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2.4 Products of vectors  (Page 4/16)

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A → × B → = − B → × A → .

The corkscrew right-hand rule    is a common mnemonic used to determine the direction of the vector product. As shown in [link] , a corkscrew is placed in a direction perpendicular to the plane that contains vectors A → and B → , and its handle is turned in the direction from the first to the second vector in the product. The direction of the cross product is given by the progression of the corkscrew.

Vector A points out and to the left, and vector B points out and to the right. In figure a we are shown the cross product of A cross B pointing up, perpendicular to both A and B. A screw turning an angle phi from A to B would move up. In figure b we are shown the cross product of B cross A pointing down, perpendicular to both A and B. A screw turning an angle phi from B to A would move down.
The corkscrew right-hand rule can be used to determine the direction of the cross product A → × B → . Place a corkscrew in the direction perpendicular to the plane that contains vectors A → and B → , and turn it in the direction from the first to the second vector in the product. The direction of the cross product is given by the progression of the corkscrew. (a) Upward movement means the cross-product vector points up. (b) Downward movement means the cross-product vector points downward.

The torque of a force

The mechanical advantage that a familiar tool called a wrench provides ( [link] ) depends on magnitude F of the applied force, on its direction with respect to the wrench handle, and on how far from the nut this force is applied. The distance R from the nut to the point where force vector F → is attached and is represented by the radial vector R → . The physical vector quantity that makes the nut turn is called torque (denoted by τ → ) , and it is the vector product of the distance between the pivot to force with the force: τ → = R → × F → .

To loosen a rusty nut, a 20.00-N force is applied to the wrench handle at angle φ = 40 ° and at a distance of 0.25 m from the nut, as shown in [link] (a). Find the magnitude and direction of the torque applied to the nut. What would the magnitude and direction of the torque be if the force were applied at angle φ = 45 ° , as shown in [link] (b)? For what value of angle φ does the torque have the largest magnitude?

Figure a: a wrench grips a nut. A force F is applied to the wrench at a distance R from the center of the nut. The vector R is the vector from the center of the nut to the location where the force is being applied. The force direction is at an angle phi, measured counterclockwise from the direction of the vector R. Figure b: a wrench grips a nut. A force F is applied to the wrench at a distance R from the center of the nut. The vector R is the vector from the center of the nut to the location where the force is being applied. The force direction is at an angle phi, measured clockwise from the direction of the vector R.
A wrench provides grip and mechanical advantage in applying torque to turn a nut. (a) Turn counterclockwise to loosen the nut. (b) Turn clockwise to tighten the nut.

Strategy

We adopt the frame of reference shown in [link] , where vectors R → and F → lie in the xy -plane and the origin is at the position of the nut. The radial direction along vector R → (pointing away from the origin) is the reference direction for measuring the angle φ because R → is the first vector in the vector product τ → = R → × F → . Vector τ → must lie along the z -axis because this is the axis that is perpendicular to the xy -plane, where both R → and F → lie. To compute the magnitude τ , we use [link] . To find the direction of τ → , we use the corkscrew right-hand rule ( [link] ).

Solution

For the situation in (a), the corkscrew rule gives the direction of R → × F → in the positive direction of the z -axis. Physically, it means the torque vector τ → points out of the page, perpendicular to the wrench handle. We identify F = 20.00 N and R = 0.25 m, and compute the magnitude using [link] :

τ = | R → × F → | = R F sin φ = ( 0.25 m ) ( 20.00 N ) sin 40 ° = 3.21 N · m .

For the situation in (b), the corkscrew rule gives the direction of R → × F → in the negative direction of the z -axis. Physically, it means the vector τ → points into the page, perpendicular to the wrench handle. The magnitude of this torque is
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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