Chapter 11 – Rotational Mechanics Notes

 

I.            TORQUE

II.           Demos –

a.    Broom balancing act

                                              i.     Move mass and rebalance

                                            ii.     What is going on?

b.    Bird

c.    Wine Rack

III.         Remember NewtonŐs 2nd law? We only dealt with forces going up and down and right and left, nothing rotated. Now we can spin!

IV.         Push in CG of broomÉnotice it doesnŐt spin. All forces cancel

V.           Push on sideÉ.it falls. Why canŐt the normal force cancel my push? What could?

VI.         Torque = force x lever arm

a.    t = Fd

b.    units??  Nm (add to equation sheet and units table)

VII.      What happens to your center of gravity when you bend over? (Have student do it without wall and with wallÉwhat happens? Why?)

VIII.    If force is exerted away from CG then a torque occurs!

IX.         Check questions:

a.    Why is it easier to get a can of paint open with a long screw driver? (long lever arm)

b.    Why is it easier to do sit ups with your arms crossed over your chest then over your head? (long lever arm)

c.    Why does a pregnant woman get back pain? What could she do to fix this? (heavy weight behind her!)

d.    See saw. Draw.

 

 

 


Where should the smaller child sit to keep the see saw balanced relative to a large adult on the right? (triangle = fulcrum = pivot pointÉthis is where you measure d from)

e.    Make meter stick a plank with small weight. Why can it go past the table? Why does it rotate if you go past a certain point?

 

 

 

X.         ROTATIONAL INERTIA

XI.         Demos:

a.    2 1 meter pipes. One with plugs in center one with plugs at end. Have student spin. What happens?

b.    Rolling disks. Rolling cans. Which gets to the bottom first? Why?

c.    Gyroscopes

d.    Bike Wheel spinning on floor

XII.      For NewtonŐs 2nd Law, inertia = mass

XIII.    For rational inertia = mass x distance (from center), takes more force if further from center = moment of inertia

XIV.    Examples:

a.    spiraling football is more steady

b.    spiraling bullet is more steady

c.    spiraling top stays upright

XV.      LAW OF ROTATIONAL INERTIA: An object rotating about an axis tends to keep rotating in the absence of an external torque.

XVI.    What is easier to start spinning, a disk or a hoop? Why? For same mass, which would have a larger moment of inertia and spin longer? Which would require more energy spin? Which has more energy at the bottom and therefore is moving faster?

 

 

XVII.     CONSERVATION OF ANGULAR MOMENTUM

XVIII.        Demo

a.     student in spinning chair with masses in hands

b.    student in spinning chair with spinning bike wheel

XIX.    The spin direction, spin speed , and mass distance have to remain constant

a.    for masses, d is smaller so speed is faster when mass is pulled in

b.    for wheel, when direction is moved, rotation is transferred to person to keep it conserved = the same

XX.      CHECK QUESTIONS:

a.    If you place any weight on a balanced seesaw, it will topple unless the weight is placed directly above the fulcrum. Why will weight added above the fulcrum not upset the balance?

b.    A basketball player wishes to balance a ball on his fingertip. Will he be more successful with a spinning ball or a stationary ball? Why?

c.    Suppose you sit in the middle of a large freely-rotating turntable at an amusement park. If you crawl toward the outer rim, does the rotational speed increase, decrease, or remain unchanged? Why?

d.    Why is it incorrect to say that when you execute a somersault and pull your arms and legs inward, your angular momentum increases?

Chapter 12 Notes

 

I.              Ancient Astronomers

a.    Think about all the things the Greeks and other great men of science had to explain about the motions of the heavens:

                                              i.     sun rises in east, sets in west

                                            ii.     sunŐs path across sky changes as seasons change

                                          iii.     stars appear to move CCW around a point in the sky

                                          iv.     some star patterns remain the same through known time

                                            v.     star do not vary in altitude like the sun

                                          vi.     stars move faster than the sun

                                        vii.     some planets seem to move ŇbackwardsÓ across the sky occasionally

 

b.   PlatoŐs Problem: How can planetŐs motion be described so that they are moving in some combination of circles rather than one perfect circle?

                                              i.     DidnŐt know objects moved in straight line unless outside force

                                            ii.     DidnŐt know need gravity to make planet move in a circle

 

c.    Geocentric view: Earth is at the center and everything revolved around it

                                              i.     Common sense at the time

                                            ii.     Earth didnŐt FEEL like it moved

                                          iii.     It workedÉcould predict everything BUT retrograde

                                          iv.     Used ŇepicyclesÓ to explain retrograde

                                            v.     As more observation didnŐt quite fit, model became more complicated

d.   Heliocentric view: Sun is at the center and Earth revolves around it

                                              i.     SIMPLE!!

                                            ii.     Predictions worked better

                                          iii.     Needed to show parallax!! (as Earth moves, an object closer to it will appear in a different position relative to the starts further away) – demonstrate with object in room

                                          iv.     CouldnŐt be shown because distance were too far, and couldnŐt see small difference UNTIL Galileo (telescope)

 

e.    Aristarchus – first Greek to postulate Heliocentric (300 BC)

 

f.     Ptolemy (100Ős)

                                              i.     Geocentric view

                                            ii.     Earth not TRUE center

                                          iii.     Epicycles explained retrograde http://csep10.phys.utk.edu/astr161/lect/retrograde/aristotle.html

 

g.   Copernicus – Brought back idea of Heliocentric, not enough evidence to prove (1500Ős)

 

h.   Tycho Brahe (1550Ős)

                                              i.     Worked at observatory for England (on a deserted island)

                                            ii.     Made VERY accurate observation and measurements for 1000Ős of stars

                                          iii.     Looked for parallax but found none

                                          iv.     Like heliocentric so made compromise: everything EXCEPT the Earth revolves around the sun and then the sun revolved around the Earth

II.           Modern view –

a.     Sun is in the center but at one Focus

 

b.   Flat like a pancake (except Pluto É.probably more evidence why not a planet)

 

c.    Galileo – 1600Ős –

                                              i.      Created telescope to show parallax and prove Heliocentric, people thought telescope was magic and changed what they really saw, excommunicated

                                            ii.     Discovered-

1.   moon not smooth

2.   height of mountain on moon

3.   milky way = thousands of stars

4.   4 new planets

5.   many new stars

6.   sunspots

 

d.   Kepler

                                              i.     Young student that worked with Tycho

                                            ii.     Three laws:

1.   Law of elliptical Orbits: The planets moe in orbits which are ellipses and have the sun at one focus (thumb tack acvity)

2.   Law of Areas – The line from the sun to the moving planets sweeps over areas that are proportional to the time intervals

3.   Law of Periods – Equation that relates period and average orbital radius. closer to sun = moves faster

a.    k = constant = 1 yr2/AU3 or 3 x 10-19 s2/m3

 

 

III.         Newton – (1700s)

a.    Knew that bodies in skies must obey same laws as those on Earth

 

b.   Saw the moon going AROUND the Earth so thought something must be pulling it down = gravity!

 

c.    Reasoned, moon must have TANGENTIAL velocity (sideways) and then is just being pulled down so it matches the curve of the Earth like a projectile

 

d.   Law of Universal Gravitation

 


                                              i.     Fg = Gm1m2/r2  (book uses 3)

 

                                            ii.     G = 6.67 x 10-11 Nm2/kg2

                                          iii.     G is universal gravitational constant – same EVERYWHERE)

                                          iv.     r = distance between centers of masses

                                            v.     Fg = mg on Earth ONLY g = Gm/r)

 

e.    All masses attract all other masses

                                              i.     Why donŐt you feel it?

                                            ii.     Why do you feel the Earth?

 

f.     Known as INVERSE SQUARE LAW = further away, force drops quickly

                                              i.     Draw graph

                                            ii.     If move 4 times further = 1/16 (inverse of 4 squared!)

                                          iii.     If move 10 times further = 1/100

                                          iv.     Why does an apple fall down due to Fg but the Earth doesnŐt fall up?

                                            v.     What has more gravity, Jupiter or Earth? The Sun or Earth? The Earth or moon? Why??

IV.        Sample problem

a.    A satellite orbits the Earth with a distance of 3 x 107 m from the center of the Earth. If it requires 1000 N of centripetal force to keep it in orbit and the Earth has a mass 6 x 10 24 kg, what is the mass of the satellite?

                                              i.     G: R = 3 x 107 m, G = 6.67 x 10-11 Nm2/kg2, m1 = 6 x 1024 kg, Fg = 1000 N

                                            ii.     F: m2 =?

                                          iii.     R: Fg = Gm1m2/r2

                                          iv.     S: m2 = Fgr2/Gm1

                                            v.     C: m2 =         (1000N)(3x107)2              

(6.67x10-11Nm2/kg2)(6x 1024 kg)

 


m2 = 2249 kg

 

b.   How fast must a satellite orbit to stay at a distance of 2  x 108 m above the center of the earth?

                                              i.     G: r = 2 x 208 m, g = 10 m/s2

                                            ii.     F: v = ?

                                          iii.     R: Fc = Fg, mv2/r = mg

                                          iv.     S: v = sqrt(rg)

                                            v.     C: v = sqrt(2 x 108 m )(10 m/s2)  = 8.92 x 104 m/s