Physics 8 Friday, September 18, 2015

Transcription

1 Physics 8 Friday, September 18, 2015 Turn in HW3. HW4 not due until 2 weeks from today! I ll write it up this weekend and hand it out on Monday. Remember to sign up for laser-cutter time! Finish reading Chapters 6 and 7 ASAP if you haven t already. This weekend s reading is Chapter 8: Force! Finally! Today: we ll try to finish discussing Ch 6 and Ch 7. While you re madly signing up for laser-cutter time, I ll show you an animation I made of an irreversible process that consists of a large number of elastic collisions. It illustrates how dissipation (from coherent to incoherent motion) works.

2 More chapter 6 key ideas Center-of-mass velocity is the velocity of the CoM of a system of objects: v cm = m 1 v 1 + m 2 v 2 + m 1 + m 2 + v x,cm = m 1v x1 + m 2 v x2 + m 1 + m 2 + An isolated system s CoM velocity cannot change! You can see this by noticing that the numerator in v cm is the system s total momentum, which you know is constant for an isolated system. If you observe this system from a camera that is moving at v cm, the system s CoM will appear to be at rest. This camera s frame-of-reference is called the ZM frame, because the system s momentum is zero as seen from that frame (i.e. as seen by that moving camera).

3 A friend and I take our little track and our two little colliding carts, and we set them up (probably in the dining car) on board a moving train (train moving north at constant velocity 30 m/s), with our little track aligned with the train axis. I push a 1 kg cart north toward my friend at 1 m/s. She pushes a 1 kg cart south toward me at 1 m/s. As seen by a camera mounted on the ceiling of the train, what is the velocity (north) of the center-of-mass of the two-cart system? (A) +30 m/s (B) -30 m/s (C) 0 m/s (D) +1 m/s (E) -1 m/s

4 A friend and I take our little track and our two little colliding carts, and we set them up (probably in the dining car) on board a moving train (train moving north at constant velocity 30 m/s), with our little track aligned with the train axis. I push a 1 kg cart north toward my friend at 1 m/s. She pushes a 1 kg cart south toward me at 1 m/s. As seen by a camera mounted on the ceiling of the train, what is the velocity (north) of the center-of-mass of the two-cart system? (A) +30 m/s (B) -30 m/s (C) 0 m/s (D) +1 m/s (E) -1 m/s As seen by a camera mounted on the train platform at Princeton Junction (looking in the window as we go by), what is the (northward) velocity of the center-of-mass of the two-cart system?

5 A friend and I take our little track and our two little colliding carts, and we set them up (probably in the dining car) on board a moving train (train moving north at constant velocity 30 m/s), with our little track aligned with the train axis. I push a 1 kg cart north toward my friend at 1 m/s. She pushes a 1 kg cart south toward me at 1 m/s. As seen by a camera mounted on the ceiling of the train, what is the velocity (north) of the center-of-mass of the two-cart system? (A) +30 m/s (B) -30 m/s (C) 0 m/s (D) +1 m/s (E) -1 m/s As seen by a camera mounted on the train platform at Princeton Junction (looking in the window as we go by), what is the (northward) velocity of the center-of-mass of the two-cart system? Is one of these two cameras watching from the zero-momentum frame of the two-cart system?

6 A friend and I take our little track and our two little colliding carts, and we set them up (probably in the dining car) on board a moving train (train moving north at constant velocity 30 m/s), with our little track aligned with the train axis. I push a 1 kg cart north toward my friend at 3 m/s. She pushes a 2 kg cart south toward me at 6 m/s. As seen by a camera mounted on the ceiling of the train, what is the velocity of the center-of-mass of the two-cart system? (Let the +x axis point north.)

7 A friend and I take our little track and our two little colliding carts, and we set them up (probably in the dining car) on board a moving train (train moving north at constant velocity 30 m/s), with our little track aligned with the train axis. I push a 1 kg cart north toward my friend at 3 m/s. She pushes a 2 kg cart south toward me at 6 m/s. As seen by a camera mounted on the ceiling of the train, what is the velocity of the center-of-mass of the two-cart system? (Let the +x axis point north.) As seen by a camera mounted on the train platform at Princeton Junction (looking in the window as we go by), what is the (northward) velocity of the center-of-mass of the two-cart system?

8 A friend and I take our little track and our two little colliding carts, and we set them up (probably in the dining car) on board a moving train (train moving north at constant velocity 30 m/s), with our little track aligned with the train axis. I push a 1 kg cart north toward my friend at 3 m/s. She pushes a 2 kg cart south toward me at 6 m/s. As seen by a camera mounted on the ceiling of the train, what is the velocity of the center-of-mass of the two-cart system? (Let the +x axis point north.) As seen by a camera mounted on the train platform at Princeton Junction (looking in the window as we go by), what is the (northward) velocity of the center-of-mass of the two-cart system? Is one of these two cameras watching from the zero-momentum frame of the two-cart system?

9 If I observe a system from its zero-momentum reference frame, what can I say about its center-of-mass velocity? (a) The center-of-mass velocity (as seen from the ZM frame) is the same as the velocity of the ZM reference frame (as seen from the Earth frame). (b) The center-of-mass velocity (as seen from the ZM frame) is zero. (c) When observing from Earth s frame of reference, a system s center-of-mass velocity will be the same as the velocity (w.r.t. Earth) of the ZM reference frame. (d) (a) and (b) (e) (a) and (c) (Use (f) (b) and (c) (Use A B.) (g) (a), (b), and (c) (Use.) C.)

10 More chapter 6 key ideas An isolated system s CoM velocity cannot change! A somewhat obscure consequence of this fact is that even in a totally inelastic collision, it is not necessarily possible to convert 100% of the initial kinetic energy into heating up, mangling, etc. the colliding objects. Momentum conservation requires that the CoM velocity cannot change, so if the CoM is moving initially, it has to keep moving after the collision. Textbook: convertible vs. translational parts of a system s kinetic energy. That idea is worth remembering, but the math is not. (Can illustrate using colliding carts.)

11 Two more facts about Galilean relativity that are interesting (but whose math details aren t worth bothering with): The momentum of a system is different when observed from different frames of reference; but changes in the total momentum of a system are the same as observed from two different inertial frames of reference. The energy of a system is different when observed from different frames of reference; but changes in the total energy of a system are the same as observed from two different inertial frames of reference. So for example suppose, while sitting on a train moving at constant velocity, I rub two sticks together for several minutes, until finally I have increased the internal (thermal) energy of the sticks by a large enough amount that the sticks begin to burn: I ve converted kinetic energy into thermal energy. An observer on the ground will calculate different kinetic energies for the sticks at any instant, but she will calculate the same total transfer of energy into the sticks. Also, observers in different inertial frames will calculate the

12 You really only need these first two equations from Ch6. The third one is in the obscure category.

13 Here s another obscure result I put on the equation sheet. I will probably put it into an extra-credit problem, but otherwise you won t need to use it. (I ll illustrate on next page.)

14 animation of Earth frame vs. ZM frame Computer programming for visual artists & graphic designers: processing.org

15 I can solve the general case of a two-body elastic collision. Here s the input: because an isolated system s momentum is constant: m 1 v 1x,i + m 2 v 2x,i = m 1 v 1x,f + m 2 v 2x,f because the collision is elastic: (v 1x,f v 2x,f ) = (v 1x,i v 2x,i ) Here s the output. It s messy! v 1x,f = (m 1 m 2 )v 1x,i + 2m 2 v 2x,i m 1 + m 2 v 2x,f = (m 2 m 1 )v 2x,i + 2m 1 v 1x,i m 1 + m 2 (HW3/XC2 was a special case of this: v 2x,i = 0.)

16 Now suppose instead I look at this collision from the zero momentum frame, which is the frame in which the center-of-mass is at rest. Here s the input: m 1 v 1x,i + m 2 v 2x,i = 0 m 1 v 1x,f + m 2 v 2x,f = 0 (v 1x,f v 2x,f ) = (v 1x,i v 2x,i ) Here s the output. It s clean! v 1x,f = v 1x,i v 2x,f = v 2x,i If you observe the collision from the ZM frame, the elastic collision just flips the sign of each velocity. That s it! Even the general inelastic case is neat and tidy in the ZM frame! v 1x,f = ev 1x,i v 2x,f = ev 2x,i

17 Zero-Momentum (ZM) frame for two-object collisions Very useful (but difficult to visualize) tool: ZM frame. Elastic collision analyzed in ZM ( ) frame: v 1i,x = v 1i,x v ZM,x, v 1f,x = v 1i,x, v 1f,x = v 1f,x + v ZM,x, v 2i,x = v 2i,x v ZM,x v 2f,x = v 2i,x v 2f,x = v 2f,x + v ZM,x Inelastic collision analyzed in ZM frame (restitution coeff. e): v 1f,x = ev 1i,x, v 2f,x = ev 2i,x Step 1: shift velocities into ZM frame, by subtracting v ZM,x Step 2: write down (very simple!!) answer in ZM frame Step 3: shift velocities back into Earth frame, by adding v ZM,x You can try this on some XC problems. Otherwise, skip it.

18 There are actually three pretty neat situations that you can analyze quite easily using the ZM frame trick: When a stationary golf ball is hit by a much more massive golf club, the golf ball s outgoing speed is 2 the incoming speed of the (end of the) club. It s easier to hit a home run off of a fastball than a slow pitch. When you drop a basketball with a tennis ball resting atop the basketball, the result is quite remarkable. I was planning to skip these as obscure, and leave them as topics for extra-credit problems. But if there is overwhelming demand, we could work them out next week in class? (A) Leave it for XC (B) Do it in class

19 Who would potentially be interested in learning/using Processing for an extra-credit programming project (e.g. animating Earth s motion around the Sun, or animating the deflection of a beam under load, or animating the bouncing of a basketball, etc.)? (a) Yes! OMG! I am so interested in this! (b) I am possibly interested, but I m busy. We ll see. (c) Realistically, I am probably not interested. (d) No thanks.

20 Here are the only two equations worth knowing from Chapter 7. By contrast, Chapter 8 will have quite a few worth knowing!

21 Hugely important: when two objects interact only with one another: p 1x = p 2x v 1x / v 2x = m 2 /m 1 a 1x a 2x = m 2 m 1 When the medicine ball and I push apart from one another, we both accelerate: in opposite directions, and in inverse proportion to our masses. Lifting an object up a height x in Earth s gravity changes its gravitational potential energy by U G = mg x I usually remember U = mgh where h is height Basketball: back & forth between 1 2 mv 2 and mgh until mechanical energy is dissipated into thermal energy

22 Problem: I release a 1 kg ball from rest, from an initial height x i = +5.0 m above the ground. (Use g 10 m/s 2.) (a) What is the ball s initial G.P.E.? (Let s define x = 0 to be U G = 0.) (b) What is the ball s initial K.E.? (c) What is the ball s G.P.E. immediately before it reaches the ground? (d) What is the ball s K.E. immediately before it reaches the ground? (e) What is the ball s speed immediately before it reaches the ground? (f) If the ball bounces elastically off of the floor, what height will it reach after bouncing? (g) If instead the ball bounces off of the floor with a restitution coefficient e = 0.9, what height will it reach after bouncing?

23 Problem: Suppose your friend s mass is about 50 kg, and she climbs up 30 flights of stairs (that s about 100 m) to check out a great rooftop view of the city s architecture. (a) By how many joules did climbing the stairs change her G.P.E.? (b) Where did this gravitational potential energy come from? I mean what source energy was converted into this G.P.E.?

24 Problem: Suppose your friend s mass is about 50 kg, and she climbs up 30 flights of stairs (that s about 100 m) to check out a great rooftop view of the city s architecture. (a) By how many joules did climbing the stairs change her G.P.E.? (b) Where did this gravitational potential energy come from? I mean what source energy was converted into this G.P.E.? (c) How many food Calories did she burn (assuming her muscles are 100% efficient at converting food into mechanical work)?

25 Read for Monday: chapter 8 (force). Finally! The force concept quantifies interaction between two objects. Forces always come in interaction pairs. The force exerted by object A on object B is equal in magnitude and opposite in direction to the force exerted by B on A: F AB = F BA The acceleration of object A is given by the vector sum of the forces acting on A, divided by the mass of A: a A = F(on A) m A The vector sum of the forces acting on an object equals the rate of change of the object s momentum: F(on A) = d p A dt

26 Read for Monday: chapter 8 (force). Finally! An object whose momentum is not changing is in translational equilibrium. We ll see later that this will be a big deal for the members of a structure! To achieve this, we will want all forces acting on each member to sum vectorially to zero. The unit of force is the newton. 1 N = 1 kg m/s 2. The force exerted by a compressed or stretched spring is proportional to the displacement of the end of the spring w.r.t. its relaxed value x 0. k is spring constant. F spring x = k(x x 0 ) When a rope is held taut, it exerts a force called the tension on each of its ends. Same magnitude T on each end. Free-body diagrams depict all of the forces acting on a given object. They are used all the time in analyzing structures!

27 Physics 8 Friday, September 18, 2015 Turn in HW3. HW4 not due until 2 weeks from today! I ll write it up this weekend and hand it out on Monday. Remember to sign up for laser-cutter time! Finish reading Chapters 6 and 7 ASAP if you haven t already. This weekend s reading is Chapter 8: Force! Finally! Today: we ll try to finish discussing Ch 6 and Ch 7.