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Question: The X and Y components of a force F acting at \({30^ \circ }\) to x-axis are respectively (A) \(\d...

The X and Y components of a force F acting at 30∘{30^ \circ } to x-axis are respectively
(A) F2,F\dfrac{F}{{\sqrt 2 }},F
(B) F2,32F\dfrac{F}{2},\dfrac{{\sqrt 3 }}{2}F
(C) 32F,12F\dfrac{{\sqrt 3 }}{2}F,\dfrac{1}{2}F
(D) F,F2F,\dfrac{F}{{\sqrt 2 }}

Explanation

Solution

Hint : Recall the concept of vectors. In physics, vectors are used to represent those physical quantities that have both magnitude and direction. Vectors are drawn by using pointed arrows or symbols called ‘hat’ or ‘cap’. Vectors are used to represent quantities like displacement, force, velocity etc.

Complete Step by Step Solution:
Step I:
Whenever two or more forces are applied on a body or a system, then the resultant forces can be calculated by adding up the magnitude of individual forces.
Step II:
It is to be noted that a single force working on an object can be resolved into two of its components. One component is horizontal and the other one is vertical. In case of two dimensions, these components are mutually perpendicular to each other. But they are parallel to the X and Y axis. To represent the forces in two dimensions, perpendicular vectors i^\widehat i and j^\widehat j are used.
Step III:
The horizontal component of the force acts along the positive x-axis and is given by cos⁡θ\cos \theta . The vertical component of the force acts along the positive y-axis and is given by sin⁡θ\sin \theta ,where θ\theta is the angle of the component of force.
Step IV:
Given θ=30∘\theta = {30^ \circ }
X component of force is given by Fx=Fcos⁡30∘{F_x} = F\cos {30^ \circ }
cos⁡30∘=32\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}
Fx=32F{F_x} = \dfrac{{\sqrt 3 }}{2}F
Step V:
The vertical component of force is given by Fy=Fsin⁡30∘{F_y} = F\sin {30^ \circ }
sin⁡30∘=12\sin {30^ \circ } = \dfrac{1}{2}
Fy=12F{F_y} = \dfrac{1}{2}F
Step VI:

Option C is the right answer.

Note:
It is to be noted that the horizontal component of force represents the rightward influence of the force applied on the object. Whereas the vertical component that is acting in a direction of Y-axis describes the upward influence of the force applied on the object.