PHYSICS 218 Final Exam SPRING, 2015 Do not fill out the information below until instructed to do so! Name: Signature: Student ID: E-mail: Section Number: You have the full class period to complete the exam. Formulae are provided. You may NOT use any other formula sheet. When calculating numerical values, be sure to keep track of units. You may use blank pages provided with this packet as scratch paper or come up front to get more. Be sure to put a box around your final answers and clearly indicate your work to your grader. All work must be shown to get credit for the answer marked. If the answer marked does not obviously follow from the shown work, even if the answer is correct, you will not get credit for the answer. Clearly erase any unwanted marks. No credit will be given if we can t figure out which answer you are choosing, or which answer you want us to consider. Partial credit can be given only if your work is clearly explained and labeled. Put your initials here after reading the above instructions: 1
SCORE Part 1 (20) Part 2 (20) Part 3 (30) Part 4 (30) Part 5 (30) Part 6 (20) Part 7 (20) Part 8 (30) Total (200) 2
Problem 1 (20 points): A hunter notices a duck when it flies over his head. The duck flies with a constant velocity V=5 m/s. The hunter immediately positions the rifle to align with the direction of duck s motion and pointing upwards at the angle of 60 degrees relative to the horizontal. The duck flies at the height H=30 m above the tip of the hunter s gun. Velocity of the bullet as it leaves the barrel is v 0 = 300 m/s. The hunter then waits until the duck reaches the point where he gets a good shot at the target. (15 pts) What should be the minimal horizontal distance D (see the Figure below) between the tip of the rifle and the duck at the time when the hunter should fire his gun in order to hit the duck? V v o θ D H (5 pts) The gun misfires and the duck continues to fly horizontally with the same velocity. Actually, in this situation the hunter has a second chance to hit the dick without having to reposition the gun. What should be the distance D between the tip of the gun s barrel and the duck when the hunter should take his second shot? 3
Problem 2 (20 points): A crate of mass M is placed on the bed of a truck and is secured using two ropes, R 1 and R 2 (see Figure), to ensure the crate remains in place in case of rapid acceleration and deceleration of the truck. An accident can cause a particularly large rate of deceleration, which can result in breakage of rope R 2. Each of the two ropes is making an angle of 30 degrees with respect to the horizontal. The coefficients of static and kinetic friction between the crate and the bed of the truck are equal to each other µ s =µ k =0.3. R 1 R 2 (5 pts) Draw the force diagram for the crate in case of a rapid deceleration of the truck (15 pts) Calculate the minimum amount of tension T min that the rope R 2 should be able to withstand for the crate to stay in place in the event of an accident. The maximum magnitude of truck s deceleration in case of an accident is a=10 m/s 2. 4
Problem 3 (30 points): On a horizontal surface, a crate of mass M is attached to an uncompressed spring with spring coefficient k, the other end of which is attached to a vertical wall. The coefficient of friction between the crate and the surface is µ. A bullet of mass m is shot horizontally into the center of the crate (see Figure below). Following the impact, the crate oscillates many times (more than 300) around the equilibrium position before finally coming to rest. It has been measured that, once the crate started moving following the impact, it has moved distance d from the equilibrium position to the right before changing its direction of motion. Assuming that the bullet embeds itself into the crate instantaneously, find the following: (10 pts) Velocity of the bullet right before it hits the crate. (10 pts) Velocity of the crate when it passes through the equilibrium position the first time after it left the equilibrium position following the impact. (10 pts) What is the total scalar distance traveled by the crate before it comes to its final rest (for example, if the crate were to move to the right a distance d 1, then return to and pass through the equilibrium position, and move to the left a distance d 2 before coming to rest, the total scalar distance would be calculated as a scalar sum d 1 +d 1 +d 2 ) 5
Problem 4 (30 points): A pickup truck has a wheelbase d (this is the distance between front and rear axles). Ordinarily, weight W F rests on the front wheels and W R on the rear wheels when the truck is parked on a level road. A box weighing W B is now s placed on the tailgate, with its center of mass positioned distance s behind the rear axle. (10 pts) How much total weight now rests on the front wheels? d (10 pts) How much total weight now rests on the rear wheels? (10 pts) How much weight would need to be placed on the tailgate to make the front wheels come off the ground? 6
Problem 5 (30 points): You are spinning a bucket full of water attached to a string in the vertical plane. The length of the string is such that the distance between the center of rotation and the bucket s center of mass is L. the mass of the bucket is m B and the mass of the water in it is m W. Acceleration due to gravity g is given. (15 pts) What should be the minimal velocity v B of the bucket at the bottom of the trajectory so the bucket with water safely makes it through the top point without spilling water onto your head? (Hint: as a first step, you may want to find the required speed at the top of the trajectory for the water not to spill) (5 pts) Assuming you maintain the stable motion required to just avoid having water spilled on you, at which point in the trajectory does the tension in the string reach its maximum value? (10 pts) What is the magnitude of tension T at that point? You can express your answer in terms of v B from the first question 7
Problem 6 (20 points): The mechanism shown in the figure is used to raise a crate of supplies from a ship's hold. The crate has total mass m. A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius R and a moment of inertia I about the axle. The crate is suspended from the free end of the rope. One end of the axle pivots on frictionless bearings; a crank handle is attached to the other end. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius r, the cylinder turns, and the crate is raised. Force F is applied tangentially to the rotating crank to raise the crate. Find the acceleration of the crate. (You can ignore the mass of the rope as well as the moments of inertia of the axle and the crank.) r 8
Problem 7 (20 points): An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of h=100 km above the lunar surface. To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 10% (so that v new =90% x v 0 ). If nothing is done to correct its orbit, with what speed will the spacecraft crash into the lunar surface? (Take the following to be given: mass of the moon M=7.3x10 22 kg, its radius R=1737.5 km, mass of the spacecraft m=100 kg, gravitational constant is G=6.67x10-11 Nm 2 /kg 2 ) 9
Problem 8 (30 points): Two solid cylinders connected along their common axis by a short, light rod have radius R=5cm and total mass M=1 kg and rest on a horizontal tabletop. A spring with force constant k=300 N/m has one end attached to a clamp and the other end attached to a frictionless ring at the center of mass of the cylinders. The cylinders are pulled to the left a distance x=1 cm, which stretches the spring, and then the system is released. There is sufficient friction between the tabletop and the cylinders for the cylinders to roll without slipping as they move back and forth on the end of the spring. (10 pts) Show that the motion of the center of mass of the cylinders is simple harmonic and find angular frequency w. [Hint: The motion is simple harmonic if a and x are related by a= - w 2 x, where w is a constant. Apply Στ=I cm α and ΣF=Ma to the cylinders in order to relate acceleration a and the displacement x of the cylinders from their equilibrium position.] (10 pts) Write the equation of motion for the center of mass of the system of the cylinders, i.e. essentially you need to find A and φ in x(t)=a cos(wt+φ) (10 pts) Use equations of motion to calculate speed v of the center of mass of the system of the cylinders as it moves through the equilibrium point. Your answer can contain w, A, and φ referred in the above questions. 10