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英文版大学物理 第五章


Example 5-2 Show that the rotational inertia of a uniform annular cylinder (or ring) of inner radius R1, outer radius R2, and mass M, is I 1 M( R12 R22 ) , 2 as stated in Table 5-1b, if the rotation axis is through the center along the axis of symmetry. Divide the cylinder into thin concentric Solution: cylindrical rings or hoops of thickness dr, each with mass R1 r dm dV dr is the mass density of the body. or
1 2 2 1 1 1 2 2 2 2
2 i i
o
ri
mi
vi
mi is the mass of the ith particle and vi is its speed.
K 1 mi vi2 1 ( mi ri2 ) 2 2 2
rotational inertia (or moment of inertia) I Kinetic Energy of Rotation
z
Rotation axis
y Body Reference line y
r s
O
Rotation axis
x
O
x
Zero angular position
1. Angular Position A reference line fixed in the body, perpendicular to the rotation axis, and rotating with the body. The angular position of reference line describes the angular position of the body:
avg 2 Hale Waihona Puke 1t 2 t1 t
The (instantaneous) angular acceleration ,, is the limit in this equation as t approaches zero.
d lim t 0 t dt
The Position s=r
y
The Speed Differentiating the equation with respect to time—with r held constant
ds d r dt dt
r s
O
Rotation axis
x
v r
Zero angular position
The unit of angular acceleration is commonly the radian per second-squared (rad/s2) or the revolution per second-squared (rev/s2).
Are Angular Quantities Vectors?
(radian measure)
Equation of motion for a rotating body: = (t)
2. Angular Displacement When the body rotates about the rotation axis from 1 to 2, the body undergoes an angular displacement y 2 1 Reference At t2 line An angular displacement in the counterclockwise direction is positive, in the clockwise direction is negative. At t1 1 2 x 3. Angular Velocity O Rotation axis Average angular velocity The angular velocity is either 2 1 avg positive or negative, depending t2 t1 t on whether the body is rotating Angular velocity counterclockwise (positive) or d lim clockwise (negative). t 0 t dt
Linear mass density Surface mass density Volume mass density
Rotational inertia depends on following factors: Mass and shape of rigid body , position of rotation axis
Chapter 5 Rotations of Rigid Bodies
5-1 The Kinematics of Rotations about a Fixed Axis 5-2 Kinetic Energy of Rotation 5-3 Calculating the Rotational Inertia 5-4 Torque 5-5 Newton’s Second Law for Rotation 5-6 Work and Rotational Kinetic Energy 5-7 Angular Momentum 5-8 Newton’s Second Law in Angular Form 5-9 Conservation of Angular Momentum 5-10 The Spinning Top
= 0 +t
at r

0
t
r
2.56 0 (15) 0.38 m / s 2 0.039g 100
Although the final radial acceleration ar = 10g is large (and alarming), the astronaut’s tangential acceleration at during the speed-up is not.
at = r = 0 Therefore a = ar = 2 r = 10g
ar (10)(9.8) 2.56 rad / s 24 rev / s r 15
(Answer)
(b) What is the tangential acceleration of the astronaut if the centrifuge accelerates at a constant rate from rest to the angular speeds of (a) in 100 s? Solution: Because the angular acceleration is constant Then
5-1 The Kinematics of Rotations about a Fixed Axis A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. A fixed axis means that the rotation occurs about an axis that does not move. z In the description of the rotational Rotation Body motion, we shall introduce angular axis equivalents of the linear quantities position, displacement, velocity, r P and acceleration. y These quantities not only describe x O the rigid body as a whole but also for every particle within that body.
0
0 0 t t
1 2
2
x x0 v0t 1 at 2 2
2 v 2 v0 2a( x x0 )
2 2 0 2 ( 0 )
avg
0
2
vavg
v v0 2
Relating with Linear and Angular Variables
z Axis z
Axis
Right-hand rule z Axis Speeding up Axis Slowing down z
5. Rotation with Constant Angular Acceleration In pure translation, motion with a constant linear acceleration is an important special case. In pure rotation, the case of constant angular acceleration is also important, and a parallel set of equations holds for this case also. Angular Linear 0 t v v at
Checkpoint 1 The Acceleration Differentiating the equation with respect to time—again with r held constant dv d v2 2
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