Transcript Rotation of Rigid Bodies - wbm

Rotational motion
Chapter 9
Rigid objects
A
rigid object has a perfectly definite
and unchanging shape and size.
 In this class, we will approximate
everything as a rigid object
Radians
 In
describing rotational motion, we will
use angles in radians, not degrees.
q in radians
s
q
r
180   rad

90 


2
rad
q in radians
 An
angle in radians is the ratio of two
lengths, so it has no units.
 We will often write “rad” as the units on
such an angle to make it clear that it’s
not in degrees
 But in calculations, “rad” doesn’t factor
into unit analysis.
Angular velocity
of change of q
 w (omega) is the symbol for angular
velocity
q
wav 
t
 Rate
q
w  lim
t  0 t
Angular velocity
 At
any instant, all points on a rigid
object have the same angular velocity.
 The units of angular velocity are rad/s.
 Sometimes angular velocity is given in
rev/s or rpm.
 1 rev is 2 radians
 Angular speed is the magnitude of
angular velocity
Angular acceleration
 Rate
of change of angular velocity
 a (alpha) is the symbol for angular
acceleration
w
a av 
t
w
a  lim
t 0 t
Angular acceleration
 The
units for angular acceleration are
rad/s2.
Comparison

x is linear position
 v is linear velocity
 a is linear
acceleration
q is angular position
 w is angular velocity
 a is angular
acceleration

Rotation with constant angular
acceleration
w  w0  at
v  v0  at
1 2
x  x0  v0t  at
2
v  v  2ax  x0 
2
2
0
 v0  v 
x  x0  
t
 2 
1 2
q  q 0  w 0 t  at
2
w  w  2a q  q0 
2
2
0
 w0  w 
q  q0  
t
 2 
Example
A
CD rotates from rest to 500 rev/min in
5.5 s.
 What
is its angular acceleration, assuming
it is constant?
 How many revolutions does the disk make
in 5.5 s?

9.52 rad/s
22.9 rev
Relating linear and angular
kinematics
 We
might want to know the linear speed
and acceleration of a point on a rotating
rigid object.
 So we need relationships between
 v and w
 a and a
Speed relationship
v  rw
 Note:
these are speeds, not velocities
Acceleration relationship
2
arad
v
2

w r
r
Change in direction
Example
 Find
the required angular speed, in
rev/min, of an ultracentrifuge for the
radial acceleration of a point 2.50 cm
from the axis to equal 400,000 times the
acceleration due to gravity.

1.25 x 104 rad/s = 1.19 x 105 rev/min
Moment of Inertia
 Rotating
objects have inertia, but is
more than just their mass.
 It depends on how that mass is
distributed.
Moment of inertia
 The
moment of inertia, I, of an object is
found by taking the sum of the mass of
each particle in the object times the
square of it’s perpendicular distance
from the axis of rotation.
I  m r  m r  ...   mi ri
2
1 1
2
2 2
i
2
Moment of Inertia
 For
continuous distributions of particles,
i.e. large objects, the sum becomes an
integral.
 The moments of inertia for several
familiar shapes with uniform densities
are given on page 215 of your book.
 Moments of inertia are given in terms of
masses and dimensions.
Kinetic energy of rotating
objects
1 2
K  Iw
2
Gravitational potential energy
of rotating objects
 Same
as for other objects, but use total
mass and position of the center of
mass.
U  MgY
Example
A
uniform thin rod of length L and mass
M, pivoted at one end, is held horizontal
and then released from rest. Assuming
the pivot is frictionless, find
 The
angular velocity of the rod when it
reaches its vertical position

Sqrt(3g/L)
On your own

An airplane propeller (I=(1/12)ML2) is 2.08 m
in length (from tip to tip) with mass 117 kg.
The propeller is rotating at 2400 rev/min
about an axis through it’s center.



What is its rotational kinetic energy?
If it were not rotating, how far would it have to drop
in free fall to acquire the same kinetic energy?
1.33 x 106 J
1.16 km
Torque
 The
measure of the tendency of a force
to change the rotational motion of a
object.
 Torque depends on the perpendicular
distance between the force and the axis
of rotation
Torque magnitude
t  Fl
t (tau) [your book uses G
(gamma)] is the magnitude of the torque
 Where
 Also
called moment
F
is the magnitude of the force
 l is the perpendicular distance between
the force and the axis of rotation.
 Also
called lever arm or moment arm
Torque magnitude
Torque magnitude
Torque Magnitude
Torque sign
 Counterclockwise
rotation is caused by
positive torques and clockwise rotation
is caused by negative torques.
 We can use this symbol to indicate
which direction is positive torque.
+
Torque Units
 The
SI-unit of torque is the Newtonmeter.
 Torque is not work or energy, so it
should not be expressed as Joules.
Torque Vector Direction
t  Fl
t  rF sin 
Visual aid for torque direction
 Torque
 Think
 The
is perpendicular to both r and F.
of a normal, right-handed screw.
torque vector points in the direction
the screw moves.
Discussion Question
 Why
are doorknobs located far from the
hinges?
Example
 Forces
F1 = 8.60 N and F2 = 2.40 N are
applied tangentially to a wheel with a
radius of 1.50 m, as shown on the next
slide. What is the net torque on the
wheel if it rotates on an axis
perpendicular to the wheel and passing
through its center?
F1
F2
You try
 Calculate
the torque (magnitude and
direction) about point O due to the force
shown below. The bar has a length of
4.00 m and the force is 30.0 N.
F
q = 60°
O
2m
Torque and angular
acceleration
t  Ia
 Only
valid for rigid objects
 a must be in rad/s2 for units to work
Example
A
torque of 32.0 N-m on a certain wheel
causes an angular acceleration of 25.0
rad/s2. What is the wheel’s moment of
inertia?
On your own
A
solid sphere has a radius of 1.90 m.
An applied torque of 960 N-m gives the
sphere an angular acceleration of 6.20
rad/s2 about an axis through its center.
Find
 The
moment of inertia of the sphere
 The mass of the sphere
Example
 An
object of mass m is tied to a light
string wound around a wheel that has a
moment of inertia I and radius R. The
wheel is frictionless, and the string does
not slip on the rim. Find the tension in
the string and the acceleration of the
object.

T=(I/(I+mR2)*mg
a=(mR2/(I+mR2))g
On your own
a
On your own
 Two
blocks are connected by a string
that passes over a pulley of radius R
and moment of inertia I. The block of
mass m1 slides on a frictionless,
horizontal surface; the block of mass m2
is suspended from the string. Find the
acceleration a of the blocks and the
tensions T1 and T2 assuming that the
string does not slip on the pulley.

a=(m2/(m1+m2+I/R2))m2g
T1=(m1/(m1+m2+I/R2))m2g
T2=((m1+I/R2)/(m1+m2+I/R2))m2g
Rigid object rotation about a
moving axis
 Combined
translation and rotation.
 Translation
of center of mass
 Rotation about the center of mass
 There
is friction, but only static friction to
keep the object from slipping
Kinetic Energy
 The
kinetic energy is the sum of
translational and rotational kinetic
energies.
1
1
2
2
K  mV  I comw
2
2
Rolling without slipping
 When
something is rolling without
slipping,
V  Rw
A  Ra
On your own
A
hollow cylindrical shell with mass M
and radius R rolls without slipping with
speed V on a flat surface. What is its
kinetic energy?

MV2
Example
A
solid disk and a hoop with the same
mass and radius roll down an incline of
height h without slipping.
 Which one reaches the bottom first?

The disk
 What
if they had different masses?
 Different radii?
Dynamics of translating and
rotating objects
 We
can use both Newton’s 2nd law and
its rotational counterpart
 F  mA
t  Ia
Example
A
uniform solid ball of mass m and
radius R rolls without slipping down a
plane inclined at an angle q. A frictional
force f is exerted on the ball by the
incline. Find the acceleration of the
center of mass.

(5/7) gsinq
Work and Power
 Work
done by a constant torque
W  tq
Work and Kinetic Energy
 Total
work done equals change in K
Power
 Power
is the rate of doing work
dW
dq
t
dt
dt
P  tw
P  Fv
Example

A uniform disk with a mass of 120 kg and a
radius of 1.4 m rotates initially with an angular
speed of 1100 rev/min. A constant tangential
force is applied at a radial distance of 0.6 m.
What work must this force do to stop the
wheel?

780 kJ

If the wheel is brought to rest in 2.5 min, what
torque does the force produce?

90.4 N-m

What is the magnitude of the force?

151 N
On your own

A playground merry-go-round has a radius of
2.40 m and a moment of inertia 2100 kg-m2
about a vertical axle through its center, and
turns with negligible friction. A child applies
an 18.0-N Force tangentially to the edge of
the merry-go-round for 15.0 s.




If the merry-go-round is initially at rest, what is its
angular speed after this 15.0-s interval?
How much work did the child do on the merry-goround?
What is the average power supplied by the child?
0.309 rad/s
100 J
6.67 W
Angular momentum
 Relationship
between angular
momentum and linear momentum is the
same as between torque and force.
t  rF
L  rp  mvr
Units
 The
units for angular momentum are
kg-m2/s
Angular momentum of rigid
objects
 Look
at L for one particle of the object
Li  mi vr  mi riw ri  mi ri w
2
 Sum
over all particles for total angular
momentum

2
L   Li    mi ri w
i
 i

L  Iw
Example
A
woman with mass 50 kg is standing
on the rim of a large disk that is rotating
at 0.50 rev/s about an axis through its
center. The disk has mass 110 kg and
a radius of 4.0 m. Calculate the
magnitude of the total angular
momentum of the woman-plus-disk
system. You can treat the woman as a
point.

5275 kg-m2/s
On your own
 Find
the magnitude of the angular
momentum of the sweeping second
hand on a clock about an axis through
the center of the clock face. The clock
hand has a length of 15.0 cm and a
mass of 6.00 g. Take the second hand
to be a slender rod rotating with
constant angular velocity about one
end.

4.71 x 10-6 kg-m2/s
Conservation of angular
momentum
 If
there is no net external torque acting
on a system, then the total angular
momentum of the system is conserved.
Li  L f
Example
A
uniform circular disk is rotating with
an initial angular speed w1 around a
frictionless shaft through its center. Its
moment of inertia is I1. It drops onto
another disk of moment of inertia I2 that
is initially at rest on the same shaft.
Because of surface friction between the
disks, they eventually attain a common
angular speed wf. Find wf.