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March 22, 2006March 22, 2006

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cm

CM

external

external

M

andM

mm

mmM

whereM

mm

dt

dMmm

dt

d

mm

aF

rrR

R

aF

rrrrF

aaF

2211

21

22112

2

22112

2

2211

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