Active vocabulary

angular momentum external force internal force moment arm

— — — —

момент імпульсу зовнішня сила внутрішня сила плече сили

moment of inertia net force torque (moment of force)

— — —

момент інерції результуюча сила момент сили

Torque

Figure 1.18 a shows a cross section of a body that is free to rotate about an axis passing through O and perpendicular to the cross section. A force is applied at point P, whose position relative to O is defined by a position vector . Vectors and make an angle with each other.

(For simplicity, we consider only forces that have no component parallel to the rotation axis; thus, is in the plane of the page.)

To determine how results in a rotation of the body around the rotation axis, we resolve into two components (Fig. 1.18 b). One component, called the radial compo­nent F_r, points along . This component does not cause rotation, because it acts along a line that extends through O. (If you pull a door parallel to the plane of the door, you do not rotate the door.) The other component of , called the tangential component , is perpendicular to and has magnitude . This component does cause rotation. (If you pull a door perpendicular to its plane, you rotate the door.)

The ability of to rotate the body depends not only on the magnitude of its tangential component but also on just how far from O it is applied. To include both these factors, we define a quantity called torque as the product of two factors and write it as

Two equivalent ways of computing the torque are

and

where is the perpendicular distance between the rotation axis at O and an extended line running through the vector (Fig. 1.18 c). This extended line is called the action line of , and is called the moment arm of . Figure 1.18 b shows that we can describe r, the magnitude of as being the moment arm of the force component

Torque, which comes from the Latin word meaning «to twist», may be merely identified as the turning or twisting action of the force . When you apply a force to an object — such as a screwdriver or pipe wrench — with the purpose of turning that object, you are applying a torque. The SI unit of torque is the Newton-meter (N×m). The torque is positive if it tends to rotate the body counterclockwise, in the direction of increasing .

The definition of torque can be rewritten as a vector cross product:

Thus torque is a vector that is directed perpendicular to the plane containing and , its direction is given by the right-hand rule.

Now let us expand the definition of torque to apply it to an individual particle that moves along any path relative to a fixed point (rather than a fixed axis). The path needs no longer be a circle.

Figure 1.19 a shows such a particle at point P in the XY plane. A single force in that plane acts on the particle, and the particle's position relative to the origin O is given by position vector . The torque acting on the particle relative to the fixed point O is a vector quantity defined as

We can evaluate the vector (or cross) product in this definition of by using the right-hand rule. To find the direction of , we slide the vector (without changing its direction) until its tail is at origin O, so that two vectors in the vector product are tail to tail as in Fig. 1.19 b. Then we use the right-hand rule for vector products in Fig. 1.19 a, sweeping the right hand fingers from (the first vector in the product) into (the second vector). The outstretched right thumb then gives the direction of In Fig.1.19 b, points along the positive direction of the Z axis.

To find the magnitude of , we use the following equation:

where is the angle between and when the vectors are tail to tail. From Fig. 1.19 b, we see that this equation can be rewritten as

where is the component of perpendicular to . From Fig. 1.19 c, we see that previous equation can also be rewritten as

where the moment arm of (the perpendicular distance between O and the action line of ).

Дата добавления: 2015-11-16