Force & Mass

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Discussion

different forces on same object (result?)
different objects with same forces (result?)
different objects with same acceleration (how?)

Lex. II.   Law II.
Mutationem motus proportionalem eſſe vi motrici impreſſæ, & fieri ſecundum lineaum rectam qua vis illa imprimitur.   The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
     
Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, ſive ſimul & ſemel, ſive gradatim & ſucceſſive impreſſa fuerit. Et hic motus quoniam in eandem ſemper plagam cum vi generatrice determinatur, ſi corpus antea movebatur, motui ejus vel conſpiranti additur, vel contrario ſubducitur, vel obliquo oblique adjicitur, & cum eo ſecundum utriuſque determinationem componitur.   If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion being always directed the same way with the generating force, if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

Newton also defined what he called "the quantity of matter" "the quantity of motion". We now call them "mass" and "momentum", respectively.

Definitio. I.   Definition I.
Quantitas materiæ est mensura ejusdem orta ex illius densitate et magnitudine conjunctim.   The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
     
Aer densitate duplicata, in spatio etiam duplicato, sit quadruplus; in triplicato sextuplus. Idem intelige de nive & pulveribus per compressionem vel liquesactionem condensatis. Et par eft ratio corporum omnium, quæ per caufas quascunque diversimode condensantur. Medii interea, si quod fuerit, interstitia partium libere pervadentis, hic nullam rationem habeo. Hanc autem quantitatem sub nomine corporis vel masse in sequentibus passim intelligo. Innotescit ea per corporis cujusque pondus: Nam ponderi proportionalem esse reperi per experimenta pendulorum accuratissime instituta, uti posthac docebitur.   Thus air of double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any caused whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.
Definitio. II.   Definition II.
Quantitas motus est mensura ejusdem orta ex velocitate et quantite materiæ conjunctim.   The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly.
     
Motus totius est summa motuum in partibus singulis; ideoque in corpore duplo majore, æ quali cum velocitate, duplus est, & dupla cum velocitate quadruplus.   The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity it is quadruple.

(Newton, interpreted by Elert) Newton's second law of motion states that …

Newton's second law of motion is more compactly written as an equation that combines these relationships

∑ F = m a

Net force equals mass times acceleration. (note: acceleration means change in speed or direction.)

So what is mass?

  1. pick an object to be the standard unit mass
  2. push mass with reproducible force (or use the principle of action-reaction)
  3. measure its acceleration
  4. push an unknown mass with the same force
  5. measure new acceleration
  6. mass is inversely proportional to acceleration

Mass …

Mass of Selected Objects
mass (kg) object   mass (kg) object
~1053 observable universe   88 the author
1.5 ~ 2 × 1042 milky way   7.72 world's smallest woman
> 6 × 1030 black hole   3 ~ 7 bowling ball
2.8 ~ 6 × 1030 neutron star   0.16 billiard ball
1.99 × 1030 sun      
1.90 × 1027 jupiter      
5.97 × 1024 earth   ~ 3 × 10−6 snowflake
6.42 × 1023 mars   2.18 × 10−8 planck mass
7.35 × 1022 moon   ~ 10−12 bacterium
1.25 × 1022 pluto   ~ 10−15 virus
1.35 × 1021 earth's hydrosphere        
5.14 × 1018 earth's atmosphere   3.95 × 10−25 uranium atom
1.84 × 1015 earth's biosphere   3.06 × 10−25 top quark
~ 150,000 blue whale   1.67 × 10−27 neutron
~ 5000 african elephant   1.67 × 10−27 proton
~ 1500 passenger car   9.11 × 10−31 electron
635 world's heaviest man   < 10−36 limit of neutrino mass
88 the author   < 10−53 limit of photon mass

So what's a force?

Force …

When more than one force acts on an object it is the net force that is important. Since force is a vector quantity, use geometry instead of arithmetic when combining forces.

For a force to accelerate an object it must come from outside it. External force. Can't pull yourself up by your own bootstraps. Anyone who says you can is engaging in hyperbole.

Rule of thumb: one newton is approximately equal to a quarter pound

Selected Forces
force (N) event, process, phenomena
10−14 langevin forces of brownian motion
10−11 molecular motors consuming ATP
10−10 breaking noncovalent bonds (denaturing proteins)
10−09 breaking covalent bonds
860 weight of the author

The concept of inertia comes in many forms.

The concepts implied in Newton's Second Law of Motion are found throughout physics.
    cause
of change		 = resistance
to change		 × rate of
change of … 		   
newton's
second law		   force   mass   velocity  
F =  m  dv
dt
rotational
dynamics		   torque   moment
of inertia		   angular
velocity		  
τ =  I  dω
dt
newtonian
fluids		   shearing
stress		   viscosity   shear  
Fx  =  η  dx/dz
A dt
thermal
conduction		   temperature
gradient		   r-factor   heat  
ΔT =  R  dq
dt
ohm's
law		   potential
difference		   electrical
resistance		   charge  
V =  R  dq
dt
faraday's
law		   potential
difference		   inductance   current  
V =  L  dI
dt

Summary

Problems

practice

  1. A supertanker doesn't come with brakes. Using engines alone, it takes a loaded supertanker 13 km (8 miles) to stop. A typical vessel of this class has a gross mass of about 150 million kilograms (150 thousand tons) and a cruising speed of 50 kph (30 mph). Determine the average stopping force applied to the ship.

    Solution …

    This problem lends itself well to standard techniques. State the given quantities and convert them to SI units as needed. Solve the appropriate equation of motion to get the acceleration. Substitute this expression into Newton's Second Law and solve. (It can also be solved using the work-energy theorem -- as it will be in a later section of this book. Both methods lead to exactly the same answer -- as they should.)

    Δs =  13 km = 13,000 m   v2 =  v02 + 2aΔs ⇒
    a =  v02
    2Δs
    		 
    m =  150 × 106 kg
    v =  0 m/s F =  ma =  mv02  =  (150 × 106 kg)(13.888 … m/s)2
    v0 =  50 km   1000 m  = 13.888 … m/s 2Δs 2(13,000 m)
    1 h 3600 s  F =  1,112,892.… N = 1.1 MN
       

    This force comes from a combination of the the reverse thrust of the propeller and the viscous drag of the water.

  2. Write something else.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. Before a transparent material can be approved for use as a windshield in an airplane, it must first pass the "bird-impact resistance" test. A genuine (dead) or artificial chicken is accelerated to aircraft cruising speeds from the mouth of an air cannonand aimed at the windshield prototype. If the material shows any significant form of damage it fails the test. Each part of this problem contains an except from a USAF News Agency report.
    1. "Aircrews … find nothing funny about bird strikes. A strategically placed feathered bullet can wreak the same amount of damage as a laser-guided missile. The impact of a four-pound bird at 500 mph lasts less than a thousandth of a second, but can generate forces exceeding 100,000 pounds." Verify this statement; that is, that show that such a collision would indeed exert the force claimed.
    2. "They fire gelatin-molded, artificial birds weighing four pounds apiece, out a compressed air, 30 foot barrel cannon, which can shoot at speeds up to 900 mph. Three lasers and photodetectors measure the artificial chicken's velocity, while three cameras shoot the impact at 6,000 frames per second." Calculate the force exerted by the cannon on the chicken.

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