Chapter 8 Rotational kinematics A
Chapter 8 Rotational kinematics
Section 8-1 Rotational motion The general motion of a rigid object will include both rotational and translational components For example The motion of a wheel on a moving bicycle A wobbling football in flight is more complex case Rotations with only one fixed point Rotations (定点转动) Rotations with fixed axis(定轴转动) pure rotation
Section 8-1 Rotational motion The general motion of a rigid object will include both rotational and translational components. For example: The motion of a wheel on a moving bicycle; A wobbling football in flight is more complex case. Rotations Rotations with fixed axis(定轴转动) Rotations with only one fixed point （定点转动） pure rotation
Two definitions of a pure rotation a Every point of the body moves in a circular path. The centers of these circles must ie on a common straight line called the axis of rotation B 口 Any reference line A perpendicular to the axis (such as AB in Fig 8-1) moves through the same angle in a given time interval
Every point of the body moves in a circular path. The centers of these circles must lie on a common straight line called the axis of rotation. Any reference line perpendicular to the axis (such as AB in Fig 8-1) moves through the same angle in a given time interval. x y z B A Fig 8-1 p Two definitions of a pure rotation:
How many freedoms are needed to describe completely for a pure rotation? 1(R How many for a rotation with only one fixed point? 3 (R In general the three-dimensional description of a rigid body requires six coordinates: three to locate the center of mass, two angles(such as latitude and longitude) to orient the axis of rotation, and one angle to describe rotations about the axis
In general the three-dimensional description of a rigid body requires six coordinates: three to locate the center of mass, two angles (such as latitude and longitude) to orient the axis of rotation, and one angle to describe rotations about the axis. How many freedoms are needed to describe completely for a pure rotation? 1(R) How many for a rotation with only one fixed point? 3(R)
8-2 The rotational variables 1. Angular displacement Fig 8-4 shows a rod rotating about the z axis. Any point p on the od will trace an arc of a circle A The angle o is the angular position of the reference line ap with y A respect to the x axis. x g
2 2 ,t 1 ,t 1 P1 P2 A x y Fig 8-4 8-2 The rotational variables Fig 8-4 shows a rod rotating about the z axis. Any point P on the rod will trace an arc of a circle. The angle is the angular position of the reference line AP with respect to the x axis. Z A P x y 1. Angular displacement
We choose the positive sense of the rotation to be counterclockwise(逆时钅 =s/r(8-1) where the s is the arc which the point P moves, and r is the radius(AP) At time ti the angular position is u, att, is 2. The angular displacement of p is Ap=p2-p during △t=t
We choose the positive sense of the rotation to be counterclockwise(逆时针). (8-1) where the s is the arc which the point P moves, and r is the radius (AP). At time the angular position is , at is . The angular displacement of P is during . 1 t 2 t 1 2 = 2 −1 2 1 t = t −t = s/r
2. Angular velocity We define the average angular velocity as (8-2) △t The instantaneous angular velocity a is △φd △→0△talt (8-3) Is aa vector quantity? The dimensions of a inverse time(t-l); its units may be radians per second rady s or revolutions per second ( rev/s)
We define the average angular velocity as (8-2) The instantaneous angular velocity is (8-3) Is a vector quantity? t av = dt d t t = = →0 lim 2. Angular velocity The dimensions of inverse time ( ); its units may be radians per second ( ) or revolutions per second ( ). −1 T rad /s rev /s
3. Angular acceleration If the angular velocity of p is not constant, then the average angular acceleration is defined as △t (8-4) The instantaneous angular acceleration is △Od C m (8-5) △t→>0△t t Its dimensions are inverse time squared (T)and its units might be rad/>2orrev/s
If the angular velocity of P is not constant, then the average angular acceleration is defined as (8-4) The instantaneous angular acceleration is (8-5) 3. Angular acceleration t av = dt d t t = = →0 lim Its dimensions are inverse time squared ( ) and its units might be or . −2 T 2 rad /s 2 rev /s
8-3 Rotational quantities as vectors Commutative addition law for any vectors A+b=b+a Can angular displacements satisfy corresponding formula??? △q1+△2△2+△1 As example, we first rotate a book△1=90° about x axis, followed△如2=90° by about z axis. but if we first rotate the book a,=90% by about z axis and then Ap =90 by about x axis, the final positions of the book are different
8-3 Rotational quantities as vectors Commutative addition law for any vectors: → → → → A+ B = B+ A 1 = 90 2 = 90 2 = 90 As example, we first rotate a book 1 = 90 about x axis, followed by about z axis. But if we first rotate the book by about z axis and then by about x axis, the final positions of the book are different. Can angular displacements satisfy corresponding formula??? Δφ1 +Δφ2 ? = Δφ2 +Δφ1
We conclude that △+△如2≠△2+△ and so finite angular displacements cannot be represented as vector quantities If the angular displacement are made infinitesimal the order of the rotations no longer affects the final outcome: that is do +do=do+do Hence do can be represented as vectors
We conclude that and so finite angular displacements cannot be represented as vector quantities. If the angular displacement are made infinitesimal, the order of the rotations no longer affects the final outcome: that is Hence can be represented as vectors. 1 + 2 2 + 1 1 2 2 1 d d d d + = + . d