1. General Physics I Solids & Fluids

2. States of Matter Solids Liquids Gasses Have definite volume
Have definite shape
Molecules are held in specific locations
by electrical forces
vibrate about equilibrium positions
can be modeled as springs connecting molecules
Has a definite volume
No definite shape
Exist at a higher temperature than solids
The molecules “wander” through the liquid in a random fashion
The intermolecular forces are not strong enough to keep the molecules in a fixed position
Has no definite volume
Has no definite shape
Molecules are in constant random motion
The molecules exert only weak forces on each other
Average distance between molecules is large compared to the size of the molecules

3. Solids are not infinitely rigid, solids will always deform if a force is applied
All objects are deformable, i.e. It is possible to change the shape or size (or both) of an object through the application of external forces
Sometimes when the forces are removed, the object tends to its original shape, called elastic behavior
Large enough forces will
break the bonds between
molecules and also the
object

4. Elastic Properties
Stress is related to the force causing the deformation
Strain is a measure of the degree of deformation
The elastic modulus is the constant of proportionality between stress and strain
For sufficiently small stresses, the stress is directly proportional to the strain
The constant of proportionality depends on the material being deformed and the nature of the deformation
The elastic modulus can be thought of as the stiffness of the material

5. Young’s Modulus: Elasticity in Length
Tensile stress is the ratio of the external force to the cross-sectional area
For both tension and compression
The elastic modulus is called Young’s modulus
SI units of stress are Pascals, Pa
1 Pa = 1 N/m2
The tensile strain is the ratio of the change in length to the original length
Strain is dimensionless

6. Shear Modulus: Elasticity of Shape
Forces may be parallel to one of the objects faces
The stress is called a shear stress
The shear strain is the ratio of the horizontal displacement and the height of the object
The shear modulus is S
A material having a large shear modulus is difficult to bend

7. Bulk Modulus: Volume Elasticity
Bulk modulus characterizes the response of an object to uniform squeezing
Suppose the forces are perpendicular to, and acts on, all the surfaces -- as when an object is immersed in a fluid
The object undergoes a change in volume without a change in shape
Volume stress, DP, is the ratio of the force to the surface area
This is also the Pressure
The volume strain is equal to the ratio of the change in volume to the original volume

8. Ultimate Strength of Materials
Notes on Moduli
Solids have Young’s, Bulk, and Shear moduli
Liquids (& gasses) have only bulk moduli, they will not undergo a shearing or tensile stress
The negative sign is included since an increase in pressure will produce a decrease in volume: B is always positive
Ultimate Strength of Materials
The ultimate strength of a material is the maximum stress the material can withstand before it breaks or factures
Some materials are stronger in compression than in tension
Linear to the Elastic Limit

9. Fluids (Liquids and Gasses)
What do we mean by “fluids”?
Fluids are “substances that flow”…. “substances that take the shape of the container”
Atoms and molecules are free to move .. No long range correlation between positions.
What parameters do we use to describe fluids?
Density
Pressure V D = m r A D = F P

10. Density & Pressure are related by the Bulk Modulus, B
LIQUID: incompressible (density almost constant)
GAS: compressible (density depends a lot on pressure) ) / V ( p D - =








Bulk modulus (Pa=N/m2)
H2O
Steel
Gas (STP) Pb

11. Pressure vs. Depth Incompressible Fluids (liquids)
Due to gravity, the pressure depends on depth in a fluid
Consider an imaginary fluid volume (a cube, each face having area A)
The sum of all the forces on this volume must be ZERO as it is in equilibrium.
There are three vertical forces:
The weight (mg)
The upward force from the pressure on the bottom surface (F2)
The downward force from the pressure on the top surface (F1) ) ( 1 2 y g p - + = r

12. Archimedes’ Principle
W2? W1
Suppose we weigh an object in air and in water.
Since the pressure at the bottom of the object is greater than that at the top of the object, the water exerts a net upward force, the buoyant force, on the object.
The buoyant force is equal to the difference in the pressures times the area. F = ( p - p ) × A = r
g(y - y )A B 2 1 2 1
liquid M gV W g F B = × r
Therefore, the buoyant force is equal to the weight of the fluid displaced.

13. Sink or Float?
The buoyant force is equal to the weight of the liquid that is displaced.
If the buoyant force is larger than the weight of the object, it will float; otherwise it will sink. F mg B y
We can calculate how much of a floating object will be submerged in the liquid:
Object is in equilibrium mg F B =

14. Sink or Float? Object is in equilibrium: F mg y mg F = V g × = r V r =
displ.
liquid V g × = r
liquid
object
displ. V r =
The Tip of The Iceberg: What fraction of an iceberg is submerged? % 90
1024
917 V
water
ice
displ. = 3
kg/m r

15. How much can my balloon lift?
Sum of the forces!!!
Start with Free-body diagram
Assume balloon in in the air He W?
Air

16. Sink More, Float More, or Something Else??
A lead weight is fastened to a large styrofoam block and the combination floats on water with the water level with the top of the styrofoam block as shown.
If you turn the styrofoam+Pb upside down, what happens?
styrofoam Pb B)
styrofoam Pb C)
styrofoam Pb
A) It sinks

17. More Fun With Buoyancy
Two cups are filled to the same level with water. One of the two cups has plastic balls floating in it. Which cup weighs more?
Cup I
Cup II
a) Cup I b) Cup II c) The Same

18. Still More Fun!
A plastic ball floats in a cup of water with half of its volume submerged. Oil (roil < rball <rwater) is slowly added to the container until it just covers the ball.
Relative to the water level, the ball moves _______.
water
oil
a) Up b) Down c) Does not move

19. Pascal’s Principle So far we have discovered (using Newton’s Laws):
Pressure depends on depth: Pdepth = rgDy
Also, P = Po + ρgΔy; where Po = atmospheric pressure = 1.013×105 N/m2
Since pressure depends on depth, an object in a liquid experiences an upward buoyant force: FB = Wliquid displaced
Pascal’s Principle addresses how a change in pressure is transmitted through a fluid.

20. Pascal’s Principle:
Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel.
Pascal’s Principle is most often applied to incompressible fluids (liquids):
Increasing P at any depth (including the surface) gives the same increase in P at any other depth
The change in pressure vs. depth depends only on g and the mass density r.

21. Pascal’s Principle (2) Consider the system shown:
A downward force F1 is applied to the piston of area A1.
This force is transmitted through the liquid to create an upward force F2.
Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid. 2 1 A F = 1 2 A F =
Check that F•d is the same on both sides. Energy is conserved!

22. Fluid Flow Fluid flow without friction
Volume flow rate: V/t = A d/t = Av (m3/s)
Continuity: A1 v1 = A2 v2
i.e., flow rate the same everywhere
e.g., flow of river

23. Bernoulli’s Equation
The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy.

24. Bernoulli’s Equation (Elevated Pipe)
As a fluid moves through a region where its speed and/or elevation above the Earth’s surface changes, the pressure in the fluid varies with these changes.
P1 + 1/2 rv12 + mgy1 = P2 + 1/2 rv22 + mgy2

25. Problem
A large bucket full of water has two equal diameter drains. The water level in the bucket is kept constant by constantly refilling it. One is a hole in the side of the bucket at the bottom, and the other is a pipe coming out of the bucket near the top, which is bent downward such that the bottom of this pipe even with the other hole, like in the picture below: Though which drain is the water spraying out with the highest speed? 1. The hole 2. The pipe 3. Same

26. End of Solids & Fluids Lecture