1. Most roller coasters are designed with elliptical loops in order to limit the centripetal accelerations experienced by the passengers. The elliptical loops are positioned with the major axis is perpendicular to the Earth. This ellipse has a major axis AM, a minor axis am, an eccentricity of å and the ride is designed to give riders a feeling of weightlessness at the top of the loop. Find the following, expressing all answers in terms of the eccentricity and the major axis.
a) Minimum velocity needed at the top of the loop. so far: Fnormal +mg = (mass x velocity top squared)/radius
b) The radial normal force at the minor axis under the condition of a)
c) Net normal force experienced by the passengers at the minor axis asthey move towards the top of the loop. Discuss the results aseccentricity approaches zero and one.
2. A pirate ship is moving along the coastline, trying to overtake a merchantship making for a harbor. At the rear of the merchant ship flies a flag. You are sure that the ship can be convinced to surrender if you can blow the flagoff with a single shot! You’ll have to set the fuse to the proper length so the cannonball explodes just as it reaches the flagpole. The cannon mounted on the front of your ship has a maximum horizontal range of 950 meters (at 45degrees), so you’ll use it to make the shot.Consider the following in making your calculations: The straight -line distancefrom your cannon to the flag is 650 meters due East. Your ship has a speed of3 knots with respect to the water surface at a heading of 45 degrees south of east. The merchant ship has a speed of 4 knots with respect to the watersurface and is heading northeast. The fuse in the cannonball burns at a rate of2 cm/second. The flag is mounted at the same height as the cannon. Hint:transform to a coordinate frame attached to the cannon
a) You’ll want to keep the trajectory low, so the other ship can’t take evasive action after your shot. If your angle of elevation is to be a maximum of 30 degrees with respect to the water surface, what length of fuse is needed?
b) How long after sighting the ship will you take your shot?
c) With respect to the direction of travel of your pirate ship, at what angle should you aim the cannon when you fire the shot?
You may need physical constant not provided in these problems. Look them up. Show all calculations.Express answers in scientific notation with 3 significant digits of accuracy.
3. In a region of space near a group of charged particles, the potential energy function can be written as a function of distance from the center of the group. In particular, the potential energy may be written asU(x)= x4/5 – 15x3 + 125x2 +100which has units of electron-volts when x is in centimeters.Knowing that the forces exerted by charged particles are conservative, answerthe following questions about an electron in the region within 20 cm of the center of the group
a) At which locations within this range will the forces acting on theelectron be in equilibrium?
b) Which of these are stable equilibrium locations?
c) What is the maximum velocity (m/s)that an electron can have if it is toremain ‘trapped’ within the region?
d) What fraction of the speed of light is this?
