Lecture20-2A

Lecture20-2A - Chapter 12 Rotational Motion II Torque: the...

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Chapter 12 Chapter 12 Rotational Motion II Rotational Motion II
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Torque depends on the choice of the origin rF sin r F Torque: the Angular Analog of Force Torque: the Angular Analog of Force Torque can be thought of as an “angular force” which causes a change in angular motion. It is defined as: In (b) F sin q is the component of the force perpendicular to the door. In (c) rsin q is the component of the moment arm perpendicular to the force, defined as the lever arm. The choice is often determined by the particular application or problem. It should be noted that: In the next chapter we will define torque as a vector via the vector product:
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Note that r is the distance to the axis of rotation, and I is not not equal to Mr 2 cm .
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Example: Moment of Inertia of a Thin Disk Example: Moment of Inertia of a Thin Disk about its Center about its Center dm M A 2 rdr Due to circular symmetry we only have to integrate along the radial direction. The differential mass element is: Performing the moment of inertia integral: I r 2 dm 0 R M A 2 r 3 dr 2 M R 2 R 4 4 1 2 MR 2 This result can be used to sum thin disks for any objects with circular symmetry such as a sphere or disk etc.
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Example: Moment of Inertia of a Cone Find the moment of inertia of a uniform solid cone of height h , density r , about its central axis. The differential element for the moment of inertia for a disk of thickness dy is: dI 1 2 r 2 dm 2 r 2 r 2 dy The dependence of r on y is: r y R h h y Performing the integral: I  2 R h 4 0 h h y 4 dy  2 R h 4 h 5 5 I M 2 R 2 h /3 R h 4 h 5 5 3 10 MR 2 Integration is a bit easier if we invert the cone. Also,
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Lecture20-2A - Chapter 12 Rotational Motion II Torque: the...

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