Centre of gravity of a flat plate about y axis is:

Explanation:

CENTRE OF GRAVITY OF A FLAT PLATE

The analytical method of locating the centre of gravity is simply an application of the principle of moments, or the Varignon’s theorem, which states that the moment of the resultant force is equal to the sum of moments of its parts. For example, consider a thin uniform irregular shape as shown in Fig. The late is divided into small elements of weights ω₁, ω₂, ω₃, ....., ω_n acting through their own centres of gravity. The total weight is W, acting through the centre of gravity of the entire plate. These gravity forces represent a parallel force system. Let the co-ordinates of the particles be (x₁, y₁), (x₂, y₂), (x_i, y_i), ... etc. of weights W₁, W₂, W_i, ... , respectively. And, the co-ordinates of centre of gravity of the entire plate be (x̅, y̅) for the weight W.

The magnitude of W is equal to the sum of the weights of various particles. Therefore, by ∑F₂ = 0, we get

W = W₁ + W₂ + .... + W_n

W = ∑W_i      

or

By taking moments of weights about y-axis, we get,

i.e. , By ∑M_y = 0, we get

W.x̅ = W₁.x₁ + W₂.x₂ + ..... + W_n.x_n

Wx̅ = ∑W_i.x_i      

or

While by taking moments about x-axis,

i.e., by ∑M_x = 0, we get

Wy̅ = W₁.y₁ + W₂.y₂ + ....... + W_n.y_n

Wy̅ = ∑W_i.y_i