4.4 Resolution of Vectors
By the head-to-tail rule, two or more vectors can be added to give a resultant vector. Its reverse process is also possible, i.e., a given vector can be divided into two or more parts. These parts are called components of the given vector. If these components are added up, their resultant is equal to the given vector.
To divide a force into its components is known as resolution of a force.
Usually, a force is resolved into two components which are perpendicular to each other. These are called its perpendicular or rectangular components.
Let us resolve a force F into its perpendicular components. A force F acting on a body at an angle θ with the x-axis is shown in Fig. 4.11(a). Imagine a beam of light placed above the vector F. As the light falls perpendicularly to the x-axis, it will cast a shadow OA of vector F onto the x-axis. We call this shadow the x-component of vector F. Similarly, if light is thrown perpendicular to the y-axis, the shadow OB of vector F on the y-axis is the y-component of F.
lightbulbA component of a vector is its effective value in a given direction.
The x and y components can be practically drawn by dropping perpendiculars from the tip of vector F onto the x and y axes respectively. The x-component of force F is denoted as F_x and the y-component as F_y.
From Fig. 4.11(b), it is evident that F is the resultant vector of components F_x and F_y. Moreover, F_x and F_y are perpendicular to each other. Therefore, F_x and F_y are called the perpendicular components of vector F.
The magnitudes of the perpendicular components can be found from the right-angled triangle OAC in Fig. 4.11(b).
OA / OC = cos θ
F_x / F = cos θ
F_x = F cos θ ............ (4.2)
AC / OC = sin θ
F_y / F = sin θ
F_y = F sin θ ............ (4.3)
lightbulbDo You Know? A tight rope walker balances himself by holding a bamboo stick. This is an application of the principle of moments.