chapter 10: rotation of a rigid object about a fixed...
TRANSCRIPT
• Exam # 3 Nov 21
• Study: Problems from Exam # 1 and Exam #2,
• chap. #7,
• conservation of energy (powerpoint),
• chp. # 9 (9.1 – 9.6),
• chp. # 10.1 – 10.3
November 15, 2013
Rotational Motion
• Angular Position and Radian
• Angular Velocity
• Angular Acceleration
• Rigid Object under Constant
Angular Acceleration
• Angular and Translational
Quantities
• Rotational Kinetic Energy
• Moments of Inertia
November 15, 2013
Angle and Radian • What is the circumference S ?
• q can be defined as the arc length s along a circle divided by the radius r:
• q is a pure number, but commonly is given the artificial unit, radian (“rad”)
r
q s
rq
rs )2( r
s2
s
q Whenever using rotational equations, you must use angles expressed in radians
November 15, 2013
Conversions • Comparing degrees and radians
• Converting from degrees to radians
• Converting from radians to degrees
180)( rad
3601 57.3
2rad
180
rad degrees
q q
360)(2 rad
)(180
)(deg radrees q
q
November 15, 2013
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
• This simplification allows analysis of the motion of an
extended object
November 15, 2013
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
• As the particle moves, the only
coordinate that changes is q
• As the particle moves through q, it
moves though an arc length s.
• The angle q, measured in radians, is
called the angular position.
November 15, 2013
Displacement • Displacement is a change of position in time.
• Displacement:
• f stands for final and i stands for initial.
• It is a vector quantity.
• It has both magnitude and direction: + or - sign
• It has units of [length]: meters.
)()( iiff txtxx
x1 (t1) = + 2.5 m x2 (t2) = - 2.0 m Δx = -2.0 m - 2.5 m = -4.5 m
x1 (t1) = - 3.0 m x2 (t2) = + 1.0 m Δx = +1.0 m + 3.0 m = +4.0 m
November 15, 2013
Angular Displacement
• The angular displacement is
defined as the angle the
object rotates through during
some time interval
• SI unit: radian (rad)
• This is the angle that the
reference line of length r
sweeps out
f iq q q
November 15, 2013
Velocity
• Velocity is the rate of change of position.
• Velocity is a vector quantity.
• Velocity has both magnitude and direction.
• Velocity has a unit of [length/time]: meter/second.
• Definition:
• Average velocity
• Average speed
• Instantaneous
velocity
avg
total distances
t
0lim
t
x dxv
t dt
t
xx
t
xv
if
avg
November 15, 2013
Average and Instantaneous
Angular Speed
• The average angular speed, ωavg, of a rotating rigid object is
the ratio of the angular displacement to the time interval
• The instantaneous angular speed is defined as the limit of the
average speed as the time interval approaches zero
• SI unit: radian per second (rad/s)
• Angular speed positive if rotating in counterclockwise
• Angular speed will be negative if rotating in clockwise
f iavg
f it t t
q q q
lim
0 t
d
t dt
q q
November 15, 2013
Average Angular Acceleration
• The average angular acceleration, a, 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
avg
f it t t
a
t = ti: i t = tf: f
November 15, 2013
Instantaneous Angular Acceleration
• The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0
• SI Units of angular acceleration: rad/s²
• Positive angular acceleration is in the counterclockwise. • if an object rotating counterclockwise is speeding up
• if an object rotating clockwise is slowing down
• Negative angular acceleration is in the clockwise.
• if an object rotating counterclockwise is slowing down
• if an object rotating clockwise is speeding up
lim
0 t
d
t dt
a
November 15, 2013
Rotational Kinematics • A number of parallels exist between the equations for
rotational motion and those for linear motion.
• 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
t
x
tt
xxv
if
if
avg
f iavg
f it t t
q q q
November 15, 2013
Example 1
• A wheel rotates with a constant angular acceleration of 5.0 rad/s2. If the angular speed of the wheel is 1.5 rad/s at t = 0
(a) through what angle does the wheel rotate between
t = 0 and t = 2.0 s? Given your answer in radians and in revolutions.
(b) What is the angular speed of the wheel at t = 2.0 s?
November 15, 2013
Example 2
• During a certain time interval, the angular position of a swinging
• door is described by , where θ is in radians
• and t is in seconds. Determine the angular position, angular speed,
• and angular acceleration of the door at the following times.
• • a) t= 0 s
• • b) t = 3.00 s
200.20.1000.5 tt q
November 15, 2013
Example 3
A rotating wheel requires 3.0 s to rotate through 37.0 revolutions. Its angular
speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant
angular acceleration of the wheel?
November 15, 2013
Relationship Between Angular and
Linear Quantities
• Every point on the rotating object has
the same angular motion
• Every point on the rotating object
does not have the same linear motion
• Displacement
• Speeds
• Accelerations
s rq
v r
a ra
November 15, 2013
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
rvorr
v arat
November 15, 2013
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
222 )(
rr
r
r
var
November 15, 2013
Home Example 4
A car accelerates uniformly from rest and reaches a speed of 22.0 m/s in
9.00 s. If the diameter of a tire is 58.0 cm, find (a) the number of
revolutions the tire makes during this motion, assuming that no slipping
occurs. (b) What is the final angular speed of a tire in revolutions per
second?
November 15, 2013
Moment of Inertia of Point Mass
• For a single particle, the definition of moment of inertia is
• m is the mass of the single particle
• r is the rotational radius
• SI units of moment of inertia are kg.m2
• Moment of inertia and mass of an object are different quantities
• It depends on both the quantity of matter and its distribution (through the r2 term)
2mrI
November 15, 2013
Moment of Inertia of Point Mass
• For a composite particle, the definition of moment of inertia
is
• mi is the mass of the ith single particle
• ri is the rotational radius of ith particle
• SI units of moment of inertia are kg.m2
• Consider an unusual baton made up of four sphere fastened to
the ends of very light rods
• Find I about an axis perpendicular to the page and passing
through the point O where the rods cross
...2
44
2
33
2
22
2
11
2 rmrmrmrmrmI ii
222222222 mbMaMambMambrmI ii
November 15, 2013
Example: Moment of Inertia
of a Uniform Rigid Rod
• The shaded area has a
mass
• dm = l dx
• Then the moment of
inertia is
/ 2
2 2
/ 2
21
12
L
yL
MI r dm x dx
L
I ML