**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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