Physics 351 Wednesday, March 1, 2017

Transcription

1 Physics 351 Wednesday, March 1, 2017 HW7 due this Friday. Over spring break, you ll read of Ch 10 (rigid body rotation). A copy of this Ch 10 is on Canvas so that you don t need to take your book with you. Midterm exam (3/20) will cover Chapters 7,8,9. Quiz #5 (on HW5) today at end of class.

2 For spinning Earth, Ω points up out of the north pole. This rotation gives position r on Earth s surface the velocity That is, as seen from space, v = Ω r dr dt = Ω r This also works for a unit vector e fixed on the spinning globe: de dt = Ω e At any given instant, you can project a given vector Q onto two different sets of axes: the space axes ˆx, ŷ, ẑ that are not rotating (they re fixed in space somewhere), and the body axes ê 1, ê 2, ê 3 that are rotating with the globe (a.k.a. the rigid body).

3 If you observe the same vector Q from two different frames: A space frame S 0 that is stationary in space. A body frame S that is attached to the spinning globe. Q = Q xˆx + Q y ŷ + Q z ẑ = Q 1 ê 1 + Q 2 ê 2 + Q 3 ê 3 Then we use the product rule for the RHS to account for dê 1 dt = Ω ê 1 which gives this incredibly useful identity: ( ) ( ) dq dq = dt dt S 0 (space) S(body) + Ω Q You ll use this often to relate dq/dt as observed in the two frames.

4 ( ) dq = dt S 0 ( ) dq + Ω Q dt S Applying this (twice) to Newton s second law (inertial frame S 0 ): ( d 2 ) r m dt 2 = F S 0 gives Newton s 2nd law as seen in rotating frame S m r = F + 2mṙ Ω + m(ω r) Ω and two well-known pseudo-forces appear. (This assumes Ω = 0.) 2nd term is Coriolis force. 3rd term is centrifugal force. (If Ω 0, then you get one more term, mr Ω, called the azimuthal force or Euler force. See HW07/q8.) Pause for two points: how to evaluate each quantity in the boxed equation; and which directions all the various vectors point.

5 The centrifugal force (which our high-school physics teachers regarded with abomination) is F centrifugal = m(ω r) Ω (Use right-hand rule to see the various vectors on globe.) Its magnitude is...?

6 The centrifugal force (which our high-school physics teachers regarded with abomination) is F centrifugal = m(ω r) Ω (Use right-hand rule to see the various vectors on globe.) Its magnitude is...? This is easier to see using ρ = r sin θ: F centrifugal = mω 2 ρ ˆρ = mω 2 r sin θ ˆρ

7 F centrifugal = m(ω r) Ω = mω 2 ρ ˆρ Another HW question from 3 years ago: I am spinning a bucket of water about its vertical axis with angular velocity Ω. Once the water has settled in equilibrium (relative to the bucket), what will be the shape of the water surface? (Let ρ be the (horizontal) distance from the axis of rotation.) (A) y 1/ρ 2 (B) y 1/ρ (C) y ρ (D) y ρ 2 (E) y ρ 3

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9 Unlike the inertial force and the centrifugal force, which you experience when traveling in cars, airplanes, etc., the Coriolis force is much less familiar from everyday experience: F coriolis = 2m v Ω (Use right-hand rule to see the various vectors on globe.) One handy mnemonic is the analogy with the magnetic force law. One more HW problem from last year: What are the directions of the centrifugal and Coriolis forces on a person moving (1) south near the North Pole? (2) east on the equator? (3) west on the equator? (4) south across the equator? (5) east near Philadelphia? (6) west near Philadelphia? (7) north near Philadelphia?

10 If you look down at Earth from space (e.g. from above the equator), in which direction do you see the land beneath you moving? (A) toward the east (from west to east) (B) toward the west (from east to west) (C) It depends on whether you re looking down at the northern or the southern hemisphere.

11 If I look down at Earth, then blink my eyes for a moment, then look down again at Earth, the land that I see after blinking is (A) east of the land that I saw before blinking (I see Philly, then a while later I see England) (B) west of the land that I saw before blinking (I see Philly, then a while later I see San Francisco)

12 If I drop an object straight down from the top of a tall building, the Coriolis force will deflect the falling object (A) toward the east (B) toward the west

13 F coriolis = 2m v Ω The cross-product of (downward toward Earth s center) with Ω points (A) toward the east (B) toward the west

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17 F coriolis = 2m v Ω A projectile of mass m is fired with initial speed v 0 horizontally and due north from a position of colatitude θ. Find the direction and magnitude of the Coriolis force in terms of m, v 0, θ, and the earth s angular velocity Ω. How does the Coriolis force compare with the projectile s weight if v 0 = 1000 m/s and θ = 60? (Ω /s, g 10 m/s 2 )

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19 What is smallest value of static friction coefficient µ for which the block can stand still on the ramp? (What s the familiar answer if A = 0?)

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22 Note that tan(θ β) = cos β sin θ sin β cos θ cos β cos θ + sin β sin θ

23 Note that tan(θ β) = cos β sin θ sin β cos θ cos β cos θ + sin β sin θ

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26 Physics 351, Spring 2017, Homework Quiz #5 ( ). 15 minutes. Work alone. Closed book, one page of handwritten notes OK. Relax! All quizzes together make up only 10% of course grade. 90% semester quiz total gets full credit. Two equal masses m, connected by a massless string, hang over two pulleys (of negligible size). The left mass moves in a vertical line, but the right mass is free to swing back and forth in the plane of the masses and pulleys. Write the Lagrangian, using coordinates r and θ. Then find the equations of motion for r and for θ, as shown. (That s all.)

27 Physics 351 Wednesday, March 1, 2017 HW7 due this Friday. Over spring break, you ll read of Ch 10 (rigid body rotation). A copy of this Ch 10 is on Canvas so that you don t need to take your book with you. Midterm exam (3/20) will cover Chapters 7,8,9. Quiz #5 (on HW5) today at end of class.