講者: 許永昌 老師 1
Contents Find the rotational axis of an UCM. Definition Levi-Civita Symbol: p153, Eq Examples Summary 2
Find the rotational axis of an UCM (uniform circular motion) If a ball whose motion is an UCM is located at position r (related to the center of this circle) with velocity v at an instant. How to describe the direction of rotational axis? A: Usually we use the right hand rule to define the direction of rotational axis. However, it is not an algebraic definition. From the right hand rule, we get: r v= v r= . r r 3
Find the rotational axis of an UCM (uniform circular motion) For this purpose, we want this operation obeys If we want this cross product is a linear operator, we can require the cross product for 3D obeys We get 4
Cross Product |A B|=ABsin . Because A B= B A, we get A B=A (B +B // ) =A B = ABsin Ĉ. Levi-Civita Symbol: ijk 5 A B 1 x 2 y 3 z 1 x 2 y 3 z
Usage Find a normal of a plane If I provide three points on a 3D plane. Find the rotational axis Get the area of the parallelogram Why? Get the volume How? Calculate the angular momentum L, torque , Lorentz force, … P22 6
The comparison of dot product and cross product Dot Product: A BCross Product: A B Symmetric: A B=B AAntisymmetric: A B = B A ScalarVector A B=AB // =A // B=ABcos .|A B|=AB =A B=ABsin . AB=iAiBiAB=iAiBi uct duct_space duct product 7
Examples p24 ~ p25 The tangential velocity: The shortest distance between two lines Medians of a Triangle Meet in the center. Used to prove that c // m. 8 Why? Conditions? mm
Summary |A B|=ABsin . Its magnitude is related to the concept of area and its direction is defined by the right hand rule. 9
Homework
Nouns 11