lecture-ch10 - Contents 10 Rotational Motion–I 1 10.1 The...

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Unformatted text preview: Contents 10 Rotational Motion–I 1 10.1 The Rotational Variables . . . . . . . . . . . . . . . . . . . . . 1 10.1.1 Angular Displacement Vector . . . . . . . . . . . . . . 2 10.1.2 Angular Velocity Vector . . . . . . . . . . . . . . . . . 3 10.1.3 Angular Acceleration . . . . . . . . . . . . . . . . . . . 3 10.1.4 Finite Rotations . . . . . . . . . . . . . . . . . . . . . . 4 10.1.5 Infinitesimal Rotations . . . . . . . . . . . . . . . . . . 5 10.2 Kinetic Energy of Rotation . . . . . . . . . . . . . . . . . . . . 6 10.3 Calculating the Rotational Inertia . . . . . . . . . . . . . . . . 6 10.3.1 (a) Hoop about central axis . . . . . . . . . . . . . . . 7 10.3.2 (b) Annular cylinder about central axis . . . . . . . . . 7 10.3.3 (c) Solid cylinder about central axis . . . . . . . . . . . 8 10.3.4 (e) Thin rod about axis through center perpendicular to length . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10.3.5 (g) The spherical shell about any diameter . . . . . . . 8 10.3.6 (f) Solid sphere about any diameter . . . . . . . . . . . 10 10.3.7 (h) Hoop about any diameter . . . . . . . . . . . . . . 11 10.3.8 Parallel-Axis Theorem . . . . . . . . . . . . . . . . . . 11 10.3.9 (d) Solid cylinder about central diameter . . . . . . . . 12 10.3.10(i) Slab about perpendicular axis through center . . . . 13 10.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 10.5 Newton’s Second Law for Rotation . . . . . . . . . . . . . . . 14 10.6 Work and Rotational Kinetic Energy . . . . . . . . . . . . . . 15 10 Rotational Motion–I We now turn to the motion of rotation, in which an object turns about an axis. 10.1 The Rotational Variables A rigid body is a body that can rotate with all its parts locked together and without any change of its shape. For a rigid body in pure rotation, every point of the body moves in a circle whose center lies on the axis of rotation 1 (or rotation axis ), and every point moves through the same angle during a particular time interval. 10.1.1 Angular Displacement Vector We define the angular displacement vector Δ vector φ to be Δ vector φ = Δ φ ˆ n where ˆ n is unit vector along the the axis of rotation and Δ φ is the angle that every point of the rigid body moves around ˆ n according to the right- hand rule. (Curl your right hand with the thumb pointing along the axis of rotation axis, your fingers then point to the positive direction of rotation angle.) In the following figure, the point Q rotates into Q ′ through the angle Δ φ and ˆ n = −−→ OP | −−→ OP | is the unit vector in the direction of the rotation axis. ˆ n Q ′ Q O P Δ φ For a small angle of rotation with | Δ φ | ≪ 1, the displacement vector −−→ QQ ′ from Q to Q ′ is almost perpendicular to −→ PQ and the length of the displace- ment vextendsingle vextendsingle vextendsingle −−→ QQ ′ vextendsingle vextendsingle vextendsingle ≃ vextendsingle vextendsingle vextendsingle −→ PQ vextendsingle...
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lecture-ch10 - Contents 10 Rotational Motion–I 1 10.1 The...

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