Coefficient of Kinetic Friction - Physics - Solved Paper, Exams of Physics

Download Coefficient of Kinetic Friction - Physics - Solved Paper and more Exams Physics in PDF only on Docsity! 1 PHYS 1121 Test 1, 2005 Question 1 (23 marks) a) Investigators at the scene of an accident see that a car has left black rubber marks ("skid marks") that are L = 22 m long on a flat section of road. The car is stationary at one end of the marks, and it is assumed that the car began to skid at the other end. The coefficient of kinetic friction between the rubber and the wheels is µk = 0.80 and the skid marks show that all four wheels begin to skid simultaneously. Calculate the speed of the car at the beginning of the skid. Express your answer in kilometres per hour. b) A bird flies at speed vb = 5.0 m.s-1 in a straight line that will pass directly above you, at a height h = 5.0 m above your head. You are eating grapes and it occurs to you that the bird might want one and so you decide to throw it a grape. Of course, you don't want to hurt the bird, so you will throw the grape so that, at some time t, it has the same position, same height and same velocity as the bird. (Hint for 1221: what will be the height and velocity of the grape when the bird takes it?) You throw the grape from a position very close to your head, with intial speed v0 and at an angle θ to the horizontal. Air resistance is assumed to be negligible. i) Should the bird be behind you, or ahead of you when you throw the grape, and by how much? Explain your answer briefly. (3-5 clear sentences should suffice.) ii) Calculate the required values of vt and θ. iii) If air resistance on the grape were not negligible, how would that change your answer to (i)? A qualitative but explicit answer is required. Question 1 a) Normal force N, friction Ff. In the vertical direction, ay = 0, so N = mg In the horizontal direction Ff = µkN = µkmg = max, so |ax| = µkg vf2 − vi2 = 2aL = − 2µkgL but vf = 0, so vi2 = 2µkgL vi = √2µkgL = √2*0.80*9.8m.s-2*22m = 18.6 ms-1 = 67 kph. b) i) There is no air resistance. Neither the bird's nor the grapes horizontal speed changes, so, if they have the same horizontal speed, they always have it. If they ever have the same horizontal position, they must always have it. So you must throw it when the bird is directly overhead. ii) From (i), vxg = v0 cos θ = vb. (a) Vertical motion under gravity, measured from y = 0 at the position of the head: vy2 = vy02 + 2ay At y = h, vy = 0, so 0 = v02sin2θ − 2gh v0 sinθ = √2gh (b) (b)/(a) → tan θ = √2ghvb so θ = tan-1 √2ghvb = tan-1 √2*9.8ms -2*5m 5 m.s-1 = 63° (a) → v0 = vb cos θ = 5 m.s-1 cos 63° = 11 ms -1. iii) Air resistance would slow the grape during flight. The grape would have greater horizontal velocity until the end of its flight, so it would cover the distance from you to bird faster than the bird would, so you would throw it after it passed overhead. 2 Question 2 (10 marks) R bucket brick v i) A physics lecturer swings a bucket in a vertical circle, about his shoulder, as shown. It executes circular motion with period T. The bucket contains a brick. Derive an expression for the maximum period T that the motion can have in order that that the brick stay in contact with the bucket. Assume that the motion has constant angular velocity. ii) Put in appropriate values to give a numerical estimate of the period. iii) Is the assumption of constant angular velocity reasonable? Comment briefly. Question 2 ( marks) R v N W i) If the brick is in contact with the bucket, then both are travelling in a circle with speed v. The centripetal acceleration is ac = v2 R down. Newton's second law for the vertical direction gives N + W = mac = mRω2 = 4π2mR T2 To remain in contact, N ≥ 0 so 4π2mR T2 − W ≥ 0 so 4π2mR T2 ≥ mg so T2 ≤ 4π2R g or T ≤ √4π2Rg ii) Put R = 0.8 m (any value between 0.5 and 1 m is okay) T ≤ ~ 2 s. iii) The bucket is likely to slow down while ascending and accelerate while descending. (So the period should be rather less than thisto have a margin of security.) 5 Not for marks: What is interesting, of course, is that at new moon the moon is actually accelerating in the direction away from the Earth and towards the sun—although not as quickly as the Earth is accelerating towards the sun. The earth travels in a nearly circular path around the sun, with approx constant radius R and approx constant centripital acceleration. The moon's path is not as circular as the Earth's: it is closer to the sun (R-r) at new moon and further from the sun (R+r) at full moon. Because r << R, it's actually hard on this scale to show that the moon's orbit is always concave towards the sun, so that's why a diagram was not called for in this question. sun earth orbit moon orbit full moon new moon 6 Question 4 (13 marks) x M m v bathroom scales Can a bathroom scale (a device usually used for measuring one's weight) be used to measure the speed of a bullet fired from a gun? A student decides to find out. When she stands on the scale, it accurately reads her mass (60 kg). She observes that, when she stands on the scale, its lid is lowered by 5.0 mm. Assume that the scale behaves like an undamped spring, with spring constant k. i) Calculate the value of the spring constant k. (Hint: be careful with units.) The student then mounts the scale vertically, and fixes a block (M = 10 kg) on its surface. Its mass is considerably greater than that of the scale. In this orientation, and with the block fixed, the scale reads zero. In a preliminary experiment, she discovers that the bulet does not penetrate through the block, and comes to rest inside it. Her research tells her that a particular model gun fires bullets at a speed of v = 400 m.s-1 (called its muzzle velocity) and that the bullets have a mass m = 6.0 g. ii) Showing all working, and using the values given, calculate the maximum compression of the scale when a bullet is fired into it at normal incidence (as shown in lower diagram). State any assumptions you make and justify any conservation laws that you use. iii) Calculate the reading on the scale at this point. (Under no circumstances should you try to answer this problem experimentally.) i) The weight of a 60 kg person is 590 N. So 590 N depresses the "spring" by 5 mm, so the spring constant is k = |F|/x = 120 kN/m. (deduct 2 marks from anyone who uses 50 N as the force.) ii) mscale< M, so the force to accelerate part of it is small so, the external horizontal forces acting on the block and bullet are negligible, so their total momentum will be conserved during their collision. The block + bullet has mass M = 10 kg + 6 g ≅ 4 kg. Let it travel at V, so pi = pf mv = (M + m)V ≅ MV, so V = mv/M In the compression of the spring in the scale, external forces do negligible work (because it is an undamped spring). At maximum compression, the block is stationary. Assume that the mechanical energy of the bullet+block is converted into potential energy of the "spring", so 1 2 MV 2 = 12 kx 2 x = V√ Mk = mvM √ Mk = mv√Mk = .006kg*400m/s√10kg*120kN/m = 2.2 mm. iii) for the spring, |F| = kx, so if 60 kg produces a deformation of 5 mm, 2.2 mm will read a "weight" of 27 kg. 7 Question 5 (15 marks) The Australian Grand Prix has been cancelled. You decide to offer an alternative event. start finish sphere spherecylinder D C B A axis of cylinder The contestants are two identical brass spheres, a brass cylinder (whose axis is horizontal so it can roll), and a toy racing car. All have the same mass. The wheels of the car are light and they turn with negligible friction on the axle. The objects roll down four tracks, which are shown in cross section in the top sketch. The tracks are straight, but inclined downwards (all at the same angle). One of the tracks is narrower than the sphere (D) on it, as shown. The friction between the track and the objects is sufficiently high that the sphere, cylinder and wheels all roll. Air resistance and other losses are negligible. They race in pairs, and are released from rest at the same time. You may use without proof Isphere = 2 5 mR 2 and Icylinder = 1 2 mR 2 i) In the first race, only A and B compete. Which will win? Explain your answer. (You may use equations if you like, but this is not required. A few clear sentences could be enough.) Hint: it may be helpful to state some general principles that will be relevant to all of (i), (ii) and (iii). ii) In the second race, B and C compete. Which will win? Explain your answer. (Here you probably will need an equation or two, plus some explanation.) iii) In the third race, C and D compete. Which will win? Explain your answer. (You may use equations if you like, but this is not required. A few clear sentences could be enough.)