Question

A plum is located at coordinates $(-2.0 m, 0, 4.0 m)$ In unit-vector notation, what is the torque about the origin on the plum if that torque is due to a force F whose only component is $F_{z} = 6N$.

Answer 1

Hint: The torque generated depends on the measure of the force and the perpendicular length between the point about which torque is measured and the point of the importance of force. The torque is a vector term. Utilizing vector product notations, we can determine the torque's direction.

Complete step by step answer:
Given the coordinates of a plum is $(-2.0 m, 0, 4.0 m)$
The position vector is $\vec{r} = -2 \hat{i} + 0 \hat{j} + 4 \hat{k}$
$\hat{i}$, $\hat{j}$ and $\hat{k}$ are the unit vectors along the x-axis, y-axis and z-axis respectively.
Force acts in the direction of the z – axis.
$F_{z} = 6N$ i.e., $\vec{F} = 6 \hat{k}$
Torque is a cross vector of position vector and force.
$\tau = \vec{r} \times \vec{F}$
Put the value of $\vec{r}$ and $\vec{F}$.
$\tau = (-2 \hat{i} + 0 \hat{j} + 4 \hat{k}) \times 6 \hat{k} $
$\implies \tau = -12 (\hat{i} \times \hat{k} ) + 0 + 24 (\hat{k} \times \hat{k})$
$\implies \tau = -12 (-\hat{j} ) + 0 + 24 (0)$
$\implies \tau = 12 \hat{j}$
Torque will act along in y-direction due to force F.
The torque is $ 12 \hat{j}$.

Note: Torque is the estimation of the force that can create an object to rotate around an axis. Force is what makes an object accelerate in linear dynamics. Similarly, torque is what produces an angular acceleration. Hence, torque can be described as the rotational equivalent of direct force. The location where the object turns is called the rotation axis. In science, torque is just the tendency of a force to bend or twist.