this line of symmetry,

Example 5.2.7 

Determine co-ordinates of centroid of the shaded region shown in Fig. 5.2.10.

Solution: 

At x = a, y = b

⸫ y = Kx2 ⇒ b = Ka2

⸫ K = b/ a2

Consider the differential element to be a vertical strip of width dx and height y as shown in Fig. 5.2.10 (a).

Area of differential element is

dA = y dx

The distance of centroid of differential element from Y-axis is x.

The distance of centroid of differential element from X-axis is y/2.

Examples for Practice

Q.1

Prove that for the part of parabolic area bounded by X axis, line x = a and parabola y2 = kx as shown in Fig. 5.2.11.

Q.2

Obtain centroidal distances from reference axes for the shaded area shown in Fig. 5.2.12.