hw5

PHYS 125 Assignment 5 Due by the end of Friday, October 21, 2021 , via Crowdmark. Please submit solutions to each question separately into the respective Crowdmark slot. Each problem will be graded on a 0-3 scale: 0=not close to solving, 1=a reasonable attempt but far from solving, 2=correct approach, but left unfinished or made mistakes, 3=solved RHK ⌘ Resnick, Halliday and Krane; Problem 2-11 means Chapter 2, Problem 11; Exercise 2-11 means Chapter 2, Exercise 11. The relevant pages with RHK problems are attached to this PDF. 1. In Lectures 12 and 13 we discussed the Euler method of solving di ↵ erential equations. a) Write a Python code that evaluates the recursion relations in the posted Lecture 13 and run it on your computer. Your code should give the same results as the one in the worksheet shown in the lecture notes. You can use the python code attached to this PDF as reference. There is a lot of help online on how to run a Python code. b) Make a plot of x ( t ) and v ( t ). Does the time dependence of x and v make sense? c) Try a di ↵ erent number of steps. Namely, instead of nsteps=40, try 5, 20, 100, and compare the plots to the nsteps=40 case. What can you conclude about the accuracy of your solution as you vary nsteps? Submit a printout of your code, a screenshot of how you run it (e.g. in the terminal), and the plots. Note that the sample code in the lectures notes and in this homework are slightly dif- ferent. The code in this homework fixes the initial and final times, so that they do not change when you vary nsteps (which is what you want). The code in the lecture notes fixed dt , so that the final time changes depending on nsteps. 2. RHK Exercise 7-10 on page 155 3. RHK Exercise 7-14 on page 156 4. RHK Exercise 7-15 on page 156 5. A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do the wheels make? 6. What is the linear speed of a point a ) on the equator, b ) on the Arctic circle (latitude 66.5 ◦ N), and c ) at a latitude of 45.0 ◦ N, due to the rotation of the Earth? The radius of the Earth is R E ⇡ 6 . 4 ⇥ 10 3 km. 7. A cooling fan is turned o ↵ when it is running at 850 rev/min. It runs 1500 revolu- tions before it comes to a stop. a ) What was the fan’s angular acceleration, assumed constant? b ) How long did it take the fan to come to a complete stop? 8. Calculate the following scalar and vector products, where ˆ i , ˆ j , and ˆ k are the usual unit vectors of Cartesian coordinates. Comment on your answers. ˆ i · ˆ j ; ˆ i · ˆ k ; ˆ j · ˆ k ; ˆ i ⇥ ˆ j ; ˆ k ⇥ ˆ i ; ˆ j ⇥ ˆ k 9. Given vectors ~ A = 2 ˆ i + ˆ j - ˆ k and ~ B = ˆ i - ˆ j + 2 ˆ k , find ~ A ⇥ ~ B . 1

# creat a file for saving the data file1 = open( "output40.txt" , "w" ) # values of g and terminal velocity g= 9.8 vt= 60 # initial and final time ti= 0 tf= 20 # number of steps nsteps= 40 # time per step dt=(tf-ti)/nsteps # set initial x, v and a x0= 0 v0= 0 a=g*( 1 -v0** 2 /vt** 2 ) # write the initial values on the screen print(ti,x0,v0,a) # save the initial values in the output file file1.write( " " .join([str(ti), str(x0), str(v0),str(a)]) + "\n" ) # start the loop: for n in range( 1 ,nsteps): # find x and v at the next time step x1=x0+v0*dt v1=v0+a*dt # current time in seconds t=n*dt # write the resultson the screen # find the acceleration at the next time step a=g*( 1 -v1** 2 /vt** 2 ) print(t,x1,v1,a) # save the results in the output file file1.write( " " .join([str(t), str(x1), str(v1),str(a)]) + "\n" ) # make "next step" x and v the current x and v x0=x1 v0=v1 # repeat nstep times print( "Finished the job." )

gen/hydrogen atomic mass ratio is 13.9. Locate the center of mass relative to the nitrogen atom. 12. Consider Sample Problem 7-3. The 6.5-kg fragment is observed at s to be moving with a velocity whose horizontal component is 11.4 m/s in the same direction as that of the launch of the original projectile and whose vertical component is 4.6 m/s downward. Find the velocity of the 3.1-kg fragment at that time. 7-4 Center of Mass of Solid Objects 13. Three thin rods each of length L are arranged in an inverted U , as shown in Fig. 7-28. The two rods on the arms of the U each have mass M ; the third rod has mass 3 M . Where is the center of mass of the assembly? t 1.42 nates of the center of mass of the box with respect to the co- ordinate system shown in Fig. 7-30. 7-5 Conservation of Momentum in a System of Particles 16. A vessel at rest explodes, breaking into three pieces. Two pieces, one with twice the mass of the other, fl y off perpen- dicular to one another with the same speed of 31.4 m/s. The third piece has three times the mass of the lightest piece. Find the magnitude and direction of its velocity immediately after the explosion. (Specify the direction by giving the angle from the line of travel of the least massive piece.) 17. Each minute, a special game warden ’ s machine gun fi res 220, 12.6-g rubber bullets with a muzzle velocity of 975 m/s. How many bullets must be fi red at an 84.7-kg animal charging to- ward the warden at 3.87 m/s in order to stop the animal in its tracks? (Assume that the bullets travel horizontally and drop to the ground after striking the target.) 18. A railway fl at car is rushing along a level frictionless track at a speed of 45 m/s. Mounted on the car and aimed forward is a cannon that fi res 65-kg cannon balls with a muzzle speed of 625 m/s. The total mass of the car, the cannon, and the large supply of cannon balls on the car is 3500 kg. How many cannon balls must be fi red to bring the car as close to rest as possible? 19. Twelve 100.0-kg containers of rocket parts in empty space are loosely tethered by ropes tied together at a common point. The center of mass of the twelve containers is originally at rest. A 50-kg lump of “ space-goo ” moving at 80 m/s collides with one of the containers and sticks to it. ( a ) Assuming that none of the tethers break, fi nd the speed of the center of mass of the twelve containers after the collision with the space goo. ( b ) Assuming instead that the tether of the struck container does break, fi nd the speed of the center of mass of the twelve containers after the collision. 7-6 Systems of Variable Mass 20. A rocket at rest in space, where there is virtually no gravity, has a mass of 2.55 10 5 kg, of which 1.81 10 5 kg is fuel. The engine consumes fuel at the rate of 480 kg/s, and the ex- haust speed is 3.27 km/s. The engine is fi red for 250 s. ( a ) Find the thrust of the rocket engine. ( b ) What is the mass of the rocket after the engine burn? ( c ) What is the fi nal speed attained? 21. Consider a rocket at rest in empty space. What must be its mass ratio (ratio of initial to fi nal mass) in order that, after fi r- ing its engine, the rocket ’ s speed is ( a ) equal to the exhaust speed and ( b ) equal to twice the exhaust speed? 22. During a lunar mission, it is necessary to make a midcourse correction of 22.6 m/s in the speed of the spacecraft, which is moving at 388 m/s. The exhaust speed of the rocket engine is 1230 m/s. What fraction of the initial mass of the spacecraft must be discarded as exhaust? 23. A rocket of total mass 1.11 10 5 kg, of which 8.70 10 4 kg is fuel, is to be launched vertically. The fuel will be burned at the constant rate of 820 kg/s. Relative to the rocket, what is the minimum exhaust speed that allows liftoff at launch? 24. A 5.4-kg toboggan carrying 35 kg of sand slides from rest down an icy slope 93 m long, inclined 26 ° below the horizon- tal. The sand leaks from the back of the toboggan at the rate of 2.3 kg/s. How long does it take the toboggan to reach the bottom of the slope? 25. A freight car, open at the top, weighing 9.75 metric tons, is coasting along a level track with negligible friction at 156 Chapter 7 / Systems of Particles Figure 7-28. Exercise 13. Figure 7-29. Exercise 14. Figure 7-30. Exercise 15. L L L M M 3 M 14. Fig. 7-29 shows a composite slab with dimensions 22.0 cm 13.0 cm 2.80 cm. Half of the slab is made of aluminum (density 2.70 g/cm 3 ) and half of iron (density 7.85 g/cm 3 ), as shown. Where is the center of mass of the slab? 11 cm 11 cm 13 cm 22 cm 2.8 cm Aluminum Iron 15. A box, open at the top, in the form of a cube of edge length 40 cm, is constructed from thin metal plate. Find the coordi- 40 cm 40 cm 40 cm O z y x