Upload
camilla-henry
View
215
Download
0
Embed Size (px)
Citation preview
Rotation and Torque
Lecture 09
Thursday: 12 February 2004
ROTATION: DEFINITIONSROTATION: DEFINITIONSROTATION: DEFINITIONSROTATION: DEFINITIONS
• Angular position: • Angular displacement: 2 – 1 =
ttt
12
12ave :locity Angular ve Ave.
Instantaneous Angular velocity:
dt
d
•Use your right hand
•Curl your fingers in the direction of the rotation
•Out-stretched thumb points in the direction of the angular velocity
What is the direction of the angular velocity?
DEFINITIONS (CONTINUED)DEFINITIONS (CONTINUED)
dt
d
ttt
:onacceleratiangular ousInstantane
:onacceleratiangular Average
12
12avg
Direction of Angular Acceleration
The easiest way to get the direction of the angular acceleration is to determine the direction of the angular velocity and then…
• If the object is speeding up, velocity and acceleration must be in the same direction.
• If the object is slowing down, velocity and acceleration must be in opposite directions.
For constant For constant
221
00
0
tt
t
221
0
021
0
020
2
)(
)(2
tt
t
x v a
Relating Linear and Angular Variables
Relating Linear and Angular Variables
rr
r
r
va
ra
rv
rs
c
t
2222
Three Accelerations
1. Centripetal Acceleration (radial component of the linear acceleration)-always non-zero in circular motion.
2. Tangential Acceleration (component of linear acc. along the direction of the velocity)-non-zero if the object is speeding up or slowing down.
3. Angular Acceleration (rate of change in angular velocity)-non-zero is the object is speeding up or slowing down.
r
vac
2
raT
Energy Considerations
Although its linear velocity v is zero, the rapidly rotating blade of a table saw certainly has kinetic
energy due to that rotation.
How can we express the energy?
We need to treat the table saw (and any other rotating rigid body) as a collection of particles with different
linear speeds.
KINETIC ENERGY OF ROTATION
KINETIC ENERGY OF ROTATION
2
221
222122
21
221
221
Where ii
iiii
ii
iii
iii
rmI
IK
rmrmK
rv
vmKK
vmK
Defining Rotational Inertia
•The larger the mass, the smaller the acceleration produced by a given force.
•The rotational inertia I plays the equivalent role in rotational motion as mass m in translational motion.
amF
•I is a measure of how hard it is to get an object rotating. The larger I, the smaller the angular acceleration produced by a given force.
Determining the Rotational Inertia of an Object
1. For common shapes, rotational inertias are listed in tables. A simple version of which is in chapter 11 of your text book.
2. For collections of point masses, we can use :
where r is the distance from the axis (or point) of rotation.
3. For more complicated objects made up of objects from #1 or #2 above, we can use the fact that rotational inertia is a scalar and so just adds as mass would.
I is a function of both the mass and shape of the object. It also depends on the axis of rotation.
Ni
iiirmI
1
2
Comparison to TranslationComparison to Translation
• x • v • a • m I
• K=1/2mv21/2I2
Force and TorqueForce and Torque
I
Torque as a Cross Product
(Like F=Ma)The direction of the Torque is always in the direction of
the angular acceleration.
• For objects in equilibrium, =0 AND F=0
sinFr
Fr
Torque Corresponds to Force
Torque Corresponds to Force
• Just as Force produces translational acceleration (causes linear motion in an object starting at rest, for example)
• Torque produces rotational acceleration (cause a rotational motion in an object starting from rest, for example)
• The “cross” or “vector” product is another way to multiply vectors. Cross product results in a vector (e.g. Torque). Dot product (goes with cos ) results in a scalar (e.g. Work)
• r is the vector that starts at the point (or axis) of rotation and ends on the point at which the force is applied.
An Example
W
Forces on “extended” bodies can be viewed as acting on a point mass (with the same total mass)
At the object’s center of mass (balancing point)
xmg
SinFr
x
Determining Direction of A CROSS PRODUCT
Fr
Angular Momentum of a Particle
• Angular momentum of a particle about a point of rotation:
• This is similar to Torques
SinFr
Fr
SinPrl
prl
Find the direction of the angular momentum vector-Right hand
rule
P
P
r
r
Does an object have to be moving in a circle to have angular momentum?
• No.
• Once we define a point (or axis) of rotation (that is, a center), any object with a linear momentum that does not move directly through that point has an angular momentum defined relative to the chosen center as
p
prL