PHYS 172: Moden Mechanics Summe 2010 Δp sys = F net Δt ΔE = W + Q sys su su ΔL sys = τ net Δt Lectue 4 The Momentum Pinciple & Pedicting Motion Read 2.6-2.9
READING QUESTION #1 Reading Question Which of the following doesn t descibe the sping foce? A. It inceases as a the sping gets stetched o compessed moe B. It points in the opposite diection of the stetch/compession C. It is a vecto D. It is constant
READING QUESTION #2 What is the coect unit fo foce? A) m/s B) (kg*m)/s C) m/s 2 D) J E) N
Today: The Momentum Pinciple Momentum Pinciple Applying Momentum Pinciple: Thown Ball Iteative Appoach to Motion
The Momentum Pinciple Only deals with the system! Δ p = F Δt net Inteactions between the suounding & the system! Only extenal foces matte! Definition : F Δt net is known as the impulse
Δ p = F Δt net The pinciple of supeposition Net foce F push F eath The Supeposition Pinciple: The net foce on an object is the vecto sum of all the individual foces exeted on it by all othe objects Each individual inteaction is unaffected by the pesence of othe inteacting objects F gavity Ignoed fiction! Fnet = F1+ F2 +... Definition of net foce:
Applying the Momentum Pinciple (p. 54) 1. Choose a system (some potion of univese). Rest of univese is called suoundings. 2. List objects in suoundings that have significant influence on system; make appopiate diagams 3. Choose initial and final times 4. Apply the momentum pinciple to the chosen system: p f = p i + F net Δt Substitute known values into the tems of the momentum pinciple. Use position update if needed. 5. Check units and easonableness of you answe
Thowing A Ball In this seies of clicke questions, we e going to answe the following questions: If you thow a ball up in the ai with a cetain initial velocity: 1. how long will it take to get to the top of its tajectoy? 2. what is the ball s position at the top of the tajectoy? we ll assume fo simplicity that thee s no ai esistance but often this is a bad assumption!
CLICKER QUESTION #1 Ou system is the ball. Significant objects in the suoundings ae the Eath (which is esponsible fo the gavitational foce on the ball). = 0, mg, 0 N You thow a ball of mass m, and just afte the ball leaves you hand its velocity is v xi, v yi,0. If we can neglect ai esistance, the magnitude of the net foce is just mg, whee g = +9.8 N/kg. What is the vecto impulse acting on the ball in the next time inteval Δt? A. B. C. D. m 0, v xi, v yi,0 0, mg, 0 Δt 0, mg,0 Δt g,0 Δt E. 0, g, 0 Δt Impluse = F Δ t = 0, mg,0 Δt net y x
system = ball CLICKER QUESTION #2 Hee v <<c. Theefoe, the system s initial momentum is m v xi v,0, yi and the impulse is 0, mg, 0 Δt. A. B. C. v xi, v yi m m,0 v xi, v yi v xi, v yi,0,0 + 0, mg, 0 Δ t, v gδt,0 What is the final momentum at the xi yi end of the time inteval Δt? E. mv, mv mgδt, 0 D. m v xi yi y p i Δp p = p + F Δt f i net x p = 0, mg,0 N p f = m vxi, vyi,0 + 0, mg,0 Δt f
y p i Δp x = 0, mg,0 N p f So we have this esult (momentum update): m v, v,0 = m v, v gδt,0 x, f y, f x, i y, i Cancel the m s: v, v,0 = v, v gδt,0 x, f y, f x, i y, i
CLICKER QUESTION #3 At the top of the ball s tajectoy, the y A. Δt = v yi component of its velocity is (momentaily) zeo. v In tems of the initial components of velocity that you gave the ball, how long does it take the ball to each its maximum height? B. C. D. E. yi Δt = g v xi Δtt = g g Δt = v Δt = Δ p = F Δt mv = mgδt y net, y 0 yi v yi yi g p i p f Δpp Δ t = v yi g
CLICKER QUESTION #4 Fom when the ball fist leaves you hand to the top of the tajectoy, what is A. v xi, v yi,0 the aveage velocity of the ball? Note that since the velocity is changing at a constant ate, due to a constant foce, the aveage velocity is v avg v v ( ) v v i + v f = 2 B. C. D. E. v, v xi yi 2 2 v xi v xi,0,0 v, 2 yi,0,0 v xi,,0 2 v yi
CLICKER QUESTION #5 When the ball left you hand its position was x i, yi,0. What is its position when it is at the top of its tajectoy? Remembe that the aveage yi velocity was v,,0, and that xi v 2 it took amount of time to get to the top. A. B. C. x x x v, 2 yi i, yi,0 + v xi, 0 i i + v v yi Δt = D. x, i, 0 g i y xi v g yi, v yi v yi yi +,0 2 g v yi E. x i, yi +, 0 2 v g yi, v yi v yi yi +,0 2 g
Inteacting Objects When objects inteact with each othe, they affect each othe s momentum (and hence velocity). GIVEN: AND KNOWING: WE CAN PREDICT: Position and momentum of a paticle NOW FORM OF INTERACTIONS (FORCES) Position and momentum of a paticle LATER basic stuctue of mechanics
Inteacting Objects It s convenient to inset a few moe steps: GIVEN AND KNOWING WE CALCULATE position and stuctue of the stength of the momentum of an inteactions: the object in system inteactions FORCES AND APPLYING momentum pinciple WE CALCULATE final position of system AND USING position update fomula WE CALCULATE the changes in momentum (and thus final momentum)
y The Sping Foce x L 0 elaxed Define s = Δ L= L L 0 F sping = k s s s>0 sping is pulling back w/ sping is pushing back w/ s < 0
The Sping Foce As sping compesses, foce gets lage.
Example: Block on Sping A 0.060060 kg block block sits at est on a sping (k s = 8N/m) N/m). How fa is the sping compessed? Follow steps in applying momentum pinciple: 1. System = block 2. Significant inteacting objects: sping, eath Ignoe: ai, Jupite, etc. s = compession y z x
Example: Block on Sping 3. Initial and final times abitay 4. Apply momentum pinciple Δ p= FnetΔt p f = pi = 0 Δ p= 0 FnetΔ t = 0 F nt e = 0 Thus we conclude F = F + F = 0 net eath sping z y s x = 0, mg,0 N + 0, k s,0 N = 0 s mg s = = k k s 7.35cm 5. Units check; easonable
Constant vs. Vaiable Foces Δpp constant t foce : x = const Δ p x Δt vaiable foce : Δt const
Non-Constant Foces Δ p = F Δt sys net F F net What do you use fo? You must: 1. Use the AVERAGE value of the net foce ove the inteval OR 2. Pick a small enough Δt that the foce is mostly constant ove inteval
Iteative Pediction of Motion Fist divide the total time inteval into many smalle time intevals (so that F is oughly constant ove the inteval). Then Calculate the (vecto) foces acting on the system Update system momentum: f i net Update position: Repeat p = p + F Δt x = x + v Δt f i ave You ll do this many times in lab this semeste.
Iteative Pediction of Motion 10 8 6 4 2 3 4 5 6 7 8 baseball tajectoy
Today: The Momentum Pinciple Momentum Pinciple Applying Momentum Pinciple: Thown Ball Iteative Method to Pedicting Motion Next Time: Non-Constant Foces Iteative Pediction of Motion Sping Foce Gavitational Foce