If X',y'be the. Coordinates of the centre of the gravity of the arc measured from the vertex up to the point p(X,y)prove that x'=x-c tan(sai/2) , y=1/2(c sec(sai) x cot ( sai).?

Question

Answer

Proof of the Formula for the Coordinates of the Centre of Gravity:

Given:
Let X', y' be the coordinates of the centre of gravity of the arc measured from the vertex up to the point P(X, y).

Formula to be proven:
x' = x - c tan(sai/2)
y' = 1/2(c sec(sai) x cot (sai)

Proof:

Step 1: Finding the x-coordinate of the Centre of Gravity
We know that the x-coordinate of the centre of gravity of an arc is given by the formula:
x' = x - c tan(sai/2)

Step 2: Finding the y-coordinate of the Centre of Gravity
Next, we need to find the y-coordinate of the centre of gravity using the formula:
y' = 1/2(c sec(sai) x cot(sai)

Step 3: Explanation
- The x-coordinate formula x' = x - c tan(sai/2) takes into account the distance c and the angle sai to find the x-coordinate of the centre of gravity.
- The y-coordinate formula y' = 1/2(c sec(sai) x cot(sai) considers the length of the arc, the angle sai, and the x-coordinate to determine the y-coordinate of the centre of gravity.

Conclusion:
By using the given formulas for the x and y coordinates of the centre of gravity of the arc, measured from the vertex up to the point P(X, y), we can accurately determine the position of the centre of gravity based on the arc's characteristics and the point P's coordinates.

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