» 211 Syllabus

Unit 2 ; Topic Five - Work & Energy Intro

-Topic 6:
      Transformed Energy : Ch.8 § 3,5 ; Ch.9 § 5 ; Ch.18 § 6
      . . . (HW due Fri.Oct.08)
      Energy transfer : Ch.10 § 4 ; Ch.11 § 1,2,3,4 ; Ch.9 § 4
      . . . (HW due Mon.Oct.11)
    Quiz Wed.Oct.13

-Topic 7:
      Rotational Energy : Ch.12 ; Ch.13
      . . . (HW due Fri.Oct.15)
  - plan for Exam 2 Mon.Oct.18

Work by Force : ch.7 prob. 1,3,4,5,6 ; 7 ; 9,11,12,13,15,17,19,21
Kinetic Energy : ch.7 prob. 34,35,36,38,40,42,43 ; 44,46,,48,49(why is d ~ KEi)
gravity PE : ch.7 prob. 55,56,57,59,61,69,72,74 ; 71,73,75,77 ; 62,63,65,70
Elastic PE : ch.7 prob. 24,25,27,28,29 , 31,32,33,51
PE functions : ch.8 prob. 1,11,12,19 ; 21,22,24,26,28,31,33 , 3,6,4,7,10,13,15 ,

1. (Ch.7 prob.81 & 82 combined) A 120 [kg] crate slides 15[m] across a rough (μ_k = 0.50) level floor,
     while a woman pulls its tow rope with 600[N] Tension at 35° angle above horizontal.
         a) calculate the Friction Force , and the Work done that it does during the slide.
         b) calculate the Work that the Tension does during the slide .
         c) . . . How much Work did the gravity Force do during the slide? . . . How much Work did the surface Normal do ?
         d) IF there are NO other Forces being applied during the slide, what is the change in the crate's KE?
2. ("real" Ch.7 prob.26) A typical mousetrap has an effective spring stiffness 250 [N/m],
     with Force and distance measured along the arc that the "mouse-squashing" bar travels.
     ... the spring is already stretched 0.2 [m] (effectively) when bought ... before it is "set"

   0) convince yourself that Fspring is parallel Δr , all along the bar's arc.       a) Calculate the Force needed to lift the mousetrap bar, to begin setting it.       b) ... calculate the Force to set, after bar moves 0.18[m] (along the "snap path" arc)       c) Compute the average Force applied while setting the mousetrap       ... for Work, we want a "distance-average", not a "time-average".       d) . . . calculate the Work done while setting the mousetrap . Explain         how the "set" mousetrap's KE can be zero, after Work has been done on it.

3. (like Ch.7 prob.29, Fig.30) A 40 [N/m] spring with relaxed length L = 0.28[m] is connected from x = −L/2 to x = +L/2 ;
     ... the midpoint of the spring is pushed from   y=0 , to   y=0.04[m] . . . then to   y=0.08[m] .
         a) Compute the Length of the spring at the first stop ... and also at the second stop .
         . . . then compute the stretch distance of the spring at the first stop , and at the second stop (notice it is NOT doubled)
         b) compute the Force vector component that the spring applies to the pushing device (triangles! one last time)
         c) what is the Force as a function of y? {compare F_y as y doubled ; treat F~yⁿ unless you love trig functions}
         d) What would be the PE formula for this "displaced-in-the-middle" spring? {to obtain your F function above}

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4. (like Ch.7, prob.91) A 95[kg] luge-rider starts at altitude 350[m] with speed 1[m/s], is going 36.111[m/s] at alt 240[m].
         a) compute the luge-rider's KE at the top , and the gravity PE at the top .
         b) compute the luge-rider's KE at the bottom , and the gravity PE at the bottom .
         c) what's the difference between them? (i.e, subtract)
         d) if the starting speed was ~zero, but friction did the same Work, how fast would the luge be going at the bottom?
5. A 16000[kg] log truck was going 20 [m/s] at the top of a 300[m] tall hill , with no brakes ; must turn left at the bottom ;
     . . . there's a "runaway truck" escape ramp 1/3 of the way down the hill, and another ramp 2/3 of the way down.
         a) compute the truck's KE at the top of the hill ... and the truck's gravity PE at the top of the hill.
         b) if there was no friction from "engine braking" , how fast would the truck be going at the bottom? (use Energy!)
         c) if the truck is only going 25[m/s] at the the first truck ramp, how much Work did the "engine braking" do?
         d) the truck takes the first ramp, runs into gravel piles that stop the truck in 50[m]. What Force did the gravel apply?
6. A neutral sodium atom will induce an attraction in a nearby helium atom {mechanism will be explained in Phys.II } ;
     the PE function is approximately   U = 4 ε ( (A/r)⁶ − (A/r)¹² ) , where r is the distance between atom centers
     . . . for He near Na , ε ≈ −1E−19[Joules] , and A ≈ 0.3E−9[m] . {we'll show how to deduce these in Units 3 & 4}
         (a) where (besides infinity) is the PE function   U(r)   equal to zero ?
         (b) calculate the Force function ... notice that Ch.8 prob.17 is incorrect .
         (c) . . . where (besides infinity) is the Force function equal to zero?
         (d) what is the Force at the place that the PE is zero? (inward, or outward? strong, or weak?)
         (e) what is the PE at the place where the Force is zero? (positive, or negative?)

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oops ... we should have done an elastic PE scenario ... sorry.

(to topic 1 summary)

USE UNITS all thru your answer ... don't just append them onto the final number !

Put symbols on a sketch, as you read . . . use meaningful phrases : "v_match" , "catch" , "ground" "turn-around"
. . . show process spans as brackets or vector arrows , with words like   "average" , "losing" , "gaining" , "t=0→2" ...

Write statements as symbols first ... each statement has a SUBJECT ... manipulate symbols before plugging numbers.
. . . keep track of adjectives such as   "initial" , "average" , "final" , "stopped" . . . "change"

Topic 2 Reminders :

(to topic 2 summary)

subscripts for "total" or "system" , "part A" ...

keep track of the SOURCE for the external Forces applied to the (passive) object

Recognize whether you are predicting based on theory . . . reasoning from causes to their effects ... (what should it do?)
      or whether you are deducing from observation . . . (what does this imply about it?)

Get un-stuck (often) by wondering "why isn't it the same as it used to be?"

Topic 3 Reminders :

it's the Sum of external Forces that cause an object's mass to accelerate subscripts for "total" or "system" , "part A" ...

once you separate vectors into components, keep each component separate from the others!
ΣF = ma , for each component separately.

Topic 4 Reminders :

Name each Force by the source of that Force ... its cause ... a function of the environment.

spring Force : F_spring = − k s ... opposite the stretch vector .

Pressure Force : F_spring = P A   generally ... in a fluid , δP = ρ g δh .

Gravity : every mass in the universe contributes to the gravity field   g   at any place of interest.
... add "nearby" contributions as vectors ... g_{by M} = G M / r²_M-to-point (toward M) . . . then use   F_{gravity, on m} = mg ;       NOTICE : m ≠ M !

Especially when the story-line is about location (rather than time), use the Work and Energy approach

Energy : a condition (status) quantity ... objects   have   Kinetic Energy at any instant , based only on conditions at that instant .
. . . Kinetic Energy is Energy of Motion . . . similar to momentum , but a scalar , NOT a vector .
      (compare momentum : carried with the object wherever it goes . . . until it is transferred to something else ... )
. . . KE might be thought of as the "amount of Force that has accumulated in the object thru the distance it has been applied for"
      (contrast momentum : the "Force that has accumulated thru the duration" that it was applied for ... )

Work : a process quantity ... Force dot displacement ... parallel components only !
. . . Force component along the displacement   does   positive Work   on   the object being pushed
      so transfers positive Energy into the object , while the displacement occurs .
. . . Force component opposite the displacement does negative Work on the object , removing Energy from the object
      . . . (obviously Forces which do negative Work did NOT cause the displacement)
=> W_{by F} = F·Δs .

Obtaining a "formula" for KE : . . . ( leaving out Σ , to improve clarity )
start with F = d p / dt . . . dot-multiply both sides with   − − · ds   (with s on the right)
      F ·ds = d p · ds / dt . . . recognize right-hand side as   d p · v . . . substitute to get a single variable , using p = m v .
      F ·ds = m d v · v , or   = d p · p /m . . . either choice will let us integrate ;
. . . integrate both sides thru a process ; long enough that v and p change , but short enough that F is essentially uniform within it
      ∫ F ·ds = m ∫ d v · v   , or   F ·ds = ¹/_m ∫ d p · p   , from start to finish .
      =>   F · Δs = (½ m v²)_final − (½ m v²)_initial .
      recognize the left-hand side as   Work   =   Δ KE ;
=> KE = ½ m v² = ½ p·v = ½ p²/m   , in any of these equivalent forms .

Important statement #6 about how the Universe works : Energy is Conserved

. . . each external Force does Work while the Force application point moves ;
      that Work ... done on (or to) the system by the external subject
. . . transfers Energy from the subject which applied that Force into the object .
      Work can be positive or negative , but are scalar quantities (no 3-d vector components).
=> KE_system,start + Σ W_{by all F} = KE_system,end .

. . . it is often useful to consider what Work the applied Forces might potentially do ;
      and ascribe that quantity to the system as Potential Energy
. . . with this perspective , Force applied along the displacement ... (by one part of the system on another part of the system)
      transforms Potential Energy into some other form of Energy, as it is actualized (made real rather than potential).
=> Σ E_start + Σ W_{by F not in PE list} = Σ E_end , for a system .

Both of the "bottom line" statements above mean that Energy is conserved ... the second makes it look more apparent .

Each kind of Force has its own fuction of location , so has a distinct Potential Energy form .

gravity (local) PE :
constant Force but only the vertical component of displacement "counts" ... parallel components !
=> PE_grav = m g h , with h measured upward (z-direction) from a "zero-height reference" point
. . . because there's no special height, the height that you call zero is your choice . . .
      you won't forget where it is if you indicate h=0 on your diagram .

spring PE :
. . . with F = − ks , average Force during the stretch is ½ of the ending Force
=> PE_elastic = ½ k s²   . . . stretch s must be measured from its relaxed length

Pressure PE :
. . . air spurts from an inflated tire with KE because the Pressure inside the tire is greater than the Pressure outside the tire.
      Work is F·ds = (P_inside − P_outside)A·ds . so :
=> PE/Volume = P   ; it is okay to measure from a non-zero reference pressure , but be consistent !

Pressure :
. . . more intense Pressure at deeper depths occurs because of gravity pulling the mass above that depth
      treat that kind of pressure as density of gravity PE :
=> PE/Volume = ρ g δh .

Friction :
always treat friction Forces by the Work they do , because they do NOT have a Potential Energy function .
=> There IS NO PE for Friction Force ... must be explicitly included as one of the "Work by Forces not included in the PE list"

Often, the Potential Energy function is easier to measure or calculate than the Force function.

We can deduce the Force (as a function of location) from a known Potential Energy function ;
. . . thinking of the Work that the Force would do during a possible (potential, imagined, future) tiny displacement . . .
      moving along the negative of that displacement, opposite the Force, is what adds to that form of Potential Energy .
=> − dU / dr = F_r , the component along that direction (r)

A space-derivative is often called a gradient , or slope . . . (distinct from a time-derivative , which is a rate slope)
. . . the second derivative of a function is called the function's curvature .

Important statement #7 about how the Universe works:

Objects are pulled / pushed by Forces in a way that reduces their Potential Energies

We should try this gradient procedure with a couple of known PE functions :

local gravity : calling the "up" component z , we have U = m g z ...
− d/dz ( m g z ) = − m g . . . is supposed to be the z-component of the Force exerted by local gravity
. . . notice that the negative sign , indicating downward direction for the Force , was obtained automatically.

local gravity : calling the "East" component x , using U = m g z ...
− d/dx ( m g z ) = 0 . . . is supposed to be the x-component of the Force by local gravity ... yes, it checks .

springs : if stretched in the + x-direction , from x = 0 , then U = ½ k x²
− d/dx ( ½ k x² ) = − k x . . . is supposed to be the x-component of the Force by the spring .
. . . the spring stiffness is called k , indicating that it is the kurvature of PE ... the second derivative of the PE function

spring : if stretched in the − y-direction , from unstretched length y = 2[m] , then U = ½ k (y − 2m)² ... or is it U = ½ k (2m − y)² . . . ?
− d/dy ( ½ k (y − 2m)² ) = − k (y − 2m) . . . or is it   − k (−1) (2m − y) . . . yes, they're the same .

Enough Practice ; let's try a new one !

Gravitation : (Universal)
we know that the Force by Gravitation is inward , with strength F = m GM/r² .
. . . it should be apparent that the appropriate varying component is r . . . "inward" is −r .
      the source mass , and the influenced mass , are constant ... let's call −m GM = c , for clarity .
Now we need to find a PE function whose r-derivative is   c r⁻²
. . . how about trying   c r⁻³ ? . . . no , that has derivative   (−3)c r⁻⁴
. . . . . lets try   c r⁻¹ . . . all we need is a negative sign in front .
=> U_Gravitation = − m GM / r .

the Gravitational Potential Energy of a planet , near the Sun , is negative .
As the planet-forming material fell inward from far away (where PE≈0) , to lower PE , it gained KE
. . . as the material lost some KE during collisions , it became trapped , unable to achieve the large distances that it once was at .
the amount of Energy an object needs , in order to achieve "infinite" distance , is called its Escape Energy
. . . the speed that corresponds to that much Kinetic Energy , which is the same for any mass , is the Escape Speed.

Because all source Masses contribute to the Gravitational Field g , at any location ,
      the Gravitational PE of some field mass is NOT due to any single source Mass .
. . . instead, each source Mass   M_i   contributes to the total Gravitational Potential , called V_Grav , at the place of interest .
      notice that a "mass m" object at the place of interest has PE = m V_Grav   when it is there ,
      but the place itself has Gravitational Potential (due to "space warp") , even if no mass is there .
=> V_Grav(x,y,z) = Σ V_i . . . . . each contribution V_i = GM_i /r_i   , depends on distance r_i from its Mass to the place (x,y,z) .