转动运动学

Directions of vectors ω and α
• ω , α are vectors • Their directions are actually given by the right-hand rule
Is it possible to have ω and α to be anti-parallel ?
Answers:
• They can be parallel if the angular speed is increasing • They can be antiparallel if the speed is decreasing
Sense of rotation
ω: out of page (Right Hand Rule) α: Into page
m 13m/s
y 2m Vf 55 o x
m
V2i
Since 18.6 m/s = 41.6 mi/hr, the driver in the northward car was lying because this is beyond the speed limit.* * The speed limit for both roads is 35 mi/h Question: Is KE conserved in this collision ? How much KE is lost to deformation ?
y 2m Vf m 13m/s m V2i 55 o x
Conservation of momentum in the x-direction 13m = 2mvf cos 55° ……………..[1] Conservation of momentum in the y-direction mv2i = 2mvf sin 55° ……………..[2] Divide [2] by [1] to eliminate vf and m to find v2i = 13 tan 55° = 18.6 m/s
identical to linear kinematics equation except for substitution: xÆθ vÆ ω aÆα tÆt
Comparison Between Rotational and Linear Equations
Can predict the future state, given the initial state and the acceleration !
– similar to the kinematic equations for linear motion – The rotational equations have the same mathematical form as the linear equation
Rotational Kinematic Equations (looks familiar ?)
Can define Angular Speed ω of a Rigid Body
• the ratio of the angular displacement to the time interval
Units:[rad/s]
instantaneous
Angular speed ω is positive if θ is increasing (counterclockwise) negative if θ is decreasing (clockwise)
• Choose a fixed reference line OX • Point P is at a fixed distance r from the origin To track P….
• It makes sense to define an angular position of a point P on the rotating CD. • This angular position is the angle swept by a fixed line relative to a reference line OX.
0
ω = ω0 + αt t = -ω0 /α = -(10.47rad/s)/(-2 rad/s2) = 5.24 s b. θ = θ0 + ω0t + (1/2)αt2 = 0 + (10.47rad/s)(5.24s) + (1/2)(-2 rad/s2)(5.24s)2 θ = 27.4 rad or about 27.4/(2π) = 4.36 revolutions
Example (See Course Website Examples) A bock of mass m1=5kg is released from A on a curved frictionless track. It hits another block of mass m2 = 10kg initially at rest. Problem: Determine the height to which m1 rises after the elastic collision.
h =0.55m.
Problem 8-26
Two automobiles of equal mass approach an intersection. One vehicle is traveling with velocity 13.0 m/s toward the east, and the other is traveling north with speed v2i. Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of 55.0° north of east. The speed limit for both roads is 35 mi/h, and the driver of the northward-moving vehicle claims that he was within the speed limit when the collision occurred. Is he telling the truth?
– The relative locations of particles making up the object e.g. A, B remain constant
A
P B
How do we track a point P on the disc ?
Defining Angular Position
We now turn our attention to Motion of
Rigid BODIES
(i.e. not point particles) • Think of a compact disc (CD). • It is a rigid object – a nondeformable object.
There are 2π radians in a 360° revolution
Sense of rotation
Helpful Conversions
• Comparing degrees and radians
• Converting from degrees to radians
If the rigid body is rotating, we can define an Angular Displacement
If the body’s angular speed is changing, we can define an Angular Acceleration α
Units:[rad/s2]
inst
Question: How does the ω and α of any two points A, B on a rotating rigid body compare ?
P
θ
r
X
• As the particle moves, the only coordinate that changes is θ • As the particle moves through θ, it moves though an arc length s. • The arc length and r are related: s=θr (θ in units of radians)
Problem 10-3 An electric motor rotating a grinding wheel at 100 rev/min is switched off. The wheel then moves with a constant negative angular acceleration of magnitude 2.00 rad/s2. (a) During what time interval does the wheel come to rest? (b) Through how many radians does it turn while slowing down? Solution: a. (100 rev/min)(2π rad/rev)(1min/60s) = 10.47 rad/s
• the angle the object rotates through during some time interval
• This is the angle that the reference line of length r sweeps out