Physics 351 Monday, April 27, 2015

Transcription

1 Physics 351 Monday, April 27, 2015 Final exam is Wednesday, May 6, from 9am-11am, in DRL A4. You can bring a sheet of your own handwritten notes. Final exam includes Ch7 (Lagrangians), Ch9 (non-inertial frames), Ch10 (rigid-body rotation), Ch13 (Hamiltonians), and will (like midterm) be mainly based on HW problems. You can submit XC work as late as Tuesday, May 12.

2 Water flows through an old plumbing pipe that is partially blocked by mineral deposits along the wall of the pipe. Through which part of the pipe is the fluid speed largest? (A) Fastest in the narrow part. (B) Fastest in the wide part. (C) The speed is the same in both parts.

3 A water-borne insect drifts along with the flow of water through a pipe that is partially blocked by deposits. As the insect drifts from the narrow region to the wider region, it experiences (A) an increase in pressure (P wide > P narrow ) (B) no change in pressure (P wide = P narrow ) (C) a decrease in pressure (P wide < P narrow ) Hint: is the speed the same or different? Is there a net force (per unit area) that causes this change in speed?

4 (P 1 A 1 )(v 1 t) (P 2 A 2 )(v 2 t) = 1 2 m(v2 2 v 2 1) + mg(y 2 y 1 ) (P 1 P 2 )(Volume) = (P 1 P 2 ) m ρ = 1 2 m(v2 2 v 2 1) + mg(y 2 y 1 )

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6 This experiment combines the equation of continuity with Bernoulli s equation. When I open the valve, compressed air will flow from left to right through the horizontal tube. The tube is wide on the left and right, but narrow in the middle. Where is the speed largest? Where is the pressure lowest? How will height of the green liquid respond to changes in pressure in the horizontal tube?

7 Once I turn on the air flow, the horizontal speed of the flowing air will be (A) fastest above tube B (B) slowest above tube B (C) the same above all tubes

8 Once I turn on the air flow, the height of the green liquid will be (A) lowest in tube B (B) highest in tube B (C) the same in all tubes

9 Pascal s principle: a pressure change applied to a confined incompressible fluid is transmitted undiminished throughout the fluid and to the walls of the container in contact with the fluid. Have you ever used one of these handy devices? You can get a mechanical advantage that is huge compared with what you can easily get with e.g. a block and tackle.

10 A container is filled with oil and fitted on both ends with pistons. The area of the left piston is 10 mm 2. The area of the right piston is mm 2. What force must be exerted on the left piston to keep the N car on the right at the same height? (A) 10 N (B) 100 N (C) 1000 N (D) N

11 Viscosity (η) describes the frictional drag force experienced e.g. between two parallel surfaces with area A, separation d, and relative velocity v F x = η A v x d = η A dv x dy Picture a molasses sandwich... For two large planes, v x of fluid varies linearly with y: fluid speed adjacent to each plane matches speed of that plane.

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13 Remarkably, the total flow (volume/time) through a cylindrical tube (assuming steady irrotational flow, no turbulence) scales as R 4. It s not surprising that it scales as 1/l, but one s first guess might have been that the flow would scale like R 2 instead. But it doesn t, because the fluid speed is largest in the center of the cylinder (where the factor 2πr is smallest).

14 A Lucite block sinks when it is dropped into a bucket of water. Suppose that the same block is supported by a string and slowly lowered (at constant speed) into a bucket of water. How do the tensions in the string compare at the four positions shown? (A) T 1 > T 2 > T 3 > T 4 (B) T 1 < T 2 < T 3 < T 4 (C) T 1 > T 2 > T 3 = T 4 (D) T 1 = T 2 = T 3 = T 4

15 A boat carrying a large boulder is floating on a small lake. The boulder is thrown overboard and sinks. As a result, the water level (with respect to the bottom of the lake) (A) rises (B) drops (C) remains the same Do you want a hint? (Hint on next page.)

16 A boat carrying a large boulder is floating on a small lake. The boulder is thrown overboard and sinks. As a result, the water level (with respect to the bottom of the lake) (A) rises (B) drops (C) remains the same Hint: What volume of water is displaced by floating the boulder inside the boat? What volume of water is displaced by sinking the boulder into the lake?

17 Morin A bead is free to slide along a frictionless hoop of radius R. The hoop is forced to rotate with constant angular speed ω around a vertical diameter. Find H in terms of θ and p θ, then write down Hamilton s equations. Is H the energy? Is H conserved?

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19 Physics 351 Monday, April 27, 2015 Final exam is Wednesday, May 6, from 9am-11am, in DRL A4. You can bring a sheet of your own handwritten notes. Final exam includes Ch7 (Lagrangians), Ch9 (non-inertial frames), Ch10 (rigid-body rotation), Ch13 (Hamiltonians), and will (like midterm) be mainly based on HW problems. You can submit XC work as late as Tuesday, May 12.