HW 3. Due: Tuesday, December 4, 2018, 6:00 p.m.

Transcription

1 Oregon State University PH 211 Fall Term 2018 HW 3 Due: Tuesday, December 4, 2018, 6:00 p.m. Print your full LAST name: Print your full first name: Print your full OSU student ID#: Turn this assignment in to box #15 (located outside of Wngr 234) OR a pdf document of the complete file, prepared as directed on the next page. I affirm and attest that this HW assignment is my own work. All the reasoning and results here are my own doing. Sign your name (full signature): Print today s date:

2 Oregon State University PH 211 Fall Term 2018 HW 3 Directions: Solve these four problems fully, just as you would if they were exam problems. Then turn in your entire solutions set (all these pages) to Box 15; OR a.pdf file to coffinc@physics.oregonstate.edu with a subject line as follows: HW3_LastName_FirstName Parts of your four solutions will be chosen and scored for HW3 credit (35 points). Full solutions to all four problems will be posted soon after the due date/time. Note: You ll do much better with these problems if you work through the Prep 9-10 sets and also the SMT3 first, because these problems are all variations on certain Prep and SMT3 problems. Note also: The real MT3 (that s the final exam) may also include variations on Prep and SMT3 problems.

3 1. Refer to the diagram. A platform (m P ) is designed to test various equipment for durability for a future mission to Mars. The platform can be accelerated by a powerful piston so that it slides along a ramp inclined at angle q. To simulate space conditions, the whole apparatus (the ramp, platform and piston) is housed in a long airless tube (not shown). But all the testing is done here on the earth s surface, where g = 9.80 m/s 2. piston platform q ramp Before testing any equipment, of course, the researchers must test the platform-piston assembly itself, as follows: The platform is initially at rest. Then at t = 0 s, it is pushed up the ramp by the piston (whose force is parallel to the ramp). The platform s acceleration (along the ramp surface) is a(t) = k( t) + 8, where a is in m/s 2 and t is in s. The other data for this test: m P = 3.62 kg q = 17.0 µ S.RP = µ K.RP = k = 4.15 m s 5/2 a. For this test: Calculate the (instantaneous) power being delivered to the platform by the piston at t = 2.10 s. b. For this test: Calculate the total average power that the platform has received from all external forces (so exclude the gravitational force) during the entire interval 0 t 2.10 s. Comments/hints: For part a, note that instantaneous power is always P(t) = F(t) v(t). In this case, with F piston.platform and v platform in the same direction, the dot product simplifies: P piston.platform (t) = [F piston.platform (t)][v platform (t)] Then Part b is asking about average power but not just by the piston; rather by all external forces. By definition, that s P ext.avg = W ext.total /D, but note that there is more than one way to calculate W ext.total over the time interval in question. One way is simply to integrate the instantaneous external power. That is, since P ext = dw ext /dt, W ext.total = P ext dt. But what else do we always know about W ext? (Recall Lab 8-III...)

4 2. You step into an elevator on the 44th floor of a tall office building. Each floor is 3.50 m above the one below it. You re alone in the elevator. You select your destination floor, then as you wait for the door to close you notice: There s a bathroom scale with a mass of 2.43 kg resting on the elevator floor. (Of course.) So you stand on the scales during the entire ride. (Of course.) Shown here is a time graph of the scale s reading of your weight during the ride. Weight (N) t (s) The elevator is guided along vertical rails but is supported essentially by a single cable. F T (N) 14,056 Shown here is a time graph of that cable s tension, F T, during the ride. 11,996 11,900 10, (For simplicity, treat the ends of both of these graphs as infinite-slope discontinuities.) Assume g = 9.80 m/s 2. Use an energy analysis to find the average total resisting force (which is probably a combination of friction by the rails and air drag) that was exerted on the elevator during your ride. (You may be able to confirm or check your result by other methods, too good idea but use energy for your main analysis.) Comments/hints: I purposely repeated the same numbers and graph from part b of the elevator problem in HW 2 (that s the first graph above), so you can use/extend that solution any way you wish just show your work, as usual. As for the issue of calculating the average resisting force, it may be tempting to just note the difference between the second graph and what you d expect from a no-resistance graph (i.e. something like the first graph) and then do something like a weighted average. But that may get you only an approximation hence my note about checking your actual result; the approximation can be a check to see what general range of value to expect. So... what s the actual solution? As always the total external work done on the elevator is the sum of the work done by all the external forces. The resisting force (a combination of rail friction and air drag is one such force; the cable force, F T, is the other). No way around it you ll have to use the second graph above to calculate the work done by the cable on the elevator. And again, note: P T = dw T /dt, so W T.total = P T dt Watch your signs and define the three-part F T (t) carefully from the graph above. 20 t (s)

5 3. Refer to this diagram: A B Mass A (6.23 kg) is attached to the left end of an ideal (massless) spring that has a stiffness of 27.4 N/m. Mass B (4.56 kg) is not attached to anything. Both masses are supported by (and can slide freely on) a level, frictionless surface. There is no air drag nor any other resistance losses. Initially: Mass A is at rest; the spring is relaxed; and mass B is moving to the left (toward mass A) at 1.53 m/s. Then mass B collides with the right (free) end of the spring. Sometime later, mass B ceases contact with the spring. Assume that no mechanical energy is lost during the collision (i.e. that this is a perfectly elastic collision) because the spring is essentially massless. a. How far is the spring is compressed when the two masses (and spring) are moving at a single common speed? b. Find the velocity of mass A after mass B is no longer in contact with the spring. c. Now refer instead to this diagram, where there are two changes to the above scenario. A B Now the surface under mass A is not frictionless (but the surface under B is still frictionless) Now the spring is still ideal (perfectly elastic) but not a linear restoring force (does not exhibit Hooke s Law). Rather, the force it exerts when compressed by a distance x is given by F = cx 3/2, where c = 18.7 N m 3/2 With those two changes, suppose that the collision experiment is repeated (same masses; same initial velocity of mass B). Assuming g = 9.80 m/s 2, what is the minimum coefficient of static friction that must exist between mass A and the surface under it so that A does not move during B s collision with the spring? Comments/hints: Here s a rare case idealized where the collisions can be modeled as completely elastic no loss of mechanical energy from before to after. So now, for each collision, there are two facts that you can use rather than just one: no change of linear momentum (that s the usual one); and no change of mechanical energy (that s the unusual one). Then in part c, Hooke s Law does not apply, so U S (1/2)kx 2. You ll have to re-derive the amount of U S stored by the spring when it s stretched or compressed by a given amount x. Recall how we originally derived U S = (1/2)kx 2.

6 4. (More practice with energy graphs...) a. The graph here shows the total potential energy that a certain object (m = 4.00 kg) would have along the r-axis. Assuming that no external work is done (i) What minimum speed must the object have at r = 4.00 m in order to reach r = 14.0 m? U (J) (ii) Assuming the result of step (i) above, 20.0 find the object s speed at r = 14.0 m. r (m) (iii) Now assume that the actual speed of the object at r = 4.00 m is 10% greater than the value you calculated in step (i) above. Use two different methods to find the average r-acceleration of the object as it travels from r = 10.0 m to r = 12.0 m. Comments/hints: For part b, reading the graph s slope is all well and good, but don t forget how else we can calculate average acceleration...

7 b. Refer to the diagram (not to scale) Starting from rest, a block of mass m slides down a slope angled at q below the horizontal. There is kinetic friction (µ K ) between the block and the slope. There is also a steady wind blowing horizontally to the left, producing a steady (unknown) air drag force F D, as indicated. When the block has traveled a distance d to the bottom of the slope, it has a speed v. Known data: m = 7.80 kg m K = q = 19.5 d = 4.36 m v = 2.34 m/s Assumptions: Define the x-axis as parallel to the sloped surface, with the origin at the block s starting point. Let U G = 0 at the bottom of the slope. Assume g = 9.80 m/s2. Show all reasoning and calculations for each step below. (i) Use axes such as those below to draw and label the graphs of E mech.total and U G (both on the same axes). For two points on each graph (where x = 0 and x = d), calculate and label the values of each graph. E mech (J) U G = 0 m K m F D q d x (m) (ii) Calculate the slope of your graph of U G. Explain how this result makes sense. (iii) Calculate the slope of your graph of E mech.total. Explain how this result makes sense. (iv) Explain how your graphs show that the block accelerated as it moved down the sloped surface. Comments/hints: Be sure to show and make sense of the units of your results.