0

00

210 0 2

2 20 0

02

2

v v a t a const

v vv t

x x v t a t

v v a x x

Newton’s First Law of Motion

Every body continues in its state of rest or uniformspeed in a straight line unless acted on by a nonzeronet force.

Preceding defines INERTIA

Preceding defines MASS (m W)

Best when observed in the absence of friction.

Newton’s Second Law of Motion

The acceleration of an object is directly proportionalto the net force acting on it and is inverselyproportional to its mass. The direction of theacceleration is in the direction of the net forceacting on the object.

Fa =

pF = a ( F = )

m

dm

dt

Newton’s Third Law of Motion

Whenever one object exerts a force on a secondobject, the second exerts an equal and oppositeforce on the first.

The Bane of Galileo: Friction

Friction is everywhere! There is little wonderwhy it played such a prominent part inMECHANICS for the ancients (e.g., Aristotle).

Galileo recognized friction as separate frommotion, so the equations of kinematics could bediscovered (using geometry).

Friction - treated as a force (though not a vector) - always opposite to the motion - sometimes related to the motion, sometimes not so related

Dynamics of Circular Motion

Galileo sez: Circular motion is not natural; straight line motion is natural

What is required for an object to move in a circle?

2

2

2

( constant, constant,

changing, changing)

angular velocity

R R

R

R

R R

vF ma m

r

va v a

rv a

v

r

F m r

What happens to an object when thecentripetal force quits?

A 0.150-kg ball on the end of a 1.10-m cord is swingin a vertical circle. What is the minimum speed thatthe ball must have to continue moving in a circle?

2

29.80 m/s 1.10 m

3.28 m/s

T R

TB RB

TB RB

F W F

vmg m

r

v gr

F W F

F W F

Banked Curves - Rotations and Inclines

A 1000-kg car rounds a curve on a flat road of radius50 m at a speed of 50 kph (14 m/s). Will the car makethe turn if the pavement is icy and s = 0.25?

2

2

2

14 m/s1000 kg 3920 N

50 m

0.24 1000 kg 9.80 m/s

2450 N

R

fr s N

s

vF m

r

F F

m g

Ffr FR the car will slide

An airplane traveling at 520 kph attempts to turn around.By banking at an angle of 38, how long will thismaneuver take?

Need to find radius of curve,then time = radius / speed

0

cos 0

cos

y

N

N

F W

F mg

mgF

2

2

0

sin 0

sin

x R

N

N

F F

vF m

r

vF m

r

2

2

2

2

2

sincos

tan

tan

144 m/s

9.80 m/s tan 382700 m

mg vm

r

vg

r

vr

g

12

2700 m

144 m/s59 s

circumferencet

vr

v

Nonuniform Circular Motion

(polar coordinates)

Easiest to describe in termsof the circle.

tan

2

2 2

( )

R

R

R

dva a

dt

va

r

a a a

a a a

Terminal Velocity

For velocity-dependent friction force (e.g., airresistance), the frictional force can equal themotive force.

2for example, consider

0 terminal velocity

D

D

F W F m a

F v

a v

Newton’s Law of UniversalGravitation

In 1687, very controversial law of mechanics

action at a distance - no direct contact (first of the force fields)

universal - applies to both terrestrial and celestial motion

verification required experiments, calculus, and history

Law of Universal Gravitation

Every particle in the universe attracts every otherparticle with a force that is proportional to the productof their masses and inversely proportional to the squareof the distance between them. This force acts along theline joining the two particles.

2

11

2

N m6.67 10

kg

1 22

G

m mF G

r

What is the force of attraction between a 50-kgperson and a 75-kg person sitting 50 cm apart?

1 22

2 22

E

E

50 kg 75 kg6.67 11 N m /kg

(0.50 m)

1.0 6 N

m mF G

r

1 212 122

12

12 21

m mˆF G r

r

ˆ ˆr r

1 12 13 14

11

F F F F

F ii

Weighing the Earth

G 2

2

2

2 2

2 2

E

E

E

F

(1798)

9.80 m/s (6.38 6 m)

6.67 11 N m /s5.98 24 kg

E

E

E

M mG mg

r

Mg G

r

g rM

G

What is g on the top of Mt Everest, 8848 m high?

22

2 22

2

EE

5.98 24 kg6.67 11 N m /kg

6380 8.8 km

9.77 m/s ( -0.3%)

E

M Mg G G

r r h

What is g on the moon?mM = 7.35E22 kg rm = 1.74E6 m

2

2 22

2

EE

E

7.35 22 kg6.67 11 N m /kg

1.74 6 m

1.01.62 m/s

6.05

Mg G

r

What is g in the Shuttle?h = 240 km

22

2 22

2

EE

5.98 24 kg6.67 11 N m /kg

6380 + 240 km

9.1 m/s -7%

E

E

MMg G G

r r h

What is the effect of the Earth’s rotationon the value of g?

2

2 2

F

(464 m/s)

6380 km

0.0337 m/s

R

E

E

W ma

vg g

r

vg

r

Why is the Space Shuttle weightless?

2

2

0

2T

R

E

W F

M m vG m

r r

G M G Mv

r r h

r

v

Kepler’s Laws and Gravitation

Kepler found three laws ofplanetary motion in Brahe’s data.He was looking for celestial music.

1) Planetary orbits are ellipses

2) Equal areas in equal time

3) T2 s3

Kepler’s Third Law from Newton’s Laws

2

2

22 2

22 3 2 3

consider circular orbits and Newton's laws

0

20

4

4

G RF F

M m mv rG v

r r T

M m m rG

r T

T r T rG M

Gravitational Field

Force fields are simpler than contact forces.Project action at a distance.Interaction only for pairs of objects.Force depends on the “charge” of an object.

2ˆ

F G Mg r

m r

Forces in nature :

gravitation - masselectromagnetic - electric chargeweak nuclear - electric chargestrong nuclear - baryon

Electric and magnetic forces combined by Faraday and Maxwell around 1840.

Electromagnetic and weak nuclear combined in 1967.

Who is next?

Principle of Equivalence

2

2

0G R

G R

F F

M m m vG

r r

Conceptually, the gravitational mass and the inertialmass are different. But, numerically, they are equalto high precision. What about the laws of mechanics?