physics 201: lecture 22, pg 1 lecture 21 goals: use free body diagrams prior to problem solving...
TRANSCRIPT
Physics 201: Lecture 22, Pg 1
Lecture 21Goals:Goals:
• Use Free Body Diagrams prior to problem solving
• Introduce Young’s, Shear and Bulk modulus
Exam 3: Wednesday, April, 18th 7:15-8:45 PM
Physics 201: Lecture 22, Pg 2
Statics
Equilibrium is established when
0 motion nalTranslatio F
0 motion Rotational
In 3D this implies SIX expressions (x, y & z)
Physics 201: Lecture 22, Pg 3
Balancing Acts
So, where is the center of mass for this construction?
Between the arrows!
Physics 201: Lecture 22, Pg 4
Example problem The board in the picture has an average length of 1.4 m,
mass 2.0 kg and width 0.14 m. The angle of the board is 45°. The bottle is 1.5 m long, a center of mass 2/5 of the way from the bottom and is mounted 1/5 of way from the end. If the sketch represents the physics of the problem.
What are the minimum and maximum masses of the bottle that will allow this system to remain in balance?
-0.1 m
-0.5 m-0.8 m
0.1 m0.0 m
0.4 m
Physics 201: Lecture 22, Pg 5
Example problem What are the minimum and maximum masses of the bottle
that will allow this system to remain in balance? X center of mass,
x, -0.1 m < x < 0.1 m
-0.1 m
-0.5 m-0.8 m
0.1 m0.0 m
0.4 m
2.0 kg
i
iiCM m
mxx CM
1.2
)(4.0)2(5.0
MM
xCM
12410
MM
kg 4M
Physics 201: Lecture 22, Pg 6
Example problem What are the minimum and maximum masses of the bottle
that will allow this system to remain in balance? Check torque condition (g =10 m/s2) x, -0.1 m < x < 0.1 m
-0.1 m
-0.5 m-0.8 m
0.1 m0.0 m
0.4 m
2.0 kg
kg 04max .M )6.0(20)3.0(40 z
Nm )1212( z
Nm 0 z
kg 61min .M
Physics 201: Lecture 22, Pg 7
Real Physical Objects Representations of matter
Particles: No size, no shape can only translate along a path Extended objects are collections of point-like particles
They have size and shape so, to better reflect their motion, we introduce the center of mass
Rigid objects: Translation + Rotation
Deformable objects:
Regular solids:
Shape/size changes under stress (applied forces)
Reversible and irreversible deformation
Liquids:
They do not have fixed shape
Size (aka volume) will change under stress
Physics 201: Lecture 22, Pg 8
Young’s modulus: measures the resistance of a solid to a change in its length.
Changes in length: Young’s modulus
L0 L
Felasticity in length
0//
strain tensilestress tensileY
LLAF
A
Physics 201: Lecture 22, Pg 9
Real Materials have a complex behavior
i/
/
strain tensile
stress tensileYLL
AF
FΔLiLAY
FLk
slope is Y
Hooke’s Law
Physics 201: Lecture 22, Pg 10
Bulk modulus: measures the resistance of solids or liquids to changes in their volume.
Changes in volume: Bulk Modulus
V0
Vi + V
F
Volume elasticity
iΔV/VF/AB
PressureF/AP
Physics 201: Lecture 22, Pg 11
Changes in shape: Shear Modulus
hx
AF
/
/
strainshear
stressshear S
Applying a force perpendicular to a surface
Physics 201: Lecture 22, Pg 12
Example values
Carbon nanotube 100 x 1010
Physics 201: Lecture 22, Pg 13
Example: A flying swing A flying swing, as shown, consists of a 5.0 m horizontal ball
and, at its end, a hanging wire 5.0 m long. The wire has a cross sectional area of 1.0x104 m2.The swing turns so that the wire makes a 45° angle with respect to the vertical. Assuming g=10 m/s2 and the wire’s Young’s modulus is 20x1010 N/m2 (steel), how much will the wire stretch?
m 8.53m )45sin0.50.5( R
yy TF N 5000
xx TRmF 2
22TTT xy
N 707T
Physics 201: Lecture 22, Pg 14
Example: A flying swing A flying swing, as shown, consists of a 5.0 m horizontal ball
and, at its end, a hanging wire 5.0 m long. The wire has a cross sectional area of 1.0x104 m2.The swing turns so that the wire makes a 45° angle with respect to the vertical. Assuming g=10 m/s2 and the wire’s Young’s modulus is 20x1010 N/m2 (steel), how much will the wire stretch?
m 8.53Rxx TRmF 2
N 707T
N 5002 RmFx rad/s 08.153.8/10
Physics 201: Lecture 22, Pg 15
Example: A flying swing A flying swing, as shown, consists of a 5.0 m horizontal ball
and, at its end, a hanging wire 5.0 m long. The wire has a cross sectional area of 1.0x104 m2.The swing turns so that the wire makes a 45° angle with respect to the vertical. Assuming g=10 m/s2 and the wire’s Young’s modulus is 20x1010 N/m2 (steel), how much will the wire stretch?
m 8.53RN 707Ti/
/YLL
AF
iLYAF /L
0.51020
101/70710
4
L
m 00018.0L
Physics 201: Lecture 22, Pg 16
To the bottom of the ocean
What is the density change at the bottom of the ocean for liquid mercury?
Facts: 1 atm = 1.013 x 105 N/m2
So 1 atm increase for every 10 m.
Thus, 10 km is 10000 m or 1000 atm & B = 3 x 1010 N/m2 for Hg
iΔV/VF/AB
BF/A
VΔV
i
2
210
28
1033.0N/m 103
N/m 10
iVΔV
Physics 201: Lecture 22, Pg 17
The ice bomb
How does this compare to the case when water freezes? On freezing the fractional change in volume is about 0.08.
B = 0.2 x 1010 N/m2 for ice (close to that of water)
2
210108
N/m 102.0
P
V
ΔV
i
28 N/m 106.1 P
atm 106.1 3P
Physics 201: Lecture 22, Pg 18
For Tuesday
Review