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
and
, 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.
The corkscrew right-hand rule can be used to determine the direction of the cross product
. Place a corkscrew in the direction perpendicular to the plane that contains vectors
and
, 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
is attached and is represented by the radial vector
. The physical vector quantity that makes the nut turn is called
torque (denoted by
, and it is the vector product of the distance between the pivot to force with the force:
.
To loosen a rusty nut, a 20.00-N force is applied to the wrench handle at angle
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
, as shown in
[link] (b)? For what value of angle
does the torque have the largest magnitude?
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
and
lie in the
xy -plane and the origin is at the position of the nut. The radial direction along vector
(pointing away from the origin) is the reference direction for measuring the angle
because
is the first vector in the vector product
. Vector
must lie along the
z -axis because this is the axis that is perpendicular to the
xy -plane, where both
and
lie. To compute the magnitude
, we use
[link] . To find the direction of
, we use the corkscrew right-hand rule (
[link] ).
Solution
For the situation in (a), the corkscrew rule gives the direction of
in the positive direction of the
z -axis. Physically, it means the torque vector
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] :
For the situation in (b), the corkscrew rule gives the direction of
in the negative direction of the
z -axis. Physically, it means the vector
points into the page, perpendicular to the wrench handle. The magnitude of this torque is
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