91Ó°ÊÓ

$$\begin{gathered} \stackrel{â‡\P Ä}{F}=\left(-8\hat{i}+6\hat{j}+0\hat{k}\right)N \\ \stackrel{â‡\P Ä}{r}=\left(3\hat{i}+4\hat{j}+0\hat{k}\right)m \end{gathered}$$

Using the concept of torque, the unknown torque value is calculated. As per the concept, the torque acting on a body is due to the tangential force acting on a body along a radial path of the object in a circular motion. Thus, the cross-vector of the force and radial vector of the object will give the torque value.

Formulae:

The position vector in a 3-D diagram,$\stackrel{â‡\P Ä}{r}=x\hat{i}+y\hat{j}+z\hat{k}$

The force vector in 3-D,$\stackrel{â‡\P Ä}{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$

The torque acting on the body due to the tangential force,

$\begin{array}{rcl}\stackrel{â‡\P Ä}{\mathrm{Ï„}}& =& \stackrel{â‡\P Ä}{r}×\stackrel{â‡\P Ä}{F}\\ & =& \left|\begin{array}{ccc}i& j& k\\ x& y& z\\ F_x& F_y& F_z\end{array}\right|\\ & =& \left(y{F}_z-z{F}_y\right)\hat{i}+\left(z{F}_x-x{F}_z\right)\hat{j}+\left(x{F}_y-y{F}_x\right)\hat{k}\end{array}$

$\left|\stackrel{â‡\P Ä}{r}×\stackrel{â‡\P Ä}{F}\right|=rF\sin \mathrm{Ï•}$ …(¾±)

$r=\sqrt{x^2+y^2}andF=\sqrt{F_x^2+F_y^2}$

$\begin{array}{rcl}\stackrel{â‡\P Ä}{\mathrm{Ï„}}& =& \stackrel{â‡\P Ä}{r}×\stackrel{â‡\P Ä}{F}\\ \stackrel{â‡\P Ä}{\mathrm{Ï„}}& =& \left|\begin{array}{ccc}i& j& k\\ 3& 4& 0\\ -8& 6& 0\end{array}\right|\\ & =& \left(0\right)\hat{i}Nm+\left(0\right)\hat{j}Nm+\left(18+32\right)\hat{k}Nm\\ & =& \left(0\right)\hat{i}+\left(0\right)\hat{j}+\left(50\right)\hat{k}Nm\end{array}$

We know,

$\begin{array}{rcl}r& =& \sqrt{x^2+y^2}\\ & =& \sqrt{64m^2+36m^2}\\ & =& 10m\\ F& =& \sqrt{F_x^2+F_y^2}\\ & =& \left(\sqrt{9+6}\right)N\\ & =& 5N\\ \mathrm{Ï„}& =& rF\\ & =& 50Nm\end{array}$

$\left|\stackrel{â‡\P Ä}{r}×\stackrel{â‡\P Ä}{F}\right|=50Nm$

Using equation (i)

$\begin{array}{rcl}\left|\stackrel{â‡\P Ä}{r}×\stackrel{â‡\P Ä}{F}\right|& =& rF\sin \mathrm{Ï•}\\ 50Nm& =& 50Nm×\sin \mathrm{Ï•}\\ \sin \mathrm{Ï•}& =& 1\\ \mathrm{Ï•}& =& {90}^o\end{array}$

The angle between the directions of $\stackrel{â‡\P Ä}{r}$and $\stackrel{â‡\P Ä}{F}$ is$\begin{array}{rcl}\mathrm{Ï•}& =& {90}^o\end{array}$