Torque: Difference between revisions

Latest revision as of 19:58, 20 January 2010

This article is about the analogue, from linear motion to angular motion, of: force

Definition

Due to a single force, about a point

Suppose ${\overline {F}}$ is a force and ${\overline {r}}$ is the radial vector from a point $A$ to the point of application of ${\overline {F}}$ . The torque due to ${\overline {F}}$ about $A$ is defined as the cross product of ${\overline {r}}$ and ${\overline {F}}$ . In other words, it is defined as:

${\overline {\tau }}={\overline {r}}\times {\overline {F}}$

The torque is a vector quantity. (More properly, torque is an alternating 2-tensor, and is treated as a vector via a (non-canonical) identification of $\mathbb {R} ^{3}$ with the space of alternating 2-tensors on $\mathbb {R} ^{3}$ ).

The magnitude of torque is given by:

$\tau =|rF\sin \theta |$

where $\theta$ is the angle between the radial vector and the force vector.

Note that the torque, as a vector quantity, depends only on the point $A$ , the line of action of ${\overline {F}}$ , and the magnitude of ${\overline {F}}$ . Thus, two forces of the same magnitude with the same line of action (Even though they may act at different points) generate the same torque.

Due to a single force, about an axis

Suppose ${\overline {F}}$ is a force and ${\overline {r}}$ is the vector from (and perpendicular to) a line $\ell$ to the point of application of ${\overline {F}}$ . The torque due to ${\overline {F}}$ about $A\ell$ is defined as the cross product of ${\overline {r}}$ and ${\overline {F}}$ . In other words, it is defined as:

${\overline {\tau }}={\overline {r}}\times {\overline {F}}$

The torque is a vector quantity. (More properly, torque is an alternating 2-tensor, and is treated as a vector via a (non-canonical) identification of $\mathbb {R} ^{3}$ with the space of alternating 2-tensors on $\mathbb {R} ^{3}$ ).

The magnitude of torque is given by:

$\tau =|rF\sin \theta |$

where $\theta$ is the angle between the radial vector and the force vector.

Due to multiple forces

The torque doe to multiple forces is defined as the vector sum of the torques due to each of the forces. When there is a continuum of forces, it is defined as a suitable integral. Fill this in later

Units and dimensions

Question	Answer
Scalar or vector?	Vector
Instantaneous or time-cumulative?	Instantaneous
MLT dimensions	$ML^{2}T^{-2}$ : MLT;1;2;-2
SI units	$J$ (Joules)

@@ Line 3: / Line 3: @@
 ==Definition==
-===Due to a single force===
+===Due to a single force, about a point===
 Suppose <math>\overline{F}</math> is a [[force]] and <math>\overline{r}</math> is the radial vector from a point <math>A</math> to the point of application of <math>\overline{F}</math>. The '''torque''' due to <math>\overline{F}</math> about <math>A</math> is defined as the cross product of <math>\overline{r}</math> and <math>\overline{F}</math>. In other words, it is defined as:
@@ Line 18: / Line 18: @@
 Note that the torque, as a vector quantity, depends only on the point <math>A</math>, the ''line of action'' of <math>\overline{F}</math>, and the magnitude of <math>\overline{F}</math>. Thus, two forces of the same magnitude with the same line of action (Even though they may act at different points) generate the same torque.
+===Due to a single force, about an axis===
+Suppose <math>\overline{F}</math> is a [[force]] and <math>\overline{r}</math> is the vector from (and perpendicular to) a line <math>\ell</math> to the point of application of <math>\overline{F}</math>. The '''torque''' due to <math>\overline{F}</math> about <math>A\ell</math> is defined as the cross product of <math>\overline{r}</math> and <math>\overline{F}</math>. In other words, it is defined as:
+<math>\overline{\tau} = \overline{r} \times \overline{F}</math>
+The torque is a vector quantity. (More properly, torque is an alternating 2-tensor, and is treated as a vector via a (non-canonical) identification of <math>\R^3</math> with the space of alternating 2-tensors on <math>\R^3</math>).
+The magnitude of torque is given by:
+<math>\tau = |rF \sin \theta|</math>
+where <math>\theta</math> is the angle between the radial vector and the force vector.
 ===Due to multiple forces===
@@ Line 25: / Line 39: @@
 ==Units and dimensions==
-The dimensions of torque are
+{| class="sortable" border="1"
+! Question !! Answer
+|-
+| Scalar or vector? || [[Quantity type::Vector]]
+|-
+| Instantaneous or time-cumulative? || [[Quantity type::Instantaneous]]
+|-
+| MLT dimensions || <math>ML^2T^{-2}</math>: [[MLT::MLT;1;2;-2]]
+|-
+| SI units || <math>J</math> (Joules)
+|}