Review - Nov. 30th 2004Review - Nov. 30th 2004

Chapters: 10, 11, 12, 13, 15, 16Chapters: 10, 11, 12, 13, 15, 16

Review of rotational variables (scalar Review of rotational variables (scalar notation)notation)

Angular position:Angular position: in radianss

r

Angular displacement:Angular displacement: 2 1

Average angular velocity:Average angular velocity: 2 1

2 1avg t t t

Instantaneous angular velocity:Instantaneous angular velocity:0

Limt

d

t dt

Average angular acceleration:Average angular acceleration: 2 1

2 1avg t t t

Instantaneous angular acceleration:Instantaneous angular acceleration:0

Limt

d

t dt

Relationships between linear and angular Relationships between linear and angular variablesvariables

Position:Position: in radianss r

Time period for rotationTime period for rotation::circumference 2 2

velocity

rT

v

Tangential acceleration:Tangential acceleration: ta r

Centripetal acceleration:Centripetal acceleration:2

2r

va r

r

dv r r

dt

Velocity:Velocity:

Kinetic energy of rotationKinetic energy of rotation2

i iI m r212K I

2 2I r dm r dV Therefore, for a continuous rigid object:Therefore, for a continuous rigid object:

h

ro ta tionaxis

c .o .m . ax is

I co m

•If moment of inertia is known If moment of inertia is known about an axis though the center of about an axis though the center of mass (mass (c.o.m.c.o.m.), then the moment of ), then the moment of inertia about any parallel axis is:inertia about any parallel axis is:

Parallel axis theoremParallel axis theorem

2comI I Mh

•It is essential that these axes are It is essential that these axes are parallel; as you can see from table parallel; as you can see from table 10-2, the moments of inertia can be 10-2, the moments of inertia can be different for different axes.different for different axes.

Some rotational inertiaSome rotational inertia

TorqueTorque

sin

sin

tr F rF

r F r F

•There are two ways to compute There are two ways to compute torque:torque:

•The direction of the force vector is The direction of the force vector is called the called the line of actionline of action, and , and rr is is called the called the moment armmoment arm..

•The first equation shows that the The first equation shows that the torque is equivalently given by the torque is equivalently given by the component of force tangential to the component of force tangential to the line joining the axis and the point line joining the axis and the point where the force acts.where the force acts.

•In this case, In this case, rr is the moment arm of is the moment arm of FFtt..

Summarizing relations for translational Summarizing relations for translational and rotational motionand rotational motion

•Note: work obtained by multiplying torque by an angle - Note: work obtained by multiplying torque by an angle - a dimensionless quantity. Thus, torque and work have the a dimensionless quantity. Thus, torque and work have the same dimensions, but you see that they are quite same dimensions, but you see that they are quite different.different.

Rolling motion as rotation and Rolling motion as rotation and translationtranslation

s RThe wheel moves with speed ds/dt

com

dv R

dt

The kinetic energy of rolling

2 212

2 2 21 12 2

2 21 12 2

P P com

com

com com r t

K I I I MR

K I MR

K I Mv K K

Torque and angular momentumTorque and angular momentum

definitionr F

•Here, Here, pp is the linear momentum is the linear momentum mv mv of the object.of the object.

sinl mvr

rp rmv

r p r mv

•SI unit is Kg.mSI unit is Kg.m22/s./s.

is defined as:l l r p m r v Angular momentum

•Torque was discussed in the previous chapter; cross Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this the end of this presentation; torque also discussed in this chapter (section 7).chapter (section 7).

Angular momentum of a rigid body about a Angular momentum of a rigid body about a fixed axisfixed axisWe are interested in the component of We are interested in the component of

angular momentum parallel to the axis of angular momentum parallel to the axis of rotation:rotation:

1 1

2

n n

z iz i i ii i

L l m v r vr dm

r r dm r dm I

In fact:In fact: L I

Conservation of angular momentumConservation of angular momentum

If the net external torque acting on a system is zero, If the net external torque acting on a system is zero, the angular momentum of the system remains the angular momentum of the system remains constant, no matter what changes take place within constant, no matter what changes take place within the system.the system.

It follows from Newton's second law that:It follows from Newton's second law that:

What happens to kinetic energy?What happens to kinetic energy?

2 22 21 1 1

2 2 22i i i i

f f f f i i if f f

I I IK I I I K

I I I

•Thus, if you increase Thus, if you increase by reducing by reducing II, you , you end up increasing end up increasing KK..

•Therefore, you must be doing some work.Therefore, you must be doing some work.

•This is a very unusual form of work that you This is a very unusual form of work that you do when you move mass radially in a do when you move mass radially in a rotating frame. rotating frame.

•The frame is accelerating, so Newton's laws The frame is accelerating, so Newton's laws do not hold in this frame do not hold in this frame

a constantL

i fL L

i i f fI I

f i

i f

I

I

EquilibriumEquilibriumA system of objects is said to be in equilibrium if:A system of objects is said to be in equilibrium if:

1.1. The linear momentum The linear momentum PP of its center of mass is of its center of mass is constant.constant.

2.2. Its angular momentum Its angular momentum LL about its center of mass, or about its center of mass, or about any other point, is also constant.about any other point, is also constant.If, in addition, If, in addition, LL and and PP are zero, the system is said are zero, the system is said

to be in to be in static equilibriumstatic equilibrium..

0net net

dPF F

dt

0net net

dL

dt

1.1. The vector sum of all the external forces that act The vector sum of all the external forces that act on a body must be zero.on a body must be zero.

2.2. The vector sum of all the external torques that act The vector sum of all the external torques that act on a body, on a body, measured about any axismeasured about any axis, must also , must also be zero.be zero.

The requirements of equilibriumThe requirements of equilibrium

1.1. The vector sum of all the external forces that act The vector sum of all the external forces that act on a body must be zero.on a body must be zero.

2.2. The vector sum of all the external torques that act The vector sum of all the external torques that act on a body, on a body, measured about any axismeasured about any axis, must also , must also be zero.be zero.

, ,

, ,

, ,

0 0

0 0

0 0

net x net x

net y net y

net z net z

F

F

F

Balance oftorques

Balance offorces

3.3. The linear momentum The linear momentum PP of the body must be zero. of the body must be zero.

One more requirement for static equilibrium:One more requirement for static equilibrium:

ElasticityElasticity

•A A stressstress, a force per unit area, produces a , a force per unit area, produces a strainstrain, or , or dimensionless unit deformation.dimensionless unit deformation.

•These various stresses and strains are related via a These various stresses and strains are related via a modulus modulus of elasticityof elasticity

Hydraulic stressTensile stress Shear stress

•All of these deformations have the following in common:All of these deformations have the following in common:

stress = modulus × strain

Tension and compressionTension and compression•The figure left shows a graph of The figure left shows a graph of stress versus strain for a steel stress versus strain for a steel specimen.specimen.

•Stress = force per unit area (Stress = force per unit area (FF//AA))

•Strain = extension (Strain = extension (LL) / length () / length (LL))

•For a substantial range of applied For a substantial range of applied stress, the stress-strain relation is stress, the stress-strain relation is linear.linear.

•Over this so-called Over this so-called elasticelastic region, region, the, the specimen recovers its original the, the specimen recovers its original dimensions when the stress is dimensions when the stress is removed.removed.

•In this region, we can write:In this region, we can write:F L

EA L

This equation, This equation, stress = stress = EE × strain × strain, is known as , is known as Hooke's lawHooke's law, , and the modulus and the modulus EE is called is called Young's modulusYoung's modulus. . The The dimensions of dimensions of EE are the same as stress, are the same as stress, i.e.i.e. force per unit area. force per unit area.

Vp B

V

•BB is called the is called the bulk modulusbulk modulus..

•VV is the volume of the specimen, and is the volume of the specimen, and VV its change in volume under a its change in volume under a hydrostatic pressure hydrostatic pressure pp..

Shear stressShear stress

F xG

A L

•GG is called the is called the shear modulusshear modulus..

Hydraulic Hydraulic stressstress

Newton's Newton's law of law of gravitationgravitation

11 2 21 22

6.67 10 N m /kgm m

F G Gr

A uniform spherical shell of matter attracts a A uniform spherical shell of matter attracts a particle that is outside the shell as if the shell's particle that is outside the shell as if the shell's mass were concentrated at its center.mass were concentrated at its center.

A uniform spherical shell of matter exerts no net A uniform spherical shell of matter exerts no net gravitational force on a particle located inside itgravitational force on a particle located inside it

Shell theoremsShell theorems

Gravitational potential energyGravitational potential energy

•But, close to the Earth's surface,But, close to the Earth's surface,

•Further away from Earth, we must choose a Further away from Earth, we must choose a reference point against which we measure reference point against which we measure potential energy.potential energy.

r1

1 22 1 1 2

1 1g

GMW GMm m r r

r r r r

21 2

GM GMg

rr r

. . gi e U W mgh

The natural place to chose as a reference point is The natural place to chose as a reference point is rr = = , since , since UU must be zero there, must be zero there, i.e.i.e. we set we set rr11 = = as our reference point. as our reference point.

1 2g

GmM GmM GmMU W

r r r

1.1. THE LAW OF ORBITS: All planets move in THE LAW OF ORBITS: All planets move in elliptical orbits, with the sun at one focus.elliptical orbits, with the sun at one focus.

Planets and satellites: Kepler's lawsPlanets and satellites: Kepler's laws

2.2. THE LAW OF AREAS: A line that connects a THE LAW OF AREAS: A line that connects a planet to the sun sweeps out equal areas in planet to the sun sweeps out equal areas in the plane of the planet's orbit in equal times; the plane of the planet's orbit in equal times; that is, the rate that is, the rate dAdA//dtdt at which it sweeps out at which it sweeps out area area AA is constant. is constant.

2 21 12 2

dA dr r

dt dt

3.3. THE LAW OF PERIODS: The square of the THE LAW OF PERIODS: The square of the period of any planet is proportional to the period of any planet is proportional to the cube of the semimajor axis of the orbit.cube of the semimajor axis of the orbit.

2 2

2 32

2 2T r

GM

Ene

rgy

Satellites: Orbits and EnergySatellites: Orbits and Energy

2

2

GMm vm

r r

•Again, we'll do the math for a Again, we'll do the math for a circular orbit, but it holds quite circular orbit, but it holds quite generally for all elliptical orbits.generally for all elliptical orbits.

•Applying Applying FF = = mama::

2

2

total

GMm GMmE K U

r r

GMmK

r

212 2 2

GMm UK mv

r

•Thus,Thus,

2

GMmE

a

Simple Harmonic MotionSimple Harmonic Motion•The simplest possible version of harmonic motion is called The simplest possible version of harmonic motion is called Simple Harmonic Motion (SHM)Simple Harmonic Motion (SHM)..

•This term implies that the periodic motion is a This term implies that the periodic motion is a sinusoidalsinusoidal function of time,function of time,

•The positive constant The positive constant xxmm is called the is called the amplitudeamplitude..

•The quantity The quantity ((tt + + )) is called the is called the phasephase of the motion. of the motion.

•The constant The constant is called the is called the phase constantphase constant or or phase anglephase angle..•The constant The constant is called the is called the angular frequency angular frequency of the motion.of the motion.

•TT is the period of the oscillations, and is the period of the oscillations, and ff is the frequency. is the frequency.

22 f

T

The velocity and acceleration of The velocity and acceleration of SHMSHM

•The positive quantity The positive quantity xxmm is called the velocity amplitude is called the velocity amplitude vvmm..

( )( ) cos

( ) sin

m

m

dx t dv t x t

dt dt

v t x t

Velocity:Velocity:

2

2

( )( ) sin

( ) cos

( ) ( )

m

m

dv t da t x t

dt dt

a t x t

a t x t

Acceleration:Acceleration:

In SHM, the acceleration is proportional to the In SHM, the acceleration is proportional to the displacement but opposite in sign; the two displacement but opposite in sign; the two quantities are related by the square of the quantities are related by the square of the angular frequencyangular frequency

The force law for SHMThe force law for SHM2 2( ) ( )F ma m x m x

•Note: SHM occurs in situations where the force is proportional Note: SHM occurs in situations where the force is proportional to the displacement, and the proportionality constant to the displacement, and the proportionality constant mm is is negative, negative, i.e.i.e.

F kx•This is very familiar - it is Hooke's law.This is very familiar - it is Hooke's law.

SHM is the motion executed by a particle of SHM is the motion executed by a particle of mass mass mm subjected to a force that is proportional subjected to a force that is proportional to the displacement of the particle but of to the displacement of the particle but of opposite sign.opposite sign.

2k m

Tm k

Mechanical energy:Mechanical energy: 212 mE U K kx

xxmm is the maximum displacement or amplitude is the maximum displacement or amplitude

Waves I - wavelength and Waves I - wavelength and frequencyfrequency

2k

kk is the is the angular wavenumberangular wavenumber..

Tra

nsv

ers

e w

ave

Tra

nsv

ers

e w

ave

2

T

is the is the angular frequencyangular frequency..

1

2f

T

v fk T

frequencyfrequency

velocityvelocity

Review - traveling waves on a Review - traveling waves on a stringstring

v

•The tension in the string is The tension in the string is ..

•The mass of the element The mass of the element dmdm is is dldl, where , where is the mass per is the mass per unit length of the string.unit length of the string.

VelocityVelocity

2 2 212 cos ( )kinetic m

dKP v y kx t

dt

2 2 212 cos ( )elastic m

dUP v y kx t

dt

Energy transfer ratesEnergy transfer rates

2 2 212

2 2 2 21 1 12 2 2

2 cos ( )

2

avg m

m m

P v y kx t

v y v y

( , ) sin( )my x t y kx t