finish momentum start spinning around
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Finish Momentum Start Spinning Around. March 22, 2006. Watsup?. Quiz on Friday Last part of energy conservation through today. Watch for still another WebAssign Full calendar for the remainder of the semester is on the website near the end of the last set of PowerPoint slides. - PowerPoint PPT PresentationTRANSCRIPT
Finish MomentumFinish MomentumStart Spinning AroundStart Spinning Around
March 22, 2006March 22, 2006
Watsup?Watsup?
• Quiz on FridayQuiz on Friday– Last part of energy conservation through Last part of energy conservation through
today.today.
• Watch for still another WebAssignWatch for still another WebAssign• Full calendar for the remainder of the Full calendar for the remainder of the
semester is on the website near the semester is on the website near the end of the last set of PowerPoint end of the last set of PowerPoint slides.slides.
• Next exam is next week!Next exam is next week!
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CM
external
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dt
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aF
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as vector mass ofcenter the
define weif m likelook wouldThis
)()(
We therefore define the center of mass as
ii
ii
CM
mM
mM
rR 1
1
Center of Mass, Coordinates
The coordinates of the center of mass are
where M is the total mass of the system
CM CM CM
i i i i i ii i i
m x m y m zx y z
M M M
Center of Mass, position
The center of mass can be located by its position vector, rCM
ri is the position of the i th particle, defined by
CM
i ii
m
M r
r
ˆ ˆ ˆi i i ix y z r i j k
Center of Mass, Example
Both masses are on the x-axis
The center of mass is on the x-axis
The center of mass is closer to the particle with the larger mass
Center of Mass, Extended Object
Think of the extended object as a system containing a large number of particles
The particle distribution is small, so the mass can be considered a continuous mass distribution
Center of Mass, Extended Object, Coordinates The coordinates of the center of mass of a
uniform object are
CM CM
CM
1 1
1
x x dm y y dmM M
z z dmM
Center of Mass, Extended Object, Position The position of the center of mass can also
be found by:
The center of mass of any symmetrical object lies on an axis of symmetry and on any plane of symmetry
CM
1dm
M r r
Density
dvdm
Vm
mass
volume-unit
Usually, is a constant but NOT ALWAYS!
Center of Mass, Example
An extended object can be considered a distribution of small mass elements, m=dxdydz
The center of mass is located at position rCM
For a flat sheet, we define a mass per unit area …
dadm
mass
eunit volum
Linear Density
dxdm
Lm
Center of Mass, Rod
Find the center of mass of a rod of mass M and length L
The location is on the x-axis (or
yCM = zCM = 0)
xCM = L / 2
A golf club consists of a shaft connected to a club head. The golf club can be modeled as a uniform rod of length L and mass m1 extending radially from the surface of a sphere of radius R and mass m2. Find the location of the club’s center of mass, measured from the center of the club head.
m2m1
LR
Motion of a System of Particles
Assume the total mass, M, of the system remains constant
We can describe the motion of the system in terms of the velocity and acceleration of the center of mass of the system
We can also describe the momentum of the system and Newton’s Second Law for the system
Velocity and Momentum of a System of Particles The velocity of the center of mass of a system of
particles is
The momentum can be expressed as
The total linear momentum of the system equals the total mass multiplied by the velocity of the center of mass
CMCM
i ii
md
dt M
vr
v
CM toti i ii i
M m v v p p
Acceleration of the Center of Mass The acceleration of the center of mass can
be found by differentiating the velocity with respect to time
CMCM
1i i
i
dm
dt M v
a a
Newton’s Second Law for a System of Particles Since the only forces are external, the net external
force equals the total mass of the system multiplied by the acceleration of the center of mass:
Fext = M aCM
The center of mass of a system of particles of combined mass M moves like an equivalent particle of mass M would move under the influence of the net external force on the system
Momentum of a System of Particles The total linear momentum of a system of
particles is conserved if no net external force is acting on the system
MvCM = ptot = constant when Fext = 0
Motion of the Center of Mass, Example A projectile is fired into the
air and suddenly explodes With no explosion, the
projectile would follow the dotted line
After the explosion, the center of mass of the fragments still follows the dotted line, the same parabolic path the projectile would have followed with no explosion
Chapter 10Chapter 10
Rotation of a Rigid ObjectRotation of a Rigid Object
about a Fixed Axisabout a Fixed Axis
Rigid Object
A rigid object is one that is nondeformable The relative locations of all particles making up
the object remain constant All real objects are deformable to some extent, but
the rigid object model is very useful in many situations where the deformation is negligible
Angular Position
Axis of rotation is the center of the disc
Choose a fixed reference line
Point P is at a fixed distance r from the origin
Angular Position, 2
Point P will rotate about the origin in a circle of radius r
Every particle on the disc undergoes circular motion about the origin, O
Polar coordinates are convenient to use to represent the position of P (or any other point)
P is located at (r, ) where r is the distance from the origin to P and is the measured counterclockwise from the reference line
Angular Position, 3
As the particle moves, the only coordinate that changes is
As the particle moves through it moves though an arc length s.
The arc length and r are related: s = r
Radian
This can also be expressed as
is a pure number, but commonly is given the artificial unit, radian
One radian is the angle subtended by an arc length equal to the radius of the arc
Conversions
Comparing degrees and radians
1 rad = = 57.3°
Converting from degrees to radians
θ [rad] = [degrees]
Angular Position, final
We can associate the angle with the entire rigid object as well as with an individual particle Remember every particle on the object rotates through
the same angle The angular position of the rigid object is the
angle between the reference line on the object and the fixed reference line in space The fixed reference line in space is often the x-axis
Angular Displacement
The angular displacement is defined as the angle the object rotates through during some time interval
This is the angle that the reference line of length r sweeps out
f i
Average Angular Speed
The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval
f i
f it t t
Instantaneous Angular Speed
The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero
lim0 t
d
t dt
Angular Speed, final
Units of angular speed are radians/sec rad/s or s-1 since radians have no dimensions
Angular speed will be positive if θ is increasing (counterclockwise)
Angular speed will be negative if θ is decreasing (clockwise)
Average Angular Acceleration The average angular acceleration, ,
of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:
f i
f it t t
Instantaneous Angular Acceleration
The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0
lim0 t
d
t dt
Angular Acceleration
Units of angular acceleration are rad/s² or s-2 since radians have no dimensions
Angular acceleration will be positive if an object rotating counterclockwise is speeding up
Angular acceleration will also be positive if an object rotating clockwise is slowing down
Angular Motion, General Notes When a rigid object rotates about a fixed axis
in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration So all characterize the motion of the
entire rigid object as well as the individual particles in the object
Directions
Strictly speaking, the speed and acceleration ( are the magnitudes of the velocity and acceleration vectors
The directions are actually given by the right-hand rule
Using this convention, is a VECTOR!
Hints for Problem-SolvingHints for Problem-Solving Similar to the techniques used in linear
motion problems With constant angular acceleration, the
techniques are much like those with constant linear acceleration
There are some differences to keep in mind For rotational motion, define a rotational axis
The choice is arbitrary Once you make the choice, it must be maintained
The object keeps returning to its original orientation, so you can find the number of revolutions made by the body
Rotational Kinematics
Under constant angular acceleration, we can describe the motion of the rigid object using a set of kinematic equations These are similar to the kinematic equations for
linear motion The rotational equations have the same
mathematical form as the linear equations
Rotational Kinematic Equations
f i t
21
2f i it t
2 2 2 ( )f i f i
1
2f i i f t
The derivations are similar to what we did with kinematics. Example:
200
0
0
2
1
constant
tt
dt
dt
C
Ctdt
dtddt
d
Comparison Between Rotational and Linear Equations
Relationship Between Angular and Linear Quantities Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion
s r
v r
a r
A dentist's drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 104 rev/min. (a) Find the drill's angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.
An airliner arrives at the terminal, and the engines are shut off. The rotor of one of the engines has an initial clockwise angular speed of 2 000 rad/s. The engine's rotation slows with an angular acceleration of magnitude 80.0 rad/s2. (a) Determine the angular speed after 10.0 s. (b) How long does it take the rotor to come to rest?
A centrifuge in a medical laboratory rotates at an angular speed of 3 600 rev/min. When switched off, it rotates 50.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.
Speed Comparison
The linear velocity is always tangent to the circular path called the tangential
velocity The magnitude is
defined by the tangential speed
ds dv r r
dt dt
Acceleration Comparison
The tangential acceleration is the derivative of the tangential velocity
t
dv da r r
dt dt
Speed and Acceleration Note
All points on the rigid object will have the same angular speed, but not the same tangential speed
All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration
The tangential quantities depend on r, and r is not the same for all points on the object
Centripetal Acceleration
An object traveling in a circle, even though it moves with a constant speed, will have an acceleration Therefore, each point on a rotating rigid object will
experience a centripetal acceleration
22
C
va r
r
Resultant Acceleration
The tangential component of the acceleration is due to changing speed
The centripetal component of the acceleration is due to changing direction
Rotational Motion Example
For a compact disc player to read a CD, the angular speed must vary to keep the tangential speed constant (vt = r)
At the inner sections, the angular speed is faster than at the outer sections