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Angular Motion

28 rotation with constant angular acceleration in

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Unformatted text preview: rotation times the angular acceleration. Again, radian measure must be used for the angular acceleration. 2.8 Rotation with Constant Angular Acceleration In pure translation, motion with a constant linear acceleration is an important special case (a falling body for example). In Newton’s Lab, we found equations that hold for such motion. In pure rotation, the case of constant angular acceleration is also important and a parallel set of equations holds for this case also. We will not derive them here, but simply write them from the corresponding linear equations. Linear Equation a 1 2 2a Equation Number (20) 1 a 2 Angular Equation (22) (24) 1 2 (26) 6 2 Equation Number (21) 1 2 (23) (25) (27) 2.9 Rotational Kinetic Energy Let us now return to our rigid body that contains a system of particles that is moving around in a circular path shown in Figure 4. Each particle of the rigid body is in motion and therefore has some kinetic energy, determined by its mass and tangential speed. If the mass of the ith particle is mi and its tangential speed is vi, the kinetic energy of the particle is 1 2 (28) We can express the total kinetic energy, K, of this rotating rigid body as the sum of the kinetic energies of the individual particles as 1 2 (29) Thus according to equation (16) we can write the total kinetic energy as 1 2 1 2 1 2 (30) where we have factored the ω2 from the sum because it is the same for every particle inside the rigid body. The quantity in parentheses is called the moment of inertia I of the rigid object. Thus we can say (31) Thus the kinetic energy of a rotating rigid object around a pivot point is 1 2 (32) By looking at this definition, we can see that it is a measure of an objects resistance to change in its angular speed. Therefore it plays a role in rotational motion identical to the role mass plays in translational motion. Notice that the moment of inertia depends not only on the mass of the object but also on how the mass is distributed the rotation axis. Although the quantity in equation (32) is commonly referred to as the rotational kinetic energy, it is not a new form of energy. It is ordinary kinetic energy because it was derived from the sum overindividual kinetic energies of the particles contained in the rigid body; It is simply just a new role for kinetic energy for us. On the storage side of the conservation equation for energy, we should now consider that the kinetic energy term should...
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