Time dilation Gamma factor

Transcription

1 Time dilation Gamma actor Quick derivation o the Relativistic Sqrt(1-v 2 /c 2 ) ormula or time, in two inertial systems The arrows are a light beam seen in two dierent systems. Everybody observes the light traveling exactly at speed c. Light travels distance c, in Moving system which has speed d v to the right in the Lab system. Time interval in system M: t M = d/c h vt LAB In the Lab, the same light beam travels the oblique distance h, in a time t LAB The Moving system moves to the right an amount equal to vt LAB Thereore h 2 = d 2 + v 2 t LAB 2 Divide all by c 2 t LAB2 = (h/c) 2 = (d/c) 2 + (v/c) 2 t LAB 2 [1-(v/c) 2 ] t LAB2 = (d/c) 2 = t M 2 where 1-(v/c) 2 = 1/ γ 2 SO: t LAB = γ t M and since γ>1.0, the Lab observer sees the process taking longer than M sees as i M s clock is running slow. Rulers in M seem to LAB to be shorter by a actor 1/ γ Gamma applies not only to space-time, but also to Momentum-Energy variables

2 i>clicker question 01 Do you have an i>clicker? A)Yes, I do B)No, I don t C)I don t even know what it is D)I have a CPS pad that does not work with i>clicker system E)Can I use my TV remote control instead? Instructions: To participate in the quiz 1) turn on your i>clicker using ON/OFF button 2) press the button with the letter corresponding to the correct answer 3) relax and wait or the results

3 i>clicker question 02 Are you registered OK in Webassign? A)Yes B)No Instructions: To participate in the quiz 1) turn on your i>clicker using ON/OFF button 2) press the button with the letter corresponding to the correct answer 3) relax and wait or the results

4 i>clicker question 03 Have you registered your clicker serial number in CHIP? A)Yes B)No Instructions: To participate in the quiz 1) turn on your i>clicker using ON/OFF button 2) press the button with the letter corresponding to the correct answer 3) relax and wait or the results

5 i>clicker question 03B Did you receive an announcement rom me last week about 172H? A)Yes B)No Instructions: To participate in the quiz 1) turn on your i>clicker using ON/OFF button 2) press the button with the letter corresponding to the correct answer 3) relax and wait or the results

6 Chapters 2-4 The Momentum Principle The momentum principle Forces Impulse System and surroundings Fundamental orces: gravity & electric orces Conservation o momentum

7 The Momentum Principle An object moves in a straight line and at constant speed except to the extent that it interacts with other objects p = γ mv γ = 1 v 1 c 2 The Momentum Principle Δ p = F Δt net Change o momentum is equal to the net orce acting on an object times the duration o the interaction Assume that F does not change during Δt What is orce F? measure o interaction. A push or a pull. deined by the momentum principle. 7

8 Measuring orce 1. Use the momentum principle not convenient The Momentum Principle Δ p = F Δt net 2. Using Hookes spring orce law F spring = k s S L 0 L s =Δ L= L L 0 Robert Hooke k S spring stiness (spring constant) units: N/m Direction o orce: toward equilibrium. Restoring orce NOTE: example uses your intuitive knowledge o weight, which is gravitational orce 8

9 Impulse The Momentum Principle Δ p = F Δt net Deinition o impulse Impulse F net Δt Note: small Δt F net ~ const Momentum principle: The change o the momentum o an object is equal to the net impulse applied to it I the orce is not constant, then this equation deines the AVERAGE orce during Δt 9

10 Predictions using the Momentum Principle The Momentum Principle Update orm o the momentum principle p pi = FnetΔt Δ p = FnetΔt p = pi + FnetΔt p, p, p = p, p, p + F, F, F Δt x z z ix iz iz x y z Short enough, F~const For components: p = p + F Δt x ix net, x p = p + F Δt y iy net, y p = p + F Δt z iz net, z 10

11 Example Δ p = F Δt net k S =500 N/m no riction NB: orce must not change during Δt But in this example, it must change, to the extent that the object moves during the one second interval. I the object is heavy enough, it won t move very ar. Force: provided by a spring stretched by ΔL=4 cm interaction duration: 1 s? Find momentum p i p i =<0,0,0> kg. m/s F = k ΔL F spring = 500(N/m)0.04(m)=20 N F spring = 20,0,0 N 1. Force: spring S p = p + F Δt 2. Momentum: i net p =< 0,0,0 > kg m/s + 20,0,0 N 1 s p =< 20,0,0> kg m/s A an blowing on the object would give more constant orce as the object moves ( ) N. s = kg. m/s 2. s = kg.m/s 11

12 Δ p = F Δt net The principle o superposition F push F earth Net orce The Superposition Principle: The net orce on an object is the vector sum o all the individual orces exerted on it by all other objects Each individual interaction is unaected by the presence o other interacting objects F gravity Ignores riction between Earth and rock Example shows ZERO net orce on object. What can we conclude? MISconception: need constant NET orce to maintain steady motion Not so.? Why do planets move? How would planets move i the sun were suddenly to disappear? 12

13 Friction F pull F riction 13 animated slide

14 System and surroundings p system System: an object [or a collection o objects] or which we calculate a momentum (car in this case) Surroundings: objects which interact with system (earth, man, air ) p = p + F Δt i net Only external orces on the deined system matter Internal orces cancel due to equal and opposite changes in momenta o components o system 14 animated slide

15 Applying the Momentum Principle to a system 1. Choose a system and surroundings 2. Make a list o objects in surroundings that exert signiicant orces on system 3. Apply the Momentum Principle 4. Apply the position update ormula i needed 5. Check or reasonableness (units, etc.) p = pi + FnetΔt r = r + v Δt i avg 6. Be sure your vector F net consists ONLY o orces acting ON the system, and includes none o the orces that the system exerts on the surroundings or on other parts o itsel. Home study: See examples in 2.3 and in

16 Example: a hockey puck A hockey puck with a mass o 0.16 kg is initially at rest. A player hits it applying avg. orce F = 400,400,0 N during Δt = 4 ms. Where would be the puck 2 seconds ater it loses contact with hockey stick? Solution: 1. Choose a system and surroundings: 2. Make a list o objects in surroundings that exert signiicant orces on system 3. Apply the Momentum Principle p = p + F Δt i net p = 0,0,0 m kg/s + 400, 400,0 N 4 10 p = 1.6,1.6, 0 m kg/s ( ) ( -3 s) Hockey stick Gravity Normal orce opposing gravity Friction Right ater impact, and neglecting riction 16

17 Example: a hockey puck A hockey puck with a mass o 0.16 kg is initially at rest. A player hits it applying orce F = 400,400,0 N during Δt = 4 ms. Where would be the puck 2 seconds ater it loses contact with hockey stick? Solution: 3. Momentum p = 1.6,1.6, 0 m kg/s Actually, this is the initial momentum or the 2-second recoil part o example 4. The position update ormula r = r + v Δt i avg p p mv v = m r = 0,0,0 m + 10,10,0 m/s 2 s r = 20,20,0 m y * Choose coordinate system origin as: position o puck at the end o interaction, which is the beginning o phase 2 10,10,0 m/s ( ) p What tacit assumptions have we made regarding riction? r x 17

18 Motion o an object under constant orce System: cart and an. One dimension. p = pi + FnetΔt Assume: v<< c Then: air F net v = vi + Δt ( ) Position update: r = r + v t t i avg i m v v vt ( ) Δt Divide t into small intervals so that v does not change (much) ( ) r = ri + v t Δt v i Need to ind area under v(t)! vi + v r = ri + t ti 2 ( ) Only i v changes at constant rate t i t t =v avg 18

19 v Motion o an object under varying orce v v i ( ) r = ri + v t Δt Area under the curve: t i t Δ t 0 ( ) ( ) r = ri + lim v t Δt t = + i t i () r r v t dt integral Home study: : iterative prediction o motion 19

20 Example: colliding students Two students are late or class and run into each other head-on. Q: Estimate the orce that one student exerts on the other during collision Simplest model: F F loor, N F air F loor, P F Earth (Gravity) System: one spherical student Surroundings: earth, loor, air, second spherical student Force: Earth, loor, air, other student unknown! 20

21 y F F loor, N F air Example: colliding students F Earth F loor, P p p = F Δt i net Strategy: p pi = FnetΔt p = γ mv r = r + v Δt i avg < 0,0 > < p,0 >=< F F F, F F > Δt ix x ix loor, P air loor, N Earth < p,0 >=< F,0>Δt pix = FΔt 21

22 y F F loor, N F air Example: colliding students F Earth F loor, P x Strategy: p pi = FnetΔt p = γ mv r = r + v Δt i avg < 0,0 > < p,0 >=< F F F, F F > Δt ix loor, P air loor, N Earth First, let s neglect air resistance relative to the huge collision orce. The running student s shoe pushes backwards on the loor, so the loor exerts a horizontal reaction orce against the student s shoe (and thereore the student). We neglect this orce, too, and our answer or the collision orce will justiy this assumption. Finally note that vertical orces balance, since there is no change in vertical momentum. This means that the support o the loor balances the student s weight. 22

23 y F F loor, N F air pix = FΔt Example: colliding students F Earth F loor, P x Strategy: p pi = FnetΔt p = γ mv r = r + v Δt i avg What is collision time? Assume: v i =5 m/s, Δx=0.05m v avg Δx Δx = Δ t = Δ t vavg Δ t = Δx ( v + v )/2 i Δ t = 0.02 s What is initial momentum? Assume: m=60 kg p ix = mv = 300 kg m/s ix Find F: p ix 300 kg m/s F = = = N Δt 0.02 s This assumes each student compresses about 2 inches during the collision. Not skull to skull! 23