Lecture 27: More on Rotational Kinematics

Transcription

1 Lecture 27: More on Rotational Kinematics Let s work out the kinematics of rotational motion if α is constant: dω α = 1 2 α dω αt = ω ω ω = αt + ω ( t ) dφ α + ω = dφ t 2 α + ωo = φ φo = 1 2 = t o 2 φ α + ωo + φo t o o = t Analogous to linear velocit with constant acceleration Analogous to linear position with constant acceleration

2 Relationships Between Angular and Linear Variables When an object is rotating, ever point in it is moving with a linear velocit We know that if the object rotates through an angle φ, the linear distance moved is given b: s = rφ It then follows that: ds dφ v = = r = rω dv dω at = = r = rα Note that there is also a radial (or centripetal) acceleration: 2 v 2 ac = = ω r r

3 Vector Nature of Rotational Quantities So far we ve developed a set of rotational kinematic quantities which are analogous to those seen in linear motion: φ α a But we know that position, velocit, and acceleration are reall vector quantities So we might epect that the same is true for the rotational variables And it is partl! ω v

4 Just as we didn t need vectors to describe one-dimensional motion, we don t need them to describe rotation about a fied ais But in general the ais of rotation is not constant For eample, imagine a football thrown with a slight wobble To handle such cases we do need vectors But there s a problem The rotation of an object through an angle φ can t be represented as a vector

5 To see wh, consider an object that undergoes two 90 o rotations, about different aes: Original Rotation about Rotation about z z z Original Rotation about Rotation about z

6 So we see that the result of two rotations about different aes can depend on the order in which the rotations are done Since for vectors we know that A + B alwas equals B + A, rotations cannot be considered vector quantities! The situation is better, though, if we consider ver small rotations: r 2 r 1 φ

7 The difference between r 1 and r 2 is r: φ ( φ ) ( 1 1) r = r 1 cos i + r sin φ j r i + r φ j We can therefore define a vector φ: = φ j = r j r So small rotations do in fact behave as vectors i.e., the order in which the are done doesn t matter

8 This is important since it allows us to define vectors associated with angular velocit and acceleration: d = A vector, since dφ is a small angle The magnitude of ω the scalar ω we discussed earlier To determine the direction of ω, use the right-hand rule: Curl the fingers of our right hand in the same direction as the rotation Your thumb then points along the direction of ω Once ω has been defined, we can also define α: = d

9 Vector Cross Product There is one additional vector operation that will be important in our discussion of rotational motion This is a form of vector multiplication that results in another vector We alread know about the dot product, where two vectors are multiplied to give a scalar The notation used is: A B = C The magnitude of C is given b C = A B sinφ

10 We still need to define the direction of C It is perpendicular to the plane in which vectors A and B lie Still need to decide between up and down We again do this with the help of the right-hand rule Begin with the fingers of the right hand pointed along the first vector in the cross product Orient our hand such that our fingers curl toward the second vector Your thumb them points in the direction of C

11 Notes on the Cross-Product For parallel vectors, the cross product is 0 The result depends on the order of the vectors in the product: A B = ( B A) Appling the cross product to the unit vectors, we find: i j = k j k = i k i = j

12 This means that for an two vectors: ( A A A ) ( B B B ) A B = i + j + k i + j + k z z ( i j) A B ( i k) ( j i) A B ( j k) ( k i) + A B ( k j) = A B + z + A B + + A B z z z = A B k A B z A B k + z z ( A B A B ) i + ( A B A B ) j + ( A B A B ) A B + A B j A B = z i j i z z z z k

13 Hand mnemonic (if ou ve seen matrices before ): The cross product can be written as the determinant of a 3 3 matri: A B = i j k A A A z B B B z