Rules of Resolution of a Force

Very Important Rules to follow for Mechanics

Nallimilli’s Law1:

If Force is away from the origin ( intersection of co – ordinate axis X&Y ) and making an angle with either of the axis, the corresponding axis component is always “cos” the opposite axis component is always “sin” using the sign symbols of corresponding quadrant.
Explanation: In above diagram P is in first quadrant, the sign symbols for both components are positive. If an angle makes with X-axis then the resolution component of P corresponding to X-axis is Pcosθ, opposite component is Psinθ related to Y-axis, if an angle makes with Y-axis then the resolution component of P corresponding to Y-axis is Pcosθ, opposite component is Psinθ related to X-axis
If angle with X-Axis Px = + Pcosθ, Py= +Psinθ ; If angle with Y-Axis Px = + Psinθ, Py= +Pcosθ

Nallimilli’s Law2:

If Force is towards the origin ( intersection of co – ordinate axis X&Y ) and making an angle with either of the axis, shift the force component into opposite quadrant and away from the intersection of co-ordinate axis, and perfectly shift the angle also with the same axis at the
initial quadrant position then implement law1
Explanation: In above diagram P is in first quadrant & towards origin, shifts P to opposite quadrant i.e. third quadrant, the sign symbols for both components are Negative. If an angle makes with X-axis then shift the angle with same X-axis in opposite quadrant, using law1 resolution component of P corresponding to X-axis is Pcosθ, opposite component is Psinθ related to Y-axis, If an angle makes with Y-axis then shift the angle with same Y-axis in opposite quadrant, using law1 resolution component of P corresponding to Y-axis is Pcosθ, opposite component is Psinθ related to X-axis
If angle with X-Axis in first quadrant after shifting same angle with X-axis in third quadrant
Px = - Pcosθ, Py = -Psinθ
If angle with Y-Axis in first quadrant after shifting same angle with Y-axis in third quadrant
Px = - Psinθ, Py = -Pcosθ
Resolution component of force P Px Py
With X-axis in first quadrant + Pcosθ +Psinθ
With Y-axis in first quadrant +Psinθ + Pcosθ
With X-axis in second quadrant -Pcosθ +Psinθ
With Y-axis in second quadrant -Psinθ + Pcosθ
With X-axis in third quadrant -Pcosθ -Psinθ
With Y-axis in third quadrant -Psinθ -Pcosθ
With X-axis in third quadrant +Pcosθ -Psinθ
With Y-axis in third quadrant +Psinθ -Pcosθ