Physics 121

Newtonian Mechanics

Lecture notes are posted on

www.physics.byu.edu/faculty/chesnel/physics121.aspx

InstructorKarine Chesnel

April 2, 2009

Review 3

Mid-term exam 3

• Friday April 3 through Tuesday April 7

• At the testing center : 8 am – 9 pm

• Closed Book and closed Notes

• Only bring: - Math reference sheet - Pen / pencil- Calculator- your CID

• No time limit (typically 2 – 3 hours)

Midterm exam 3

Review: ch 9 – ch 13

Ch. 10 Rotation of solid• Moment of inertia• Rotational kinematics• Rolling motion• Torque

Ch. 12 Static equilibrium and elasticity• Rigid object in equilibrium• Elastic properties of solid

Ch. 11 Angular momentum• Angular momentum• Newton’s law for rotation• Isolated system• Precession motion

Ch. 9 Linear Momentum & collision• Center of mass• Linear momentum• Impulse• Collisions 1D and 2D

Ch. 13 Universal gravitation• Newton’s law of Universal gravitation• Gravitational Field & potential energy • Kepler’s laws and motion of planets

Linear Momentum & Impulse

Review 3 4/2/09

• Newton’s second law Fdt

pd Vmp

• The linear momentum of a particle is the product of its mass by its velocity

Units: kg.m/s or N.s

cstp

• For an isolated system 0

dt

pd

• The impulse is the integral of the net force, during an abrupt interaction in a short time

f

idtFI

Modeling of an impulse

t1 t2t

F

avgF

t

tFdtFIf

i avg .

Ip

• According to Newton’s 2nd law:

Collisions

iiff pppp ,2,1,2,1

1. Conservation of linear momentum

2,22

2,11

2,22

2,11 2

1

2

1

2

1

2

1iiff VmVmVmVm

2. Conservation of kinetic energy

(2)

Elastic collision

iiff VmVmVmVm ,22,11,22,11

(1)

V1,i

V2,i

V1,

f

V2,f

Review 3 4/2/09

Collisions 1D

if Vmm

mmV ,1

21

21,1

if V

mm

mV ,1

21

1,2

2

• If one of the objects is initially at rest:

iif Vmm

mV

mm

mmV ,2

21

2,1

21

21,1

2

• Combining (1) and (2), we get expression for final speeds:

iif Vmm

mV

mm

mmV ,1

21

1,2

21

12,2

2

V1,i

V1,f V2,f

V2,i

Collisions 2D

• 3 equations

• 4 unknow parametersV1,i

x

y V1,f

V2,f

Inelastic collision: the kinetic energy K is NOT conserved

Review 3 4/2/09

Center of Mass

m1

m2

m3

m4

m5

m6

iirm

M OC

O

C

y

x

z

Ensemble of particles

r1

r6

Ch.9 Momentum and collision 03/05/09

dmr

M OC

Solid objectC

O

dm

rP

Solid characteristics

M OC dmr

• The center of mass is defined as:

C

O

dm

rdVdm

Ctot VMp

FaM C

• The moment of inertia of the solid about one axis:

dmrI 2

2' MRII

Review 3 4/2/09

Rotational kinematics

dt

d

dt

d

• Solid’s rotation

Angular position

Angular speed

Angular acceleration

RVt

• Linear/angular relationship

Velocity

Acceleration

• Tangential

• Centripetal

Rat 2RaC

For any point in the solid

• Rotational kinetic energy2

2

1 IK

Review 3 4/2/09

Motion of rolling solid

P

C

R

Non- sliding situation

• The kinetic energy of the solid is given by the sum of the translational and rotational components:

Ksolid = Kc + Krot

22

2

1

2

1 IVMK Csolid

222

2

1

2

1 IMRK solid

22 )(2

1 IMRK solid

cstKU sol If all the forces are conservative:

Review 3 4/2/09

Torque & angular momentum

Fr

The torque is defined as

F

The angular momentum is defined as

dt

Ld

Deriving Newton’s second law in rotation

angular momentum Linear momentum

prL

sinFr

When a force is inducing the rotation of a solid about a specific axis:

For an object in pure rotation

Inet IL

Review 3 4/2/09

Precession

sin.mgR

Top view

LdL d

IL

is the projection of the angular momentum in the horizontal

plane

hL

The angular momentummoves along a cone

h

pLdt

d

The precession speed is

mg

Side view

L

If an object spinning at very high speed is experiencing a torque

in a direction different than its angular momentum L, then it will precess about a second axis

Review 3 4/2/09

Solving a problem

Static equilibrium

• Define the system

• Locate the center of mass (where gravity is applied)

• Identify and list all the forces

0

F• Apply the equality

• Choose a convenient point to calculate the torque (you may choose the point at which most

of the forces are applied, so their torque is zero)

• List all the torques applied on the same point.

0

• Apply the equality

Review 3 4/2/09

Example 3Beam and cable tension

0

F

0

• We do not know the force R that the hinge applies to the beam.P is a convenient point to calculate the torque

00

RR

gMCPW

2/MgDW

TQPT

sinTDT

sin2

MgT 0 WTR

RP

• Find the tension on the cable

T

Mg C Q

Ch.10 Rotation of Solids 3/24/09

Example 3Beam and cable tension

0

F

• Find the magnitude and direction for the force R

exerted by the wall on the beam

• in the horizontal direction

0cos TRx

tan2cos

MgTRx

• in the vertical direction

0sin TMgRy

2sin

MgTMgRy

tantan x

y

R

R 2tan12

Mg

R

RP

T

Mg C Q

Ch.10 Rotation of Solids 3/24/09

Gravitational laws

rMg ur

mMGrgmF

2

)(

Any object placed in that field experiences a gravitational force

Any material object is producing a gravitational field

rM ur

MGrg

2

)( M

r

ur

m

Fg

The gravitational field created by a spherical object is centripetal

(field line is directed toward the center)

The gravitational potential energy is

r

mMGU g

Ug

0 r

Review 3 4/2/09

Kepler’s Laws

“The orbit of each planet in the solar system is an ellipse with the Sun as one focus ”

First Law

0LcstL

“The line joining a planet to the sun sweeps out equal areas during equal time intervals as the planet travels along its orbit.”

Second Law

cstm

LdtdA

20

“The square or the orbital period of any planet is proportional to the cube of the semimajor axis of the orbit”

Third Law

32

2 4R

GMT

S

Review 3 4/2/09

Kepler’s laws

First law

0

• Physical observations (Brahe and Kepler, early 17th ) showed that orbits are elliptical

• This phenomenon could be demonstrated later (late 17th )using the Newton’s laws of motion

The motion of a body orbiting around another body under the only influence gravitational force must be in a plane

r

V

L0

L0

L0

Solar system

0LcstL

Review 3 4/2/09

0LcstL

cstVmr

0

.L

dt

drrm

cstm

L

dt

dA

20

Kepler’s laws

Second law

cstdt

rdrm

V

r

2/.drrdA

The area swept by the radius during the time interval dt is

“The line joining a planet to the sun sweeps out equal areas during equal time intervals as the planet travels along its orbit.”

Review 3 4/2/09

Case of circular orbit

2

2

r

mMG

r

Vmma S

c

r

GM

T

r S

22

32

2 4R

GMT

S

Kepler’s laws

Third law

Fg

2r

mMGma S

r

Applying Newton’s law of motionWith gravitational force

T

rrV

2Also

r

GMV S2

“The square or the orbital period of any planet is proportional to the cube of the semi-major axis of the orbit”

sS

KGM

24

The proportionality constant is

Solar system Ks =2.97 10-19 s2/m3

Review 3 4/2/09