1)A physics professor reaches for a well bucket of mass m, the professor hits a lever and the bucket falls into the well. The bucket is suspended from a rope of negligible mass which is wrapped around a sold cylinder of mass M and radius R. The cylinder is mounted on a frictionless horizontal axis over the well. (a) Use the dynamical equations of motion to find an expression for the acceleration of the vucket as it falls down the well shaft, (b) Three seconds after the bicket started its descend, the professor heard the splash of the well ater. If the mass of the bicket is 4 kg and the mass of the cylinder is 12 kg, find the depth of well. (Take the speed of the sound in air as 340 m/s. (c) What is the speed of the buckert just before it hits the water? 2) The cutting blade assembly on a radial-arm saw has a mass of 4 kg. It is pulled along a pair of frictionless horizontal rails aligned with the x-axis by a force F(x). The position of the blade assembly as a function of time is : x=(0.18 m/s^2)t^2 - (0.030 m/s^3)t^3. (a) Draw a free-body diagram for the blade assembly. Identify the source of each force in your diagram. (b) Find the net force acting on the blade assembly as a function of time. (c) Sketch a graph for velocity vs. time - for what values of time is the net force positive? Negative? Zero? 3) Two prolific painters concoct a hair-brain scheme for painting a spherical-shaped observatory dome. One painter has invented a pair of roller skates that will eject paint only when the wheels of the skates are in contact with a surface. The daredevil of the two, wearing the roller skates, starts with negligible velocity at the top of the dome and coasts down over the dome's surface. As long as the wheels are in contact, the surface of the dome will receive a coat of paint. The more sensible painter is positioned with a net some distance from the base of the observatory to break the daredevils fall. The mass of the painter, including the skates, is 80 kg and the radius of the dome is 8 m. A test run is made to demonstrate how much of the dome can be painted in one pass. Assume that the painters mass is constant over this one run. Neglect friction and air resistance. (a) Describe in angles (relative the vertical) the portion of the dome that the daredevil would paint and the portion that would remain unpainted. (b) If the net is removed, what velocity would the painter have when he impacts the ground? (c) What impulse would the ground deliver to the painter in order to stop him? Give the direction relative to the vertical. (d) A person can just survive a full body collision if the average force is less than 72,000 Newtons. If it takes the ground .05 seconds to stop the painter, is it likely he will survive the fall? Explain with physics principles. (e) Where should the assistant place the net from the dome to break the daredevils fall? Identify physics principle to use. 4) An accident occured when the driver of an automobile traveling down an 8% grade slope hit a parked car. A photograph of the accident scene revealed skid marks leading directly to the parked car. Fortunately, no one was injured by this rear end collision. The police officers responding to the call took a statement from the driver, measured the length of the skid marks, and performed a number of skid tests at the accident site. In their final report, the officers reported that the skid marke were 30m long, that the coefficient of friction was .45, and that the driver recalled slamming on her breaks and sliding into the parked car. A footnote in the report indicates that the skid tests were performed over the first 6 m of the skid marks. Based on the evidence, did the driver exceed the 25mph speed limit or should she challenge the incriminating evidence?(a) Assume that the car is a point mass and draw a free-body diagram showing all the forces acting on it as it slides down the hill. (b) Find the expression for the acclereation of the car after the driver slammed on the breaks. (c) The police made the assumption that the coefficient of friction over the first 6m was the same as that of the rest of the skid marks. Using this assumption, calculate the acceleration of the car. How does it compare with the acceleration due to gravity? (d) Find an expression for the time it takes the car to stop after the driver depresses the brakes.(e) Assume an initial velocity of 25 mph, how far would the car have traveled before stopping? (f) If the case goes to court, the judge would want to know the minimum velocity of the car before the driver applied the breaks. Calculate the minimum initial velocity of the car. (g) Based on the evidence in the accident report and the assumptions used by the police, do you recommend the driver be charged with speeding?