Question

In unit-vector notation, what is the net torque about the origin on a flea located at coordinates $\left(0,-4.0\text{m},5.0\text{m}\right)$when forces $\overrightarrow{{\mathbf{F}}_{\mathbf{1}}}\mathbf{=}\mathbf{}\left(3.0\text{N}\right)\hat{\mathbf{k}}$and $\overrightarrow{{\mathbf{F}}_{\mathbf{2}}}\mathbf{=}\left(-2.0\text{N}\right)\stackrel{\mathbf{\wedge }}{\mathbf{J}}$act on the flea?

Short answer

Net torque on flea is$-2.0\hat{i}\text{N}.\text{m}$

Step-by-step solution

Identification of given data

Position of flea is.$\left(0m,-4m,5m\right)$

$F_x=0\text{N},F_y=-2.0\text{N},F_z=3.0\text{N}$

To understand the concept

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,$\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

The force vector in 3-D,$\overrightarrow{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$

The torque acting on the body due to the tangential force,$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$

Determining the net torque about the origin on the flea

We know that,
$$\begin{gathered} \overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F} \\ =\left|\begin{array}{ccc}i& j& k\\ x& y& z\\ F_x& F_y& F_z\end{array}\right| \\ =\left(yFz-z{F}_y\right)\hat{i}+\left(z{F}_x-x{F}_x\right)\hat{j}+\left(x{F}_y-y{F}_x\right)\hat{k} \end{gathered}$$

After plugging the values, we get

$\begin{array}{rcl}\overrightarrow{\tau }& =& \left\{\left(-4\times 3\right)\text{N.m}-\left(5\times -2\right)\text{N.m}\right\}\hat{i}+0\text{N.m}+0\text{N.m}\\ & =& -2.0\hat{i}\text{N.m}\\ & & \end{array}$