Vector form of torque
Vector Form of Torque
Torque, often referred to as the moment of force, is a measure of the force that can cause an object to rotate about an axis. Just like force, torque is a vector quantity, which means it has both magnitude and direction. Understanding the vector nature of torque is crucial for solving problems in rotational dynamics.
Definition of Torque
The torque ($\vec{\tau}$) produced by a force $\vec{F}$ is given by the cross product of the position vector $\vec{r}$ (from the axis of rotation to the point of application of the force) and the force vector itself:
$$ \vec{\tau} = \vec{r} \times \vec{F} $$
The magnitude of the torque is given by:
$$ |\vec{\tau}| = rF\sin\theta $$
where:
- $r$ is the magnitude of the position vector $\vec{r}$
- $F$ is the magnitude of the force $\vec{F}$
- $\theta$ is the angle between $\vec{r}$ and $\vec{F}$
The direction of the torque vector is perpendicular to the plane formed by $\vec{r}$ and $\vec{F}$, following the right-hand rule.
Right-Hand Rule
The right-hand rule is a mnemonic for understanding the direction of the vector resulting from a cross product. For torque, point your fingers in the direction of $\vec{r}$, then curl them toward $\vec{F}$; your thumb will point in the direction of $\vec{\tau}$.
Units of Torque
The SI unit of torque is the newton-meter (Nm). It is important to note that, despite the unit's resemblance to that of energy (joule), torque is not energy. It is a measure of rotational tendency, not work.
Table of Differences and Important Points
| Aspect | Description |
|---|---|
| Nature | Torque is a vector quantity. |
| Formula | $\vec{\tau} = \vec{r} \times \vec{F}$ |
| Magnitude | $ |
| Direction | Perpendicular to the plane of $\vec{r}$ and $\vec{F}$, determined by the right-hand rule. |
| Units | Newton-meter (Nm) |
| Dependency | Depends on the magnitude of the force, the distance from the axis, and the angle between them. |
Examples
Example 1: Calculating Torque
A force of 10 N is applied at the end of a wrench 0.5 m long. If the force is applied at a right angle to the wrench (i.e., $\theta = 90^\circ$), what is the torque?
Using the formula for the magnitude of torque:
$$ |\vec{\tau}| = rF\sin\theta = (0.5\ \text{m})(10\ \text{N})\sin(90^\circ) = 5\ \text{Nm} $$
The torque is 5 Nm, and it acts perpendicular to the plane formed by the wrench and the force.
Example 2: Direction of Torque
Consider a situation where you are using a wrench to tighten a bolt. The position vector $\vec{r}$ extends from the bolt to your hand, and the force $\vec{F}$ is applied in the direction of your pull. To determine the direction of the torque, use the right-hand rule. Point your fingers in the direction of $\vec{r}$ and curl them in the direction of $\vec{F}$. Your thumb will point outwards, indicating the direction of the torque, which is into the page if the bolt is being tightened.
Example 3: Zero Torque
If the force is applied directly along the line of the position vector (i.e., $\theta = 0^\circ$ or $\theta = 180^\circ$), the torque will be zero because $\sin\theta = 0$. This means that no matter how large the force is, if it is applied in the direction of $\vec{r}$ or directly opposite to it, it will not cause rotational motion.
Understanding the vector form of torque is essential for analyzing situations involving rotational motion, such as the operation of machinery, the stability of structures, and the motion of celestial bodies. Mastery of this concept is fundamental for students and professionals in physics, engineering, and related fields.