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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. - PowerPoint PPT Presentation
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Rotational motion
Chapter 9
Rigid objectsA 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.
in radians
r
s
rad 180 rad 2
90
in radiansAn 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 velocityRate of change of (omega) is the symbol for angular
velocity
tav
tt
0
lim
Angular velocityAt 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 radiansAngular speed is the magnitude of
angular velocity
Angular accelerationRate of change of angular velocity (alpha) is the symbol for angular
acceleration
tav
tt
0
lim
Angular accelerationThe units for angular acceleration are
rad/s2.
Comparison x is linear position v is linear velocity a is linear
acceleration
is angular position is angular velocity is angular
acceleration
Rotation with constant angular acceleration
200 2
1attvxx
020
2 2 xxavv
tvv
xx
2
00
atvv 0
200 2
1tt
020
2 2
t
2
00
t 0
ExampleA 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/s22.9 rev
Relating linear and angular kinematicsWe might want to know the linear speed
and acceleration of a point on a rotating rigid object.
So we need relationships betweenv and a and
Speed relationship
Note: these are speeds, not velocities
rv
Acceleration relationship
rr
varad
22
Change in direction
ExampleFind 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 InertiaRotating objects have inertia, but is
more than just their mass. It depends on how that mass is
distributed.
Moment of inertiaThe 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.
2222
211 ...
iiirmrmrmI
Moment of InertiaFor 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
2
2
1 IK
Gravitational potential energy of rotating objectsSame as for other objects, but use total
mass and position of the center of mass.
MgYU
ExampleA 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
TorqueThe 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
Where (tau) [your book uses (gamma)] is the magnitude of the torque 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
Fl
Torque magnitude
Torque magnitude
Torque Magnitude
Torque signCounterclockwise 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 UnitsThe SI-unit of torque is the Newton-
meter.Torque is not work or energy, so it
should not be expressed as Joules.
Torque Vector Direction
Fl
sinrF
Visual aid for torque direction
Think of a normal, right-handed screw.
The torque vector points in the direction the screw moves.
Torque is perpendicular to both r and F.
Discussion QuestionWhy 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?
F2
F1
You tryCalculate 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.
O 2 m
F
= 60°
Torque and angular acceleration
Only valid for rigid objects must be in rad/s2 for units to work
I
ExampleA 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 ownA 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
ExampleAn 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 ownTwo 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))m2gT1=(m1/(m1+m2+I/R2))m2gT2=((m1+I/R2)/(m1+m2+I/R2))m2g
Rigid object rotation about a moving axisCombined 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 EnergyThe kinetic energy is the sum of
translational and rotational kinetic energies.
22
2
1
2
1 comImVK
Rolling without slippingWhen something is rolling without
slipping,
RV
RA
On your ownA 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
ExampleA 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 objectsWe can use both Newton’s 2nd law and
its rotational counterpart
mAF
I
ExampleA uniform solid ball of mass m and
radius R rolls without slipping down a plane inclined at an angle . A frictional force f is exerted on the ball by the incline. Find the acceleration of the center of mass.
(5/7) gsin
Work and Power
Work done by a constant torque
W
Work and Kinetic Energy
Total work done equals change in K
PowerPower is the rate of doing work
dt
d
dt
dW
P
FvP
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-go-
round? What is the average power supplied by the child?
0.309 rad/s 100 J 6.67 W
Angular momentumRelationship between angular
momentum and linear momentum is the same as between torque and force.
mvrrpL
rF
UnitsThe units for angular momentum are
kg-m2/s
Angular momentum of rigid objects
Look at L for one particle of the object
2iiiiiii rmrrmvrmL
Sum over all particles for total angular momentum
iii
ii rmLL 2
IL
ExampleA 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 ownFind 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.
fi LL
ExampleA uniform circular disk is rotating with an
initial angular speed 1 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 f. Find f.