Monday, April 25, 2011
Dr. Jae hoon Yu
• Angular Momentum Conservation
• Similarity between Linear and Angular Quantities
• Conditions for Equilibrium
• How to Solve Equilibrium Problems?
• Equilibrium Problem Exercises
• Elastic Properties of Solids
• Density and Specific Gravity
Today’s homework is homework #11, due 10pm, Friday, May 6!!

• Planetarium extra credit sheets
– Tape ONLY one corner of the ticket stub on a sheet with your name and ID on it
• I must be able to see the initial of the start lecturer of the show
– Bring it by the beginning of the class Monday, May 2
• Last quiz next Wednesday, May 4
– Covers from Ch. 11 through what we finish this Wednesday
• Final comprehensive exam
– Time: 11am, Monday, May 9
– Location: SH103
– Covers: Chapter 1.1 through what we finish Monday, May 2+ appendices
– A 100 problem package will be distributed in class this Wednesday
– Review: Wednesday, May 4 th , in the class after the quiz
• Attendance will be taken
• Reading assignments: 12 – 5 and 12 – 6
Monday, April 25, 2011 PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
2

Remember under what condition the linear momentum is conserved?
Linear momentum is conserved when the net external force is 0.
By the same token, the angular momentum of a system is constant in both magnitude and direction, if the resultant external torque acting on the system is 0.
 r
 r
  p c n ur
 e t
 d L r
L
 d t c n dp

0
What does this mean?
Angular momentum of the system before and after a certain change is the same.
r
L i
 r
L f o
Three important conservation laws for isolated system that does not get affected by external forces
Monday, April 25, 2011
K i
 i f f r p
 r p r r
L i

L f f
PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
Linear Momentum
Angular Momentum
3

A star rotates with a period of 30 days about an axis through its center. After the star undergoes a supernova explosion, the stellar core, which had a radius of 1.0x10
4 km, collapses into a neutron star of radius 3.0km. Determine the period of rotation of the neutron star.
What is your guess about the answer?
The period will be significantly shorter, because its radius got smaller.
Let’s make some assumptions:
1. There is no external torque acting on it
2. The shape remains spherical
3. Its mass remains constant
Using angular momentum conservation
L

 f
The angular speed of the star with the period T is
 
2

Thus
 f

I i


I f
2

2 mr
2

T i
T
T f

2

 f
Monday, April 25, 2011
 r f
2 r i
2
T i 

4

2

.
PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
4

Consider a planet of mass M p moving around the Sun in an elliptical orbit.
D
C
S r dr
A
B
Since the gravitational force acting on the planet is always toward radial direction, it is a central force
Therefore the torque acting on the planet by this force is always 0.
ur
L
 r
Since torque is the time rate change of angular momentum L , the angular momentum is constant.
Because the gravitational force exerted on a planet by the Sun results in no torque, the angular momentum L of the planet is constant.
 r u
  r r u
  r
 r
 dt ur d L 
0
 ur
L
0 const r r M p r v M p r r r v
Since the area swept by the motion of the planet is dA

1
2 r r
 
1
2 r r
 
L
2 M p dt dA dt

L
2 M p
This is Keper’s second law which states that the radius vector from the Sun to a planet sweeps out equal areas in equal time intervals.
Monday, April 25, 2011 PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
5

All physical quantities in linear and rotational motions show striking similarity.
Quantities Linear Rotational
Mass Mass
M
Length of motion Distance
Speed
Acceleration
Force Force v a


 r
 t v r t
 r
Work
Power
Momentum
Work r
L
 r r r
  u
 r r
WFd
Moment of Inertia
 2 r
Torque
Work





 
 t
 
 t

 u
 
Kinetic Energy
Monday, April 25, 2011
Kinetic
K

1
2
Rotational
PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu

1
2
I

6

What do you think the term “An object is at its equilibrium” means?
An object is either at rest ( Static Equilibrium ) or its center of mass is moving at a constant velocity ( Dynamic Equilibrium ).
When do you think an object is at its equilibrium?
Translational Equilibrium: Equilibrium in linear motion
Is this it?
 ur
F

0
The above condition is sufficient for a point-like object to be at its translational equilibrium. For an object with size, however, this is not sufficient. One more condition is needed. What is it?
F d d
CM
-F
Let’s consider two forces equal in magnitude but in opposite direction acting on a rigid object as shown in the figure. What do you think will happen?
The object will rotate about the CM. The net torque acting on the object about any axis must also be 0.
 r
 
0
Monday, April 25, 2011
For an object to be at its static equilibrium , the object should not have linear or angular speed. v

0
 
0
7
Dr. Jaehoon Yu

To simplify the problem, we will only deal with forces acting on x-y plane, giving torque only along z-axis. What do you think the conditions for equilibrium are in this case?
The six possible equations from the two vector equations turns to three equations.
 ur
F

0


F x
F y


0
0
AND
 r
 
0
  z

0
What happens if there are many forces exerting on an object?
r
5
O r’
O’
Monday, April 25, 2011
If an object is at its translational static equilibrium, and if the net torque acting on the object is 0 about one axis, the net torque must be 0 about any arbitrary axis.
Why is this true?
Because the object is not moving in the first place, no matter where the rotational axis is, there should not be any motion. This simply is a matter of mathematical manipulation.
PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
8

When is the center of gravity of a rigid body the same as the center of mass?
Under the uniform gravitational field throughout the body of the object.
CM
CoG
Let’s consider an arbitrary shaped object
The center of mass of this object is at x
CM


 m i m i x i 
 m i
M y
CM


 m i m i y i 
 m i
M
Let’s now examine the case that the gravitational acceleration on each point is g i x i y i
Since the CoG is the point as if all the gravitational force is exerted on, the torque due to this force becomes
 m
1 g
1
 m
2 g
2
 
 x
CoG
 m
1 g
1
If g is uniform throughout the body
Monday, April 25, 2011 x
1
 m
2 g
2 x
2
 m
1
 m
2
Dr. Jaehoon Yu
 
 
 gx
CoG x
Generalized expression for different g throughout the body

 m
1
 x
1


 m
2 m i m i x i x
2

 
 g x
CM 9

1. Identify all the forces and their directions and locations
2. Draw a free-body diagram with forces indicated on it with their directions and locations properly noted
3. Write down the force equation for each x and y component with proper signs
4. Select a rotational axis for torque calculations  Selecting the axis such that the torque of one of the unknown forces become 0 makes the problem easier to solve
5. Write down the torque equation with proper signs
6. Solve the equations for unknown quantities
Monday, April 25, 2011 PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
10

A uniform 40.0 N board supports the father and the daughter each weighing 800 N and
350 N, respectively, and is not moving. If the support (or fulcrum) is under the center of gravity of the board, and the father is 1.00 m from CoG, what is the magnitude of the normal force n exerted on the board by the support?
1m x Since there is no linear motion, this system
F
M
F g n
M
B g
D
M
D g is in its translational equilibrium


F x
F y


0 n  
Therefore the magnitude of the normal force n

40 .
0

800

350


1190 N

0
Determine where the child should sit to balance the system.
The net torque about the fulcrum by the three forces are
Therefore to balance the system the daughter must sit
Monday, April 25, 2011
 
M g
B

0 0

M
F g

1 .
00

M
D g
 x

0 x 
M
F g

1 .
00 m 
M
D g
PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
800
350

1 .
00 m

2 .
29 m
11

F
M
F g
1m
Rotational axis
Determine the position of the child to balance the system for different position of axis of rotation.
n x/2 x
D
M
F g
M
B g
The net torque about the axis of rotation by all the forces are


M
B g
 x / 2

M
F g


1 .
00
 x / 2

 n
 x / 2

M
D g
 x / 2

0
Since the normal force is
The net torque can
 be rewritten
Therefore x 

 n 
M

M

B
M g
B
 g x
B
/
 g
2

M

F
M
M g
F
F g g


M
D

1 .
00 g

M
D g

 x

/ x
2
/

2

M
M
F g

1 .
00

M
D g
 x

0
D g
 x / 2
What do we learn?
M
F g
M
D g

1 .
00 m 
800
350

1 .
00 m

2 .
29 m
No matter where the rotation axis is, net effect of
Monday, April 25, 2011 PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu

A person holds a 50.0N sphere in his hand. The forearm is horizontal. The biceps muscle is attached 3.00 cm from the joint, and the sphere is 35.0cm from the joint. Find the upward force exerted by the biceps on the forearm and the downward force exerted by the upper arm on the forearm and acting at the joint. Neglect the weight of forearm.
O d
F
U
F
B l mg
Since the system is in equilibrium, from the translational equilibrium condition
From the rotational equilibrium condition


F x
F y
 


F
B
0

F
U

F
U

0


F
B mg
 d


0 mg
 l
Thus, the force exerted by the biceps muscle is
F
B
F
B

 d
 mg
 mg
 l
 d l
50 .
0

35 .
0
3 .
00

583 N

0
Force exerted by the upper arm is
Monday, April 25, 2011
F
U

F
B
 mg

583

50 .
0

533 N
PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
13

A 5.0 m long ladder leans against a wall at a point 4.0m above the ground. The ladder is uniform and has mass 12.0kg. Assuming the wall is frictionless (but ground is not), determine the forces exerted on the ladder by the ground and the wall.
FBD
F
W
F
Gy
O
F
Gx mg
First the translational equilibrium, using components

F x

F
Gx

F
W

0

F y
  mg

F
Gy

0
Thus, the y component of the force by the ground is
F
Gy
 mg  
N

118 N
The length x
0 is, from Pythagorian theorem x
0

5.0
2 
4.0
2 
3.0
m
Monday, April 25, 2011 PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
14

From the rotational equilibrium
 
O
  mg x
0
Thus the force exerted on the ladder by the wall is
2

F
W
4.0

0
F
W
 mg x
0
2
 
44 N
4.0
4.0
The x component of the force by the ground is

F x

F
Gx

F
W

0 Solve for F
Gx
F
Gx

F
W

44 N
Thus the force exerted on the ladder by the ground is
F
G

F
2
Gx

F
2
Gy
The angle between the ground force to the floor



44
2 
118
2 
130 N tan

1


F
Gy
F
Gx


 tan

1

118

 44 

70 o
Monday, April 25, 2011 PHYS 1443-001, Spring 2011
Dr. Jaehoon Yu
15