Angular Motion

The net torque will cause angular acceleration of the

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Unformatted text preview: ional motion, we would expect the angular speed of the rigid body to change. The net torque will cause angular acceleration of the rigid body. We describe this situation with a new analysis model, the rigid body under a net torque, and investigate this model. Suppose that we have a rotating body again that consists as a collection of particles. The rigid body will be subject to a number of forces applied at various locations on the rigid body at which individual particles will be located. Therefore, we can imagine that the forces on the rigid body are exerted on individual particles of the rigid body. We will calculate the torque on the object due to the torque resulting from these forces around the rotation axis of the rotating object. Any applied force can be represented by its radial component and its tangential component. The radial component of an applied force provides no torque because its line of action goes through the rotation axis. Therefore, only the tangential components of an applied force contributes to the torque. 10 On any given particle, described by index variable i, which the rigid object, we can use Newton’s’ second law to describe the tangential acceleration of the particle: a (43) where the t subscript refers to the tangential components. Let us multiply both sides of this expression by ri, the distance of the particle from the rotation axis: a (44) Using equation (19) along with equation (38), and the fact that r is perpendicular to F, we can rewrite equation (44) as (45) Now, let us add up the torques on all the particles of the rigid object: (46) The left side of equation (46) is the net torque on all the particles in the rigid body. The net torque associated with internal forces is zero; however, to understand why, recall that Newton’s third law tells us that the internal forces occur in equal and opposite pairs that cancel each other out. The torque due to each action—reaction force pair is therefore zero. On summation of all torques, we see that the net internal torque vanishes. The term on the left, then, reduces to the net external torque. On the right, we adopt the rigid body model by demanding that all particles have the same acceleration α. Therefore equation (46) becomes (47) where the torque and angular acceleration no longer have subscripts because they refer to quantities associated with the rigid object as a whole rather than to individual particles. We recognize the quantity in parentheses as the moment of inertia I. Therefore, (48) That is, the net torque acting on the rigid object is proportional to its an...
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This document was uploaded on 03/20/2014 for the course PHYS 215 at Lafayette.

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