Angular Motion

Therefore in energy versions of system models we

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Unformatted text preview: be the sum of the changes in both translational and rotational kinetic energy. Therefore, in energy versions of system models, we should keep in mind the possibility of rotational kinetic energy. Equation (31) gives the amount of inertial of a collection of particles. For an extended, continuous object, we can calculate the moment of inertia by dividing the object into many small, elements of mass Δmi. Then, the moment of inertia is approximately 7 ∆ (33) where ri is the perpendicular distance to the element of mass Δmi , from the rotational axis. If we now suppose that the element of mass approaches zero, then we can say that ∆ lim ∆ (34) If you have had calculus, then you should know that equation (34) is (35) If we have an object that is homogenous and has a constant density, then we can rewrite equation (28) as (36) where the density ρ=m/V. Because not everyone has had calculus, the moments of inertia will be given to you instead of you having to use equation (35) and (36) to derive them yourself. These equations are given to you in Table 1. 2.10 Torque When you push on a door, the door rotates about an axis through the hedges. The tendency of a force to rotate the object about some axis is measured by a vector quantity called torque. (37) Torque is the cause of changes in rotational motion and is analogous to force, which causes changes in translational motion. Consider a wrench that is pivoted about the axis through the origin. The applied force F generally can act at an angle φ with respect to the position vector r locating the point of application of the force. We then find the magnitude of the torque resulting from the force as Figure 5. Torque diagram sin (38) It is very important to recognize that torque is defined only when a reference axis is specified, from which the distance r is determined. We can interpret equation (38) in two different ways. Looking at the force components we see that the component that lies parallel to the wrench will not cause a rotation of the wrench around the pivot point because it line of action passes right through the pivot point. Similarly you cannot open a door by pushing on the hinges! Therefore, only the perpendicular component of the force causes rotation of the wrench about the pivot. 8 The second way to interpret equation (38) is to associate the sine function with the distance r so that we can write sin (39) where the quantity d is called the moment arm (or lever arm) of the force F, this represents t...
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This document was uploaded on 03/20/2014 for the course PHYS 215 at Lafayette.

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