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TORQUE REVIEW
Math topic
: torqueKey terms: force, angle, cross product
In the torque applet (static drawing below) you were asked to make changes in the length and position of the wrench and in the force vector. As a result you were able to make observations about the resulting effect on the magnitude of the torque, designated in green. From this experiment you probably have an intuitive appreciation for torque. This intuition suggests that we should be able to get mathematical models to calculate torque in general. In the following section "What is torque?" we present a mathematical review of torque.
What is Torque? - The Math behind the Observations
:Torque is a measure of how much a force acting on an object causes that object to rotate. This measurement can be calculated mathematically in terms of the cross product of specific vectors.
Terms and notation for Torque:
Let r = the vector representing the distance and direction from the axis of the bolt to the point where the force is applied.
Let F = the force vector acting on the wrench (represented by the red arrow in the applet)
Torque is defined as
t
= r x FThus, torque is the cross product of the distance vector (the distance from the pivot point to the point where the force is applied) and the force vector.
From the properties of cross products, we can conclude:
The (magnitude of the torque, t ) = | r | * | F | sin q , where q is the angle determined by r and F.
The direction of the torque vector, t , perpendicular to both r and F, can be found by using the right hand rule. This rule enables us to choose the correct one of the two perpendicular directions. If we put our fingers of our right hand in the direction of r, and curl them to the direction of F, then the thumb points in the direction of the torque vector.
Example from Stewart Calculus 3 Text: p.668
A bolt is tightened by applying a 40-N force to a 0.25-meter wrench. Find the magnitude of the torque about the center of the bolt. The angle between the axis of rotation and the force is 75o.
Solution: As indicated above the magnitude of the torque vector is
|
t | = | r x F | = (0.25) (40) sin 75o @ 9.66 Newton-meters.Excellent Additional Sites on Torque:
To see more sites on torque, some with extensive tutorials, click on the following: (The following includes links to sites that are not located at the United States Naval Academy. The USNA is not responsible for the content found on these sites. The content of these sites does not reflect the opinions, standards, policy, or endorsement of the Naval Academy or the United States Government.)
- University of Guelph - Complete tutorial on torque, referenced by various universities.
- Syracuse University - nice JAVA applet for cross product vector.
- Torque Wrench Repair
USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil