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