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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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