Download Chapter 8 Rotational Dynamics continued

Chapter 8
Rotational Dynamics
continued
8.7 Center of Gravity
DEFINITION OF CENTER OF GRAVITY
The center of gravity of a rigid body is the point at
which its weight can be considered to act when
the torque due to the weight is being calculated.
8.7 Center of Gravity
When an object has a symmetrical shape and its weight is distributed
uniformly, the center of gravity lies at its geometrical center.
8.7 Center of Gravity
General Form of xcg
Balance point is under xcg
Center of Gravity, xcg , for 2 masses
W1x1 + W2 x2
xcg =
W1 + W2
8.7 Center of Gravity
Example: The Center of Gravity of an Arm
The horizontal arm is composed
of three parts: the upper arm (17 N),
the lower arm (11 N), and the hand
(4.2 N).
Find the center of gravity of the
arm relative to the shoulder joint.
W1x1 + W2 x2 + W3 x3
xcg =
W1 + W2 + W3
⎡⎣17 ( 0.13) + 11( 0.38) + 4.2 ( 0.61) ⎤⎦ N ⋅ m
=
= 0.28 m
(17 + 11+ 4.2) N
8.7 Center of Gravity
Finding the center of gravity of an irregular shape.
8.7 Rigid Objects in Equilibrium
EQUILIBRIUM OF A RIGID BODY
A rigid body is in equilibrium if it has zero translational
acceleration and zero angular acceleration. In equilibrium,
the sum of the externally applied forces is zero, and the
sum of the externally applied torques is zero.
Note: constant linear speed or constant rotational speed
are allowed for an object in equilibrium.
8.7 Rigid Objects in Equilibrium
Example: A board length L lies against a wall. The
coefficient of friction with ground is 0.650. What is smallest
angle the board can be placed without slipping?
1. Determine the forces acting on the board.
θ
8.7 Rigid Objects in Equilibrium
Example: A board length L lies against a wall. The
coefficient of friction with ground is 0.650. What is smallest
angle the board can be placed without slipping?
θ
1. Determine the forces acting on the board.
Forces:
Forces
P
G y ground normal force
Gy = W
Gx ground static frictional force
P wall normal force
W gravitational force
L/2
Gy
θ
Gx
2. Choose pivot point at ground.
W
P = Gx = µG y = µW
8.7 Rigid Objects in Equilibrium
Example: A board length L lies against a wall. The
coefficient of friction with ground is 0.650. What is smallest
angle the board can be placed without slipping?
θ
1. Determine the forces acting on the board.
Forces:
Forces
P
G y ground normal force
Gy = W
Gx ground static frictional force
P wall normal force
W gravitational force
L/2
Gy
W
P = Gx = µG y = µW
θ
P⊥
P
Gx
2. Choose pivot point at ground.
3. Find components normal to board
P⊥ and W⊥ are forces producing torque
L/2
Gy
θ
Gx
W
W⊥
8.7 Rigid Objects in Equilibrium
Example: A board length L lies against a wall. The
coefficient of friction with ground 0.650. What is smallest
angle the board can be placed without slipping?
4. Net torque must be zero for equilibrium
Torque:
τ P = + P⊥ L = ( Psin θ ) L
τ W = −W⊥ ( L / 2 ) = − (W cosθ ) ( L / 2 )
Forces:
Gy = W
P = Gx = µG y = µW
τ W + τ P = 0 ⇒ ( Psin θ ) = (W cosθ ) / 2
W = 2Psin θ / cosθ
θ
P⊥
P
L/2
θ
W⊥
8.7 Rigid Objects in Equilibrium
Example: A board length L lies against a wall. The
coefficient of friction with ground 0.650. What is smallest
angle the board can be placed without slipping?
4. Net torque must be zero for equilibrium
Torque:
τ P = + P⊥ L = ( Psin θ ) L
τ W = −W⊥ ( L / 2 ) = − (W cosθ ) ( L / 2 )
Forces:
Gy = W
P = Gx = µG y = µW
τ W + τ P = 0 ⇒ ( Psin θ ) = (W cosθ ) / 2
W = 2Psin θ / cosθ
5. Combine torque and force equations
W = 2Psin θ / cosθ = 2 µW tan θ
tan θ = 1 (2 µ ) = 1 (1.3) = 0.77
θ = 37.6°
θ
P⊥
P
L/2
θ
W⊥
Chapter 8 Rotational Kinematics/Dynamics
Angular motion variables
(with the usual motion equations)
displacement
velocity
acceleration
θ = s / r (rad.)
ω = v / r (rad./s)
α = a / r (rad./s 2 )
Uniform circular motion
centripetal acceleration
centripetal force
v2
aC = = ω 2 r
r
mv 2
FC = maC =
r
I Moment of Inertia
torque
 
(θ :  between F & r)
τ = rF sin θ
Newton's 2nd Law
τ Net = I α
rot. kinetic energy
K rot = Iω
angular momentum
1
2
L = Iω
2
8.8 Newton’s Second Law for Rotational Motion About a Fixed Axis
τ = FT r
= maT r
= mα r 2
( )
= mr α
2
= Iα
I = mr 2
Moment of Inertia
of the airplane
Moment of Inertia, I =mr2, for a point-mass, m, at the
end of a massless arm of length, r.
τ = Iα
Newton’s 2nd Law for rotations
8.8 Newton’s Second Law for Rotational Motion About a Fixed Axis
Break object into N
individual masses
τ Net = ∑ (mr )i α
2
i
Net external
torque
τ2
Moment of
inertia
( )
= (m r )α
τ 1 = m1r12 α
2
2 2

2
τ N = mN rN α
(
)
8.8 Newton’s Second Law for Rotational Motion About a Fixed Axis
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR
A RIGID BODY ROTATING ABOUT A FIXED AXIS
τ Net = I α
Requirement: Angular acceleration
must be expressed in radians/s2.
I = ∑ (mr 2 ) i
i
8.8 Newton’s Second Law for Rotational Motion About a Fixed Axis
dual
pulley
Example: Hoisting a Crate
The combined moment of inertia of the dual
pulley is 50.0 kg·m2. The crate weighs 4420 N.
A tension of 2150 N is maintained in the cable
attached to the motor. Find the angular
acceleration of the dual Pulley (radius-1 =
0.600m, radius-2 = 0.200 m).
Tension 1
Dual Pulley
Motor
Tension 2
up is positive
Forces on Pulley 
P Post
1 = 0.600m

T1
Axis

WP
 2 = 0.200m
Forces on Crate

T2′

ay

T2
Crate

W
magnitude mg
8.8 Newton’s Second Law for Rotational Motion About a Fixed Axis
2nd law for linear motion of crate
∑F
y
= T2 − mg = ma y
a y =  2α
2nd law for rotation of the pulley
I = 46kg ⋅ m 2
mg = 4420 N
T1 = 2150 N

P
1 = 0.600m

Dual Pulley
T1
 2 = 0.200m
Motor
Axis
T2 − mg = m 2α
T11 − T2  2 = Iα
T2 = m 2α + mg & T2 =
T11 − Iα
2
m22α + Iα = T11 − mg 2
α=

WP

T2 = T2 ,down

T2′ = T2 ,up (+)
Crate
T11 − mg 2
2
=
6.3
rad/s
m22 + I

ay
T2 = m 2α + mg = ( 451(.6)(6.3) + 4420 ) N = 6125 N
mg
8.9 Angular Momentum
DEFINITION OF ANGULAR MOMENTUM
The angular momentum L of a body rotating about a
fixed axis is the product of the body’s moment of
inertia and its angular velocity with respect to that
axis:
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Angular Momentum: kg·m2/s
8.9 Angular Momentum
PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM
The angular momentum of a system remains constant (is
conserved) if the net external torque acting on the system
is zero.
Angular momentum, L
Li = I iω i ; L f = I f ω f
⇒
No external torque
⇒ Angular momentum conserved
L f = Li
I f ω f = I iω i
I = ∑ mr 2 , r f < ri
Moment of Inertia
decreases
I f < Ii
Ii
>1
If
Ii
ω f = ωi;
If
Ii
>1
If
ω f > ω i (angular speed increases)
8.9 Angular Momentum
From Angular Momentum Conservation
(
)
ω f = Ii I f ω i
⇒
because I i I f > 1
Angular velocity increases
Is Energy conserved?
K f = 12 I f ω 2f
= I f Ii I f
(
)
= Ii I f
I iω 2i
1
2
2
ω 2i
( )( )
= (I I )K ⇒
i
f
1
2
i
K i = 12 I iω 2i ;
Kinetic Energy increases
Energy is NOT conserved because pulling in the arms does (NC) work
on the mass of each arm and increases the kinetic energy of rotation.
8.9 Angular Momentum
Example: A Satellite in an Elliptical Orbit
An artificial satellite is placed in an
elliptical orbit about the earth. Its point
of closest approach is 8.37x106 m
from the center of the earth, and
its point of greatest distance is
25.1x106 m from the center of
the earth.The speed of the satellite at the
perigee is 8450 m/s. Find the speed
at the apogee.
I A = mrA2 ; I P = mrP2
ω A = vA rA ; ω P = vP rP
Gravitational force along r (no torque) ⇒ Angular momentum conserved
I Aω A = I Pω P
mrA2 ( vA rA ) = mrP2 ( vP rP ) ⇒ rA vA = rP vP
vA = ( rP rA ) vP = ⎡⎣(8.37 × 106 ) / (25.1× 106 ) ⎤⎦ (8450 m/s ) = 2820 m/s
8.9 The Vector Nature of Angular Variables
Right-Hand Rule: Grasp the axis of rotation with
your right hand, so that your fingers circle the axis
in the same sense as the rotation.

ω
Your extended thumb points along the axis in the
direction of the angular velocity.
Vector Quantities in Rotational Motion
Angular Acceleration
Angular Momentum
Torque

 Δω
α=
Δt


L = Iω


 ΔL
τ=Iα=
Δt

ω
Changes Angular Momentum
If torque is perpendicular to the angular momentum, only the direction of the
angular momentum changes (precession) – no changes to the magnitude.
Rotational/Linear Dynamics Summary
linear
rotational
x displacement θ
velocity
v
ω
a acceleration α
m
F
point m inertia
force/torque
I = mr 2
τ = Fr sin θ
Uniform circular motion
aC = vT2 r
FC = mvT2 r
Potential Energies
U G = mgy
or U G = −GmM E /r
U S = 12 kx 2
linear


F = ma
W = F(cosθ )x
K = 12 mv 2
W ⇒ ΔK


p = mv


FΔt = Δp
rotational


τ = Iα
Wrot = τφ
K rot = 12 Iω 2
Wrot ⇒ ΔK rot


L = Iω


τΔt = ΔL
Conservation laws
If WNC = 0,
If Fext = 0,
Conserved: E = K + U Psystem
If τ ext = 0,
= ∑ p L = Iω