Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a...
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Transcript of Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a...
Chapter 10 - Rotation• In this chapter we will study the rotational motion of
rigid bodies about a fixed axis. To describe this type of motion we will introduce the following new concepts:
• Angular displacement (symbol: Δθ)• Avg. and instantaneous angular velocity (symbol: ω )• Avg. and instantaneous angular acceleration (symbol: α )• Rotational inertia, also known as moment of inertia
(symbol I )• The kinetic energy of rotation as well as the rotational
inertia;• Torque (symbol τ )• We will have to solve problems related to the above
mentioned concepts.
The Rotational Variables• In this chapter we will study the rotational motion of rigid
bodies about fixed axes.• A rigid body is defined as one that can rotate with all its
parts locked together and without any change of its shape.• A fixed axis means that the object rotates about an axis
that does not move, also called the axis of rotation or the rotation axis.
• We can describe the motion of a rigid body rotating about a fixed axis by specifying just one parameter.
• Every point of the body moves in a circle whose center lies on the axis of rotation, and every point moves through the same angle during a particular time interval.
Consider the rigid body of the figure.
We take the z-axis to be the fixed axisof rotation. We define a reference linethat is fixed in the rigid body and isperpendicular to the rotational axis.
The angular position of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position.
In Fig. 11-3 , the angular position is measured relative to the positive direction of the x axis.
r
s
From geometry, we know that is given by (in radians) (10-1)
Here s is the length of arc (arc distance) along a circle and between the x axis (the zero angular position) and the reference line; r is the radius of that circle.
This angle is measured in radians (rad) rather than in revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension. Because the circumference of a circle of radius r is 2r, there are 2 radians in a complete circle: rad
r
rrev _2
23601 0
(10-2)
1 rad = 57,30 = 0,159 rev (10-3)
Angular Displacement• If the body of Fig. 10-3 rotates about the
rotation axis as in Fig. 10-4 , changing the angular position of the reference line from 1 to 2, the body undergoes an angular displacement given by:
Δθ = θ2 – θ1 (10-4)• This definition of angular displacement
holds not only for the rigid body as a whole but also for every particle within that body.
• An angular displacement in the counter-clockwise direction is positive, and one in the clockwise direction is negative.
Angular Velocity • Suppose (see Fig. 10-4 ) that our
rotating body is at angular position 1 at time t1 and at angular position 2 at time t2. We define the average angular velocity of the body in the time interval t from t1 to t2 to be
tttavg
12
12 (10-5)
Instantaneous angular velocity, ω:
dt
d
tt
0
lim (10-6)
Unit of ω: rad/s or rev/s
Angular Acceleration • If the angular velocity of a rotating body is not
constant, then the body has an angular acceleration, α
• Let 2 and 1 be its angular velocities at times t2 and t1, respectively. The average angular acceleration of the rotating body in the interval from t1 to t2 is defined as:
tttavg
12
12 (10-7)
• Instantaneous angular acceleration, α:
dt
d
tt
0
lim (10-8)
Do Sample Problems 10-1 & 10-2, p244-246
Unit: rad/s2 or rev/s2
Consider a point on a rigid body rotating about
a fixed axis. At 0 the reference line that connects
the origin with point is on the -axis (point ).
P
t
O P x A
Relating the Linear and Angular Variables :
During the time interval , point moves along arc
and covers a distance . At the same time, the reference
line rotates by an angle .
t P AP
s
OP
Aθs
O
The arc length and the angle are connected by the equation
where is the distance . The speed of point is , .
s
d rds dr OP P v rs r
dt dt dt
Relation between Angular Velocity and Speed :
v rcircumference 2 2 2
The period of revolution is given by .speed
r rT T
v r
2
T
1
Tf
2 f
(10-17)
(10-18)
(10-19) (10-20)
rO
The acceleration of point is a vector that has two
components. The first is a "radial" component along
the radius and pointing toward point . We have
enountered this component in
P
O
The Acceleration :
Chapter 4 where we
called it "centripetal" acceleration. Its magnitude is2
2r
va r
r
The second component is along the tangent to the circular path of and is thus
known as the "tangential" component. Its magnitude is
The magnitude of the acceleration vector is
t
P
d rdv da r r
dt dt dt
2 2 .t ra a a
ta r
(10-8)
(10-22)
(10-23)
When the angular acceleration is constant we can derive simple expressions
that give us the angular velocity and the angula
r posi
Rotation with Constant Angular Acceleration :
tion as a function of
time. We could derive these equations in the same way we did in Chapter 2.
Instead we will simply write the solutions by exploiting the analogy between
translational and rotat
Translational Motion Rotational Motion
ional motion using the following correspondence
b
etween the two motions.
x
0
2
0
0
2
0
2 20
2 20
(
eq. 1)
(eq. 2)
2
2
2
2
o
o
t
tt
v
a
v v at
atx x v t
v v a x x
0 (eq. 3)(10-6)
Ori
iv
mi
1 2 3
Consider the rotating rigid body shown in the figure.
We divide the body into parts of masses , , ,..., ,....
The part (or "element") at has an index and mass .i
i
m m m m
P i m
Kinetic Energy of Rotation :
2 2 21 1 2 2 3 3
The kinetic energy of rotation is the sum of the kinetic
1 1 1energies of the parts: ....
2 2 2K m v m v m v
22
2 2 2 2
1 1 The speed of the th element .
2 2
1 1 The term
rotational i
is known as
nertia moment of inerti
2 2
or about the axis of rotation. Tha e axis
i i i i i ii i
i i i ii i
K m v i v r K m r
K m r I I m r
of
rotation be specified because the value of for a rigid body depends on
its mass, its shape, as well as on the position of the rotation axis. The rotational
inertia of an object describ
mu
e o
st
s h
I
w the mass is distributed about the rotation axis.2
i ii
I m r21
2K I 2 I r dm
(10-31)
(10-32) (10-33)(10-34)
(10-35)
2The rotational inertia . This expression is useful for a rigid body that
has a discreet distribution of mass. For a continuous distribution of mass the s
i ii
I m rCalculating the Rotational Inertia :
2
um
becomes an integral, .I r dm
In the table below we list the rotational inertias for some rigid bodies.2 I r dm
(10-10)
We saw earlier that depends on the position of the rotation axis.
For a new axis we must recalculate the integral for . A simpler
method takes advantage parallel-axis t of the
I
I
Parallel - Axis Theorem :
com
Consider the rigid body of mass shown in the figure.
We assume that we know the rotational inertia
about a rotation axis that passes through the center
of mass and is perpen
heorem.
dicular t
M
I
O o the page.
The rotational inertia about an axis parallel to the axis through that passes
through point , a distance from , is given by the equation
I O
P h O2
comI I Mh (10-36)
A
com
We take the origin
to coincide with the center of mass of the rigid body shown
in the figure. We assume that we know the rotational inertia
for an axis that i
O
I
Proof of the Parallel - Axis Theorem :
s perpendicular to the page and passes
through . O
We wish to calculate the rotational inertia about a new axis perpendicular
to the page and passing through point with coordinates , . Consider
an element of mass at point with coordinates
I
P a b
dm A x
2 2
2 22
2 2 2 2
, . The distance
between points and is .
.
2 2 . The second
and third integrals are zero. The first
y r
A P r x a y b
I r dm x a y b dm
I x y dm a xdm b ydm a b dm
Rotational Inertia about :P
2 2 2com
2 2
integral is . The term .
Thus the fourth integral is equal to
I a b h
h dm Mh
2
comI I Mh
Do Sample Problems 10-6 to 10-8, p254-256
In fig. a we show a body that can rotate about an axis through
point under the action of a force applied at point a distance
from . In fig. b we resolve into two com
Tor
ponents, radia
que:
O F P
r O F
l and
tangential. The radial component cannot cause any rotation
because it acts along a line that passes through . The tangential
component sin on the other hand causes the rotation of the
o
r
t
F
O
F F
bject about . The ability of to rotate the body depends on the
magnitude and also on the distance between points and .
Thus we define as torque
The distance is
s
in
k n
.
now
t
trF rF r
O F
F
r
F
r P A
as the and it is the
perpendicular distance between point and the vector .
The algebraic sign of the torque is assigned as follows:
If a force tends to rotate an objec
moment a
t in the count
rm
O F
F
erclockwise
direction the sign is positive. If a force tends to rotate an
object in the clockwise direction the sign is negative.
F
r F (10-13)
(10-39)
(10-41)(10-40)
Draaimoment
For translational motion, Newton's second law connects
the force acting on a particle with the resulting acceleration.
There is a similar relationship between the torqu
Newton's Second Law for Rotation :
e of a force
applied on a rigid object and the resulting angular acceleration.
This equation is known as Newton's second law for rotation. We will explore
this law by studying a simple body that consists of a point mass at the end
of a massless rod of length . A force is
m
r F
applied on the particle and rotates
the system about an axis at the origin. As we did earlier, we resolve into a
tangential and a radial component. The tangential component is responsible
for the
F
2
rotation. We first apply Newton's second law for . (eq. 1)
The torque acting on the particle is (eq. 2). We eliminate
between equations 1 and 2: .
t t t
t t
t
F F ma
F r F
ma r m r r mr I
(compare with: )F ma I (10-45)
We have derived Newton's second law for rotation
for a special case: a rigid body that consists of a point
mass at the end of a massless rod of length . We will
now
m r
Newton's Second Law for Rotation :
derive the same equation for a general case. O
12
3
i
ri
net
Consider the rod-like object shown in the figure, which can rotate about an axis
through point under the action of a net torque . We divide the body into
parts or "elements" and label them. The
O
1 2 3
1 2 3
1 1 2 2
3 3
elements have masses , , ,...,
and they are located at distances , , ,..., from . We apply Newton's second
law for rotation to each element: (eq. 1), (eq. 2),
(eq
n
n
m m m m
r r r r O
I I
I
21 2 3 1 2 3
1 2 3 net
. 3), etc. If we add all these equations we get
... ... . Here is the rotational inertia
of the th element. The sum ... is the net torque applied.
T
n n i i i
n
I I I I I m r
i
1 2 3he sum ... is the rotational inertia of the body.
Thus we end up with the equation nI I I I I
net I (10-15)(10-42)
Rotational Motion
Translational Motion
x
v
Analogies Between Translational and Rotational Motion
0 0
2
0 0
2
2
0
2 20
20 0
2
2
2
2
o
o
a
v v at
atx x v t
v v a x x
t
tt
2 2
2
2
mvK
m
F ma
F
P Fv
IK
I
I
P (10-18)