bbexplain all answers in all parts

Problem 4 (Conservation of Angular Momentum) A meteor of mass m = 2.1 1013 kg is approaching earth as shown on the sketch. The radius of the earth is r = 6.37 106 m . The mass of the earth is m = 5.981024 kg . Suppose the meteor e e has an initial speed of v0 =1.0101 m s . Assume that the meteor started very far away from the earth. Suppose the meteor just grazes the earth. The initial moment arm of the meteor ( h on the sketch) is called the impact parameter. You may ignore all other gravitational forces except the earth. The effective scattering area for the meteor is the area h2 . This is the effective target size of the earth as initially seen by the meteor. a) Draw a force diagram for the forces acting on the meteor. b) Can you find a point about which the gravitational torque of the earth's force on the meteor is zero for the entire orbit of the meteor? c) What is the initial angular momentum and final angular momentum (when it just grazes the earth) of the meteor? d) Apply conservation of angular momentum to find a relationship between the meteor's final velocity and the impact parameter h . e) Apply conservation of energy to find a relationship between the final velocity of the meteor and the initial velocity of the meteor. f) Use your results in parts d) and e) to calculate the impact parameter and the effective scattering cross section.

Problem A piranha-infested river runs from west to east, as depicted in Figure 2.4. Sarah Connor is living off the grid 3 km north of the river and does not have access to fresh water. Following a recent impaling, she is unable to walk. Her survival is vital for the future of humanity. Each day Kyle Rees, who lives 2 km north of the river and 12 km west of Sarah Connor, must travel from his hideout with a bucket, which he fills with water from the river. He must save as much energy as possible for the fight against the machines. What is the minimum distance he needs to travel to get to Sarah's house via the river?

9. Problem A person travels from Newcastle to Oxford by coach. Traffic is free-flowing and the coach's speed is only limited by whether the road is flat (63 mph), uphill (56 mph) or downhill (72 mph). The coach ride takes 4 hours from Newcastle to Oxford, but the return journey, which follows the same roads, takes an hour longer. How many miles is the coach ride between Newcastle and Oxford? Hint Split the journey into 3 sections of length x, y and z. What happens to the downhill sections on the return journey? What are you trying to find in terms of x, y and z? Do you care what each of them are? The numbers have been chosen carefully - can you tidy up awkward fractions?

10. Problem It's finally time for a battle to end the long-running dispute between pirates and ninjas. They face off at 90 m. The pirate limps towards the ninja at 2 m s1 , while the ninja glides towards the pirate at twice the speed. It is only a matter of time before they collide and crush the loyal parrot which repeatedly flies back and forth at a constant speed of 8 m s1 , elastically bouncing off the two. What is the total distance the parrot travels before being crushed? Hint One approach would be to find the positions where the parrot changes direction. Since we are given the speed, but asked to find distance, what else would it be useful to know?

Problem Set 11: Due Tues Nov 23 at 4:00 pm. Problem 1: (Moment of Inertia) A 1" US Standard Washer has inner radius r 1 =1.35102 m and an outer radius 3 r2 = 3.10 102 m . The washer is approximately d = 4.010 m thick. The density of the washer 3 3 is = 7.810 kg m . Calculate the moment of inertia of the washer about an axis passing through the center of mass and show that it is equal to 2 I = 1 m r0 + ri 2 ).

Part Two: Added Mass Consider the effect of a brass weight clipped to the ruler. The weight is shaped like a washer with an outer radius ro = 0.016 m and an inner radius ri = 0.002 m; it has a mass mw = 0.050 kg. It is clipped to the ruler so that the inner hole is over the 0.500 m mark on the ruler, or l = 0.479 m from the pivot point. The clip has a mass mc = 0.0086 kg and you may assume its center of mass is also over the 0.500 m mark on the ruler. If you treat the washer and clip as point masses, 3 then, as was discussed in the notes for Experiment 08, the combined unit (ruler, weight and clip) has a moment of inertia about the pivot point r 2 2 IP = m (a + b2 ) + m l + (m + m )d 2 r c w 12 where d = l for this situation. The restoring torque that tries to return the pendulum to a vertical position will be = (ml r c w + m d m d g + ) sin (ml r c w + m d m d g + ) 1. Use these two expressions to derive an equation of motion for the pendulum and calculate its period T in the small amplitude (sin ) approximation. Express your answer algebraically in terms of the variables a, b, d, l, mr, mw, mc, and g. 2. Evaluate your result numerically and compare with the value you measured in your experiment. 3. If you treat the brass object washer as a point mass, its moment of inertia about the pivot point P is Iw,P = mwl 2 . If the brass object is a washer with an inner radius ri and outer radius r , then moment of inertia about its center of mass given by I = 1 mr + ri 2 ) . If o w 2 w ( o 2 the washer is a solid disc with radius r , the moment of inertia about its center of mass 2 given by I = 1 mr .When this is taken into account, what is the new (and more w 2 w accurate) expression for Iw,P ? How many percent does this differ from the simpler expression Iw,P = mwl 2 ?

Problem 3: Stall Torque of Motor The following simple experiment can measure the stall torque of a motor. (See sketch.) A mass m is attached to one end of a thread. The other end of the thread is attached to the motor shaft so that when the motor turns, the thread will wind around the motor shaft. The motor shaft without thread has radius r0 =1.2 103 m . Assume the thread winds evenly effectively increasing the radius of the shaft. Eventually the motor will stall.

a) Suppose a mass m = 5.0 102 kg stalls the motor when the wound thread has an outer radius of rf = 2.4 103 m . Calculate the stall torque. b) Suppose the motor has an unloaded full throttle angular frequency of 1 = 2 f0 = 2 (6.0 10 Hz) (unloaded means that the motor is not applying any 0 torque). Suppose the relation between angular frequency and the applied torque of the motor is given by the relation =0 b where b is a constant with units 2 s kg m . Using your result from part a), calculate the constant b . Make a graph of vs. .

c) Graph the power output of the motor vs. angular frequency . At what angular frequency is the power maximum? What is the power output at that maximum? Briefly explain the shape of your graph. In particular, explain the power output at the extremes = 0 and = stall . d) What torque does the motor put out at its maximum power output?

Problem 4 (Conservation of Angular Momentum) A meteor of mass m = 2.1 1013 kg is approaching earth as shown on the sketch. The radius of the earth is r = 6.37 106 m . The mass of the earth is m = 5.981024 kg . Suppose the meteor e e has an initial speed of v0 =1.0101 m s . Assume that the meteor started very far away from the earth. Suppose the meteor just grazes the earth. The initial moment arm of the meteor ( h on the sketch) is called the impact parameter. You may ignore all other gravitational forces except the earth. The effective scattering area for the meteor is the area h2 . This is the effective target size of the earth as initially seen by the meteor. a) Draw a force diagram for the forces acting on the meteor. b) Can you find a point about which the gravitational torque of the earth's force on the meteor is zero for the entire orbit of the meteor? c) What is the initial angular momentum and final angular momentum (when it just grazes the earth) of the meteor? d) Apply conservation of angular momentum to find a relationship between the meteor's final velocity and the impact parameter h . e) Apply conservation of energy to find a relationship between the final velocity of the meteor and the initial velocity of the meteor. f) Use your results in parts d) and e) to calculate the impact parameter and the effective scattering cross section.