Angular Motion

In some problems this approach may be easier than

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Unformatted text preview: he perpendicular distance from the rotation axis to the line of action of F. In some problems this approach may be easier than that of resolving the force into components. If two or more forces are acting on a rigid object each has a tendency to produce rotation about a pivot pointing. Thus we can say that the net torque acting on the rigid body about an axis through the pivot point is (40) From this definition of torque, we see that the rotating tendency increases as F increases and as d increases. 2.11 Torque vs Force Torque differs from force in a very important way. Torque is calculated or measured about a pivot point. To say that a torque is 20 N is meaningless. You need to say that torque is 20 Nm about a particular point. Torque can be calculated about any pivot point, but its value depends on the point chosen. In practice, we measure or calculate torques about the same point from which we measure an objects’ angular position θ (an thus its angular velocity ω and angular acceleration α). This assumption is built into the equations of rotational dynamics. 2.12 The Rigid Body in Equilibrium Now consider two forces of equal magnitude and opposite direction applied to a rigid object. The force directed to the right tends to rotate the object clockwise about an axis perpendicular through the pivot point, whereas the force directed to the left tends to rotate it counterclockwise about that axis. Because the forces are of equal magnitude and act at the same perpendicular distance from the pivot point, there torques are equal in magnitude. Therefore, the net torque on the rigid object is zero. With no net torque, no changes occur in rotational motion and the rotational motion of the rigid body remains in its original state. This is an equilibrium situation. We now have two conditions for complete equilibrium on an object which can be stated as follows: 1. The net external force must equal zero: 0 2. (41) The net external torque must be zero about any axis: 0 (42) The first condition is a statement of translational equilibrium. The second condition is a statement of rotational equilibrium. In the special case of static equilibrium, the object is at rest, so it has no translational or angular speed. 9 Table 1: Moments of Inertia for Symmetrical Objects Rotating about One Axis 2.13 The Rigid Body under a Net Torque Now let us consider the situation where the net torque on a rigid body is not zero. Just like with Newton’s second law for translat...
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