Chapter 12Chapter 12 Equilibrium and ElasticityEquilibrium and Elasticity

Equilibrium and ElasticityEquilibrium and Elasticity

I.I. EquilibriumEquilibrium- Definition- Definition- Requirements- Requirements- Static equilibrium- Static equilibrium

II. Center of gravityII. Center of gravity

III. ElasticityIII. Elasticity - Tension and compression- Tension and compression - Shearing- Shearing - Hydraulic stress- Hydraulic stress

I. EquilibriumI. Equilibrium

- Definition:- Definition:

0,0 LP

Example:Example: block resting on a table, hockey puck sliding block resting on a table, hockey puck sliding across a frictionless surface with constant velocity, theacross a frictionless surface with constant velocity, therotating blades of a ceiling fan, the wheel of a bike traveling rotating blades of a ceiling fan, the wheel of a bike traveling across a straight path at constant speed. across a straight path at constant speed.

An object is in equilibrium if:An object is in equilibrium if:

- The linear momentum of its center of mass is constant.The linear momentum of its center of mass is constant.

- Its angular momentum about its center of mass is Its angular momentum about its center of mass is constantconstant

- Static equilibrium:- Static equilibrium:Objects that are not moving Objects that are not moving either in TRANSLATION or either in TRANSLATION or ROTATIONROTATION

Example:Example: block resting on a table.block resting on a table.

.

Stable static equilibrium:Stable static equilibrium:If a body returns to a state of static equilibrium after having If a body returns to a state of static equilibrium after having been displaced from it by a force been displaced from it by a force marble at the bottom of a marble at the bottom of a spherical bowl. spherical bowl. Unstable static equilibrium:Unstable static equilibrium:A small force can displace the body and end the equilibriumA small force can displace the body and end the equilibrium

(1) Torque about supporting (1) Torque about supporting edge by edge by FFgg=0=0 because line of because line of

action of action of FFgg through rotation axis through rotation axis

domino in equilibrium. domino in equilibrium.(1) Slight force ends equilibrium (1) Slight force ends equilibrium line of action of line of action of FFgg moves to moves to

one side of supporting edge one side of supporting edge torque due to torque due to FFgg increases increases

domino rotation.domino rotation.(3) Not as unstable as (1) to topple it one needs to rotate it through beyond balance position in (1).

.

-- Requirements of equilibrium:Requirements of equilibrium:

0dt

PdFcteP net

0,0 dt

LdL net

Balance of forces Balance of forces translational translational equilibriumequilibrium

Balance of torques Balance of torques rotational rotational equilibriumequilibrium

- Vector sum of all external torques that act on the body, Vector sum of all external torques that act on the body, measured about any possible point must be zero.measured about any possible point must be zero.

- Vector sum of all external forces that act on body must be Vector sum of all external forces that act on body must be zero.zero.

- Equilibrium:Equilibrium:

Balance of forcesBalance of forces FFnet,x net,x = F= Fnet,ynet,y = F = Fnet,znet,z =0 =0

Balance of torquesBalance of torques ττnet,x net,x = = ττnet,ynet,y = = ττnet,znet,z =0 =0

II. Center of II. Center of gravitygravity

cogcog = Body’s point where the gravitational force “effectively” acts.= Body’s point where the gravitational force “effectively” acts.

Gravitational force on extended bodyGravitational force on extended body vector sum of the vector sum of the gravitational forces acting on the individual body’s elements gravitational forces acting on the individual body’s elements (atoms) .(atoms) .

- This course initial assumption:This course initial assumption: The center of gravity is at The center of gravity is at the center of mass.the center of mass.

If If gg is the same for all elements of a body, then the is the same for all elements of a body, then the body’s Center Of Gravity (body’s Center Of Gravity (COGCOG) is coincident with the ) is coincident with the body’s Center Of Mass (body’s Center Of Mass (COMCOM).).

Assumption valid for every day objects Assumption valid for every day objects “ “gg” varies only slightly ” varies only slightly along Earth’s surface and decreases in magnitude slightly with along Earth’s surface and decreases in magnitude slightly with altitude.altitude.

neti

gicoggcog FxFx

Each forceEach force FFgigi produces a produces a

torquetorque ττii on the element of on the element of

mass about the origin O, mass about the origin O, with moment armwith moment arm xxii..

i i

giiinetgiii FxFxFr

icomiicog

iii

iicog

iiii

iiicog

igii

igicog

xmxM

x

mxmxgmxgmxFxFx

1

A baseball player holds a A baseball player holds a 36-oz36-oz bat (weight = bat (weight = 10.0 N10.0 N) with one ) with one hand at the point hand at the point OO . The bat is in equilibrium. The weight of the . The bat is in equilibrium. The weight of the bat acts along a line 60.0 cm to the right of bat acts along a line 60.0 cm to the right of OO. Determine the . Determine the force and the torque exerted by the player on the bat around an force and the torque exerted by the player on the bat around an axis through axis through OO. .

A uniform beam of massA uniform beam of mass mmbb and lengthand length ℓℓ supports blocks supports blocks

with masseswith masses mm11 and and mm22 at two positions. The beam rests at two positions. The beam rests

on two knife edges. For what value ofon two knife edges. For what value of x x will the beam be will the beam be balanced atbalanced at PP such that the normal force atsuch that the normal force at OO is zero?is zero?

A circular pizza of radius A circular pizza of radius RR has a circular piece of radius has a circular piece of radius RR/2 /2 removed from one side as shown in Figure. The center of removed from one side as shown in Figure. The center of gravity has moved from gravity has moved from CC to to CC’’ along the along the xx axis. Show that axis. Show that the distance from the distance from CC to to CC’’ is is RR/6/6. Assume the thickness and . Assume the thickness and density of the pizza are uniform throughout.density of the pizza are uniform throughout.

Pat builds a track for his model car out of wood, as in Pat builds a track for his model car out of wood, as in Figure. The track is Figure. The track is 5.00 cm5.00 cm wide, wide, 1.00 m1.00 m high and high and 3.00 m3.00 m long and is solid. The runway is cut such that it forms a long and is solid. The runway is cut such that it forms a parabola with the equation parabola with the equation

Locate the horizontal coordinate of the center of gravity of Locate the horizontal coordinate of the center of gravity of this track.this track.

9

)3( 2x

y

Find the mass Find the mass mm of the counterweight needed to balance of the counterweight needed to balance the the 1 500-kg1 500-kg truck on the incline shown in Figure. Assume truck on the incline shown in Figure. Assume all pulleys are frictionless and massless.all pulleys are frictionless and massless.

A A 20.0-kg20.0-kg floodlight in a park is supported at the end of a floodlight in a park is supported at the end of a horizontal beam of negligible mass that is hinged to a pole, as horizontal beam of negligible mass that is hinged to a pole, as shown in Figure. A cable at an angle of shown in Figure. A cable at an angle of 30.0°30.0° with the beam with the beam helps to support the light. Find (a) the tension in the cable and helps to support the light. Find (a) the tension in the cable and (b) the horizontal and vertical forces exerted on the beam by (b) the horizontal and vertical forces exerted on the beam by the pole.the pole.

States of MatterStates of Matter

• Matter is characterised as being solid, Matter is characterised as being solid, liquid or gasliquid or gas

• Solids can be thought of as crystalline Solids can be thought of as crystalline or amorphousor amorphous

• For a single substance it is normally the For a single substance it is normally the case thatcase that – solid state occurs at lower temperatures solid state occurs at lower temperatures

than liquid state andthan liquid state and– liquid state occurs at lower temperatures liquid state occurs at lower temperatures

than gaseous statethan gaseous state

Solids and Fluids

• Solids are what we have assumed all Solids are what we have assumed all objects are up to this point (rigid, objects are up to this point (rigid, compact, unchanging, simple shapes)compact, unchanging, simple shapes)

• We will now look at bulk properties of We will now look at bulk properties of particular solids and also at particular solids and also at fluidfluid

• Fluids include both liquids and gasesFluids include both liquids and gases– fluids assume the shape of their containerfluids assume the shape of their container

Deformation of Solids

• All states of matter (S,L,G) can be deformedAll states of matter (S,L,G) can be deformed

– it is possible to change the shape and volume of it is possible to change the shape and volume of solids and solids and

– the volumes of liquids and gasesthe volumes of liquids and gases

• Any external force will deform matterAny external force will deform matter– for solids the deformation is usually small in for solids the deformation is usually small in

relation to its overall size when ‘everyday’ relation to its overall size when ‘everyday’ forces are appliedforces are applied

Stress and StrainStress and Strain• When force is removed, the object will usually When force is removed, the object will usually

return to its original shape and sizereturn to its original shape and size– Matter is elasticMatter is elastic

• We characterise the elastic properties of solids in We characterise the elastic properties of solids in terms of stress (amount of force applied) and terms of stress (amount of force applied) and strain (extent of deformation) that occur.strain (extent of deformation) that occur.

• The amount of stress required to produce a The amount of stress required to produce a particular amount of strain is a constant for a particular amount of strain is a constant for a particular materialparticular material– this constant is called the elastic modulusthis constant is called the elastic modulus

strain

stressModulus Elastic

Young’s Modulus (length elasticity)Young’s Modulus (length elasticity)

• FF creates a tensile creates a tensile stressstress of F/A. of F/A. – Units of tensile stress are PascalUnits of tensile stress are Pascal– 1 Pa = 1 N/m1 Pa = 1 N/m22

• Change in length created Change in length created

by stress isby stress isDL/LDL/Loo (no unit) (no unit)

This quantity is the This quantity is the

tensile straintensile strain

Lo

AF

Young’s modulusYoung’s modulus

• Young’s modulus Young’s modulus is the ratio of tensile is the ratio of tensile stress to tensile strain.stress to tensile strain.

• Y Y has units of has units of PaPa

• Young’s modulus applies to rod or wire Young’s modulus applies to rod or wire under tension (stretching) or compressionunder tension (stretching) or compression

LA

FL

LL

AFY o

o

/

/

What is happening in detail?What is happening in detail?

• Bonds between atoms are compressed or Bonds between atoms are compressed or put in tensionput in tension

Elastic behaviourElastic behaviour

• Notice that the Stress is proportional to Notice that the Stress is proportional to the Strainthe Strain– This is similar to the relation we had This is similar to the relation we had

between spring force and its extensionbetween spring force and its extension

|F| = kx|F| = kx– We can identify We can identify kk withwith YA/LYA/Loo

F = kxF = kx soso F/A = kx/A=YDL/LF/A = kx/A=YDL/Loo

Limits of elastic behaviourLimits of elastic behaviour

Shear ModulusShear Modulus

• Shear modulus characterises a body’s Shear modulus characterises a body’s deformation under a sideways (tangential) deformation under a sideways (tangential) forceforce

hx

AFS

/

/

h

F

x

A

Bulk ModulusBulk Modulus

• Response of body to Response of body to uniformuniform squeezing squeezing

• Bulk modulus is ratio of the Bulk modulus is ratio of the changechange in in the normal force per unit area to the the normal force per unit area to the relative volume changerelative volume change

F

VV

PVV

AFB

/

/

/

Values for Y, S and BValues for Y, S and B

– Note that liquids do not have Note that liquids do not have YY defined. defined. SS for for liquids is called liquids is called viscosity. viscosity.

– In liquids and gasesIn liquids and gases SS and and BB are strongly are strongly dependent on temperature (more later)dependent on temperature (more later)

Y (N/mY (N/m22)) S (N/mS (N/m22)) B (N/mB (N/m22))

AlAl 7.0 x 107.0 x 101010 2.5 x 102.5 x 101010 7.0 x 107.0 x 101010

WaterWater -- -*-* 0.21 x 100.21 x 101010

TungstenTungsten 35 x 1035 x 101010 14 x 1014 x 101010 20 x 1020 x 101010

GlassGlass 7 x 107 x 101010 3 x 103 x 101010 5.2 x 105.2 x 101010

Example - Young’s modulusExample - Young’s modulus

• How much force must be applied to a 1 m long How much force must be applied to a 1 m long steel rod that has end area 1 cmsteel rod that has end area 1 cm22 to fit it inside a to fit it inside a 0.999 m long case? (0.999 m long case? (YYsteelsteel=20x10=20x101010N/mN/m22))

1 cm2

DensityDensity

• Density of a pure solid, liquid, or gas is its Density of a pure solid, liquid, or gas is its mass per unit volumemass per unit volume

• DensityDensity r r m/V m/V (units are kg/m(units are kg/m33))

PressurePressure

• Pressure is the force per unit areaPressure is the force per unit area

P P F/A F/A

Pressure at any point can be measured Pressure at any point can be measured by placing a plate of known area in by placing a plate of known area in (e.g.) a liquid and measuring the force (e.g.) a liquid and measuring the force (compression) on a spring attached to (compression) on a spring attached to it.it.