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Unformatted text preview: Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. — Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body , in which each particle maintains a Fxed spatial relationship to the other particles. This is an approximation, because in a real object the atoms actually oscillate about their average "equilibrium" positions in thermal motions. Here we will ignore these oscillations. The motion of a rigid body, like that of any system of particles, consists of two parts: motion of the CM and motion relative to the CM. ¡or a rigid body the latter motion consists entirely of rotation , with each particle moving in a circle about some point on an axis of rotation passing through the CM. The circles described by the particles of a rigid body do not all lie in the same plane (although they are in parallel planes). We must therefore generalize our description of circular motion to three dimensions. Vectors in circular motion One new feature in the description is that the angular velocity becomes a three- dimensional vector , denoted by ω . It is oriented perpendicular to the plane of the circle, in a direction given by a right-hand rule : Direction of angular velocity The vector ω is perpendicular to the plane of the circle followed by the particle. Curl the Fngers of the right hand the way the particle moves around the circle. The thumb points in the direction of ω . This “right hand rule” is the Frst of many in physics; most of them arise from the properties of the vector product, to be discussed in the next section. In our earlier discussion of circular motion we had a the relation v = r ω between the linear and angular speeds. Now we will show the correct relation among the vectors. PHY 53 1 Rotations 1 The position vector r and the velocity vector v lie in the plane of the circle. The situation is as shown. Note that the three vectors are mutually perpendicular. The vector product The vector relationship among r , v and ω involves a new quantity, the product of two vectors which yields another vector. The vector product of vectors A and B is written C = A × B . (Because of the notation it is often called the "cross" product.) Its deFnition in terms of components is as follows: C x = A y B z − A z B y C y = A z B x − A x B z C z = A x B y − A y B x . It is usually more practical to give rules, in arrow representation, for the magnitude and direction of the result. Let the arrows representing A and B be placed tail to tail, so that the angle between them is θ . Then we have: Rules for vector product: C = A × B Magnitude: C = AB sin θ ....
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This note was uploaded on 10/19/2011 for the course PHYSICS 53L taught by Professor Mueller during the Spring '07 term at Duke.
- Spring '07