Gravity and Circular Motion

Centripetal Acceleration

• We can have acceleration with constant

speed

• If a=v/ t we find the direction of a is

towards the center of the circle

0vvv

Magnitude of ac

• We can also express this in terms of period

of the rotation since v=d/t

• v=d/t=2r/T

• So we also have

r

vac

2

2

24

T

rac

Ex 1: find the acceleration

• A 2kg mass is swung at the end of a

0.5 m rope with a period of rotation of

0.75s. Find a

2

24

T

rac

22

75.0

5.04

s

mac

235

smac

Ex 2: find force

• The Earth travels in its orbit at a speed

of about 30 km/s. What force is

necessary to keep it in its orbit?

r

vac

2

m

smkg

F11

224

105.1

300001098.5

NF 22106.3

r

mvmaF c

2

Exercises

• Start P. 108-9 #1-15

• Prep Lab 4-1 p. 110-113

• Start Corrections

Ex 1: Horizontal UCM • Find the acceleration (as a factor of g) for

Shrek, while going in a circle of radius 2.5m

if the period of rotation is 1.2s?

2

24

T

rac

2

2

2.1

5.24

s

mac

268s

mac gac 0.7

Ex 1: Horizontal UCM • Find the acceleration (as a factor of g) for the

Starship 2000, while going in a circle of

radius 4.0m if the period of rotation is 2.9s?

2

24

T

rac

2

2

4 4.0

2.9c

ma

s

228s

mac gac 3

Ex 2: Horizontal UCM

• How fast can a car go around a curve

with a radius of curvature of 123m if

the tires’ coefficient of friction is 0.15?

ms

mv 1238.915.0 2

maFnet

smv 13

r

mvFF Nf

2

r

mvmg

2

grv

Ex 3: More Horizontal Loop

• What is the radius for a one second loop if

there are 3 washers orbiting and nine

dangling?

• Forces on the 3 washers:

0 yF

0 Tyg FF

3

1sin 5.19

0sin93 mgmg

• x-Forces on the 3 washers:

maFx

cTx maF

2

243cos9

T

rmmg

2

2

4

cos3

Tgr

mr 70.0

cos34

2

2

gT

r

Exercises

• P. 119 #1-3

• Lab 2-1

• Last day of Term 3: all late work in, esp. Ch 3 corrections, “Test Yourself” for reTest

Orbital Motion

What if there was no Gravity?

Newton’s

Gravitation

• Newton’s most

original contribution!

• All objects in the

universe exert a

gravitational pull on

each other

• F depends on:

– d

– m

Gravitational Definitions

• This is where most people mix up problems

• Mass, Force or Field?

– 25N

– Weight

– 5kg

– 9.8N/kg

– Apparent Weight

– Gravitational acceleration

– 25lb!?

– 1.63m/s2

Universal gravitation:

– Where G is the universal gravitational constant

The problem in determining “G” is in measuring

the feeble force between two ordinary (and

“weighable”) objects

Ex 1: Find the force of attraction between Kelsey

(50 kg) and Alex (80 kg) if they sit 2 m apart.

2r

GMmFg

2r

GMmFg

2

2

211

)2(

50801067.6

m

kgkgkg

Nm

Fg

NFg

8107.6

(:directly measure tosmall Too

Cavendish’s “Weigh the

Earth” Experiment

• If we can get a measurable force for two known

masses, we can find G

• Once G is known, we can “weigh” anything!

• Find G if Cavendish calculated the force between

the 2.5kg mass and the 0.55kg mass (when their

centers are 10cm apart) to be:

NFg

9102.9

2r

GMmFg

Mm

rFG

g

2

kgkg

mN

55.05.2

1.0102.929

2

2111069.6

kgNmG

• Ex 2: find the gravitational field strength

“g” at an altitude of 130 km. Careful!

2r

GMg

256

242

211

)103.11038.6(

1098.51067.6

mm

kgkg

Nm

g

kgNg 41.9

mgFg

Ex. 3: find g on the moon.

2r

GMg

26

222

211

)1074.1(

1035.71067.6

m

kgkg

Nm

g

kgNg 62.1

Ex 4: find mass

• Use the Earth’s orbital velocity to find

the mass of the Sun

2

211

112

4

1067.6

105.1100.3

kgNm

ms

m

M

2

2

r

GMm

r

mv

G

rvM

2

kgM 30100.2

maF

Ex 5: find speed

• How fast would you need to orbit a 10

Solar mass black hole at a distance of

100km?

m

kgkg

Nm

v5

312

211

1000.1

1098.11067.6

2

2

r

GMm

r

mv

smv 81015.1

r

GMv

Exercises

• P. 121 #1-11

• Hand in lab Tuesday

Ex 1: Vertical Loop

• What is Bryden’s apparent weight in a 7.5

m/s rollercoaster at the bottom of a vertical

loop with a radius of curvature of 5.5m?

m

smkg

kgNkgFN

5.5

5.7808.980

2

maFnet

NFN 1600

r

mvFF gN

2

r

mvmgFN

2

Ex 2: Top of a Vertical Loop

• What is the minimum speed for the

rollercoaster at the top of the same vertical

loop to maintain contact with the rails?

• Will he make it?

maFnet

r

mvFg

2

mgr

mv

2

rgv

28.95.5 smmv smv 3.7

Ex 3: More Vertical Loop

• What is the minimum height for the

rollercoaster to maintain contact with the

rails at the top of the loop?

0E kfpfpi EEE

2

2

12 mvrmgmgh

g

rgrh

22

rh

2

5 mh 14

Ex 4: More Vertical Loop

• What is the maximum speed for the

motorcycle to maintain contact with the road

at the top of the crest (radius of curv. 35m)?

maFnet

r

mvFg

2

mgr

mv

2

rgv

28.935 smmv smv 5.18

Exercises

• P. 124 #1-6

Energy

Gravitational Potential Energy

• Switching from relative potential energy -mgh

• Absolute Grav Ep=-GMm/r

• This is negative since gravity is always an

attractive force. We have to put energy into the

system to separate the masses.

• To find the energy needed to lift a mass off the

surface of a planet, simply subtract the potential

energy before and after

Ex 1: How much

potential energy?

• What is the gravitational energy of a 55kg “little prince” standing on the surface of a 100m wide, 3x109kg asteroid?

r

GMmEp

m

kgkgkg

mN

Ep50

55100.31067.6 92

211

JEp 22.0

Ex 2: E

• How much energy is required

to lift off a 2900 Ton Saturn V

rocket 100 km straight off the

surface of the planet?

r

GMmEp

0

11

rrGMmEp

66 1038.6

1

1048.6

1GMmEp

TJEp 8.2

Ex 4: Escape velocity

• Find the escape velocity for:

– the Earth

– the Moon

– a 10 km diameter comet

• This is the minimum velocity for which

kinetic energy just balances potential, or:

0 pkT EEE

pk EE r

GMmmv 2

2

1

r

GMv

2

• The Earth:

r

GMv

2

skmv 1.11

• The Moon:

r

GMv

2

skmv 4.2

• The comet: mass?

3

3

4rVM

r

GMv

2

33 50003

41000 m

mkg

M

kgM 141024.5

m

kgGv

5000

1024.52 14

smv 7.3

• The comet continued:

could you jump clear

off the surface?

smvesc 7.3

2r

GMg

214

5000

1024.5

m

kgGg

20014.0 smg

maFnet

m

mgmg

m

FFa EcNg 2

Ec gga 2

26.19s

ma

advv 22

0

2

ms

madv 5.06.1922 2

smv 4.4

• The lesson? Be careful when jumping

around on a comet!

Exercises

• P. 127 #1-3

Orbital Energy

Orbital Energy

• If we want to put a satellite into orbit, we

need to give it a tangential velocity to

prevent it from falling back to the surface.

• Orbital energy is then the sum of an objects

potential and kinetic energy.

Eo=Ep+Ek

=-GMm/r+½mv2

• But using Newton’s second law we can

rewrite:

• Since v2=GM/r we get

Ek=½m(GM/r)=½GMm/r=-½Ep

Eo=Ep+ -½Ep=½Ep

So Eo=½Ep To find how much energy is required, we can

subtract initial Ep from orbital energy.

Ex 2: How much

energy?

• How much energy is needed to put a 13 Ton

ISS component into orbit at an altitude of

400km?

pipfpio EEEEE 2

1

if r

GMm

r

GMmE

2

if rrGMmE

1

2

1

mmGMmE

66 1038.6

1

1078.62

1

GJE 430

Ex 3: Geosynchronous

Orbit • How much energy is needed to put a 1.24

Ton communications satellite into

geosynchronous orbit?

• First find r

gac

22

24

r

GM

T

r

2324 GMTr

32

2

4

GMTr

3

2

2

4

86400

sGMr

kmr 250,42

pipfpio EEEEE 2

1

if r

GMm

r

GMmE

2

mmkgkg

kgNmE

67

242

211

1038.6

1

1022.42

112401098.51067.6

if rrGMmE

1

2

1

GJE 72

Combo Question

• What is the tension in the cable

for an 85kg load?

• The “Orange Stinger” revolves

with a period of 7.0s at a radius

of 7.5m, and an angle of 70°

maF

2

2

cos

4

T

rmFT

cTx maF

θ NFT 1500

Combo Question

• What is the tension in the cable

for an 85kg load?

• The “Orange Stinger” now

revolves with a period of 4.0s at

a radius of 8.1m, and an angle of

55°

maF

2

2

cos

4

T

rmFT

cTx maF θ

NFT 2960

Review

• Chapter Review Questions p. 127-130 1-27

(Bonus 28-32)

• Test Yourself p. 131-133 1-12