11 rotation The rotational variables Relating the linear and angular variables Kinetic energy of rotation and rotational inertia Torque, Newton’s second law for rotation Work and rotational kinetic energy

11-1, the rotational variables: (1) Translation and rotation: In pure translation （平动）, every point of the body moves in a straight line, and every point moves through the same linear distance during a particular time interval. In rotation about the fixed axis （定轴转动）, every point moves through the same angle during a particular time interval. Rigid body （刚体） : is a body (that can rotate) with all its parts locked together and without any change in its shape.

(2) The rotation of a rigid body about a fixed axis: Angular position: to describe the rotation of a rigid body about a fixed axis, called the rotation axis,we assume a reference line is fixed in the body, perpendicular to that axis and rotating with the body. We measure the angular position  of this line relative to a fixed direction (x). Unit of  : radian( rad) 弧度 Always:  =  (t)   o s r x ss

Angular displacement: a body that rotates about a rotation axis, changing its angular position from  1 to  2, undergoes an angular displacement: Where   is positive for counterclockwise rotation and negative for clockwise rotation.

Angular velocity and speed: Average angular velocity: (Instantaneous) angular velocity:  avg and  are vectors, with directions given by the right-hand rule. They are positive for counter- clockwise rotation and negative for clockwise rotation. The magnitude of the body’s angular velocity is the angular speed. 

Angular acceleration: Average angular acceleration: (Instantaneous) angular acceleration: Both  avg and  are vectors. When  is positive:  >0, the direction is the same as  ;  <0, the direction is reversed to . 

(3) Rotation with constant angular acceleration See page 221: Table 11-1 equations of motion for constant linear acceleration and for constant angular acceleration

v θ θ R x ΔSΔS 0 ω,ω, Δ (1) Circular motion: description by linear variables Speed: Distance: S=r  or  s=r   11-5 relating the linear and angular variables

O X R  切向 内法向 A B

 Radial acceleration: Tangential acceleration:

(2) Relating the linear and angular variables Home work: 5E, 29p

11-6, 11-7 kinetic energy of rotation and rotational inertia (1)rotational kinetic energy: Proof: treat the rotation rigid body as a collection of particles with different speeds. We can then add up the kinetic energies of all the particles to find the kinetic energy of the body as a whole. In this way we obtain the kinetic energy of a rotating body, (Compare with kinetic energy of particle)

Let: Then: I is rotational inertia (or moment of inertia)( 转动惯量） O riri  z mimi vivi

(2) Rotational inertia : For rigid body with continuously distributed mass: Where r and r i represent the perpendicular distance from the axis of rotation to each mass element in the body. dm r M For particle: I = m r 2

See page 227 table 11-2 some rotational inertias (calculate (e): 1.thin rod about axis through center perpendicular to length; 2. About axis through the end of the rod perpendicular to length.) Sample problem:11-5 Unit: kg.m 2

(3) Parallel-Axis Theorem: Where I is the rotational inertia of a body of mass M about a given axis. I com is the rotational inertia of the body about a parallel axis that extends through the body’s center of mass. h is the perpendicular distance between the given axis and the axis through the center of mass. M is the mass of rigid body. C h M I Com I 平行

Proof of the parallel-axis theorem: (by yourself) Question: What does the rotational inertia relate to? Answer: Mass and its distribution.

11-8 torque O ×  z F P  r Unit: N.m Where F t is the component of perpendicular to, and  is the angle between and. Torque is a turning or twisting action on a body about a rotation axis due to a force.

(1)Torque is a vector for rotation about fixed axis, its direction always along the axis, either positive or negative. (see page 230) (2) If several torque act on a rigid body that rotate about a fixed axis, the net torque is the sum of individual torque. (3) The net torque of internal forces is zero. Caution:

11-9 Newton’s second law for rotation ( 转动定律） Compare with: We obtain: For rotation about fixed axis: Proof of equation:

f i Treat the rigid body as a collection of particles, F i and f i are the external and internal forces of mass element  m i,thus: Then: O ω,αω,α riri  z F i mimi Δ  v

For whole rigid body: That is:

Sample problem: 11-7, 11-8

11-10 Work and rotational kinetic energy From: For rigid body: From initial angular position to final angular position, the work is: (1) work:

Caution: (1) If torque is constant, then: (2) When several torques act on the rigid body, the net work is sum of individual work. (3) This work is scalar also. (2) Work-kinetic energy theorem:

(4) The gravitational potential energy of rigid body: (1) U can not change when the rigid body rotate about axis that extends through the body’s center of mass. (2) The conservation of Mechanical energy and conservation of energy can also be used for rigid body. (3) power:

Sample problem: 11-9, Home work: 29p, 65p, 66p