Mechanics Problems - Force and Circular Motion at a Constant Speed Problems

No Radial Force Components

Note: Each problem begins with a list of forces necessary to solve the context-rich problem. These are for the benefit of the instructor. Delete the list before using the problems in your class.

1. Weight, Normal: Just before finals you decide to visit an amusement park set up in the Metrodome. Since it is a weekend, you invite your favorite niece along. She loves to ride on a Ferris wheel, and there is one at the amusement park. The Ferris wheel has seats on the rim of a circle with a radius of 25 m. The Ferris wheel rotates at a constant speed and makes one complete revolution every 20 seconds. While you wait, your niece who has a mass of 42 kg, rides the Ferris wheel. To kill time you decide to calculate the total force (both magnitude and direction) on her when she is one quarter revolution past the highest point. Because the Ferris wheel can be run at different speeds, you also decide to make a graph which gives the magnitude of the force on her at that point as a function of the period of the Ferris wheel.
   
   
2. Weight, Normal: While watching TV, you see a circus show in which a performer drives a motorcycle around the inside of a vertical ring. You wonder how far the cycle would fall if the rider made a mistake at the top of the loop and fell off the track and decide to calculate it. You determine that the cycle is going around at a constant speed and that it takes at most 4.0 seconds to get around the ring. At this speed, the motorcycle just barely loses contact with the ring at the top when it is upside down and is in free fall.
   
   
3. Weight, Normal, Friction: The producer of the last film you worked on was so impressed with the way you handled a helicopter scene that she hired you again as technical advisor for a new "James Bond" film. The scene calls for 007 to chase a villain onto a merry-go-round. An accomplice starts the merry-go-round rotating in an effort to toss 007 (played in this new version by Billy Crystal) off into an adjacent pool filled with hungry sharks. You must determine a safe rate of rotation such that the stunt man (you didn't think Billy would do his own stunts did you?) will not fly off the merry-go-round and into the shark-infested pool. (Actually they are mechanical sharks, but the audience doesn't know that.) You measure the diameter of the merry-go-round as 50 meters. You determine that the coefficient of static friction between 007's shoes and the merry-go-round surface is 0.7 and the coefficient of kinetic friction is 0.5.
   
   
4. Weight, Normal, Friction: A new package moving system in the new, improved post office consists of a large circular disc (i.e. a turntable) which rotates once every 3.0 seconds at a constant speed in the horizontal plane. Packages are put on the outer edge of the turntable on one side of the room and taken off on the opposite side. The coefficient of static friction between the disc surface and a package is 0.80 while the coefficient of kinetic friction is 0.60. If this system is to work, what is the maximum possible radius of the turntable?
   
   
5. Weight, Normal, Friction: You are driving with a friend who is sitting to your right on the passenger side of the front seat. You would like to be closer to your friend and decide to use your knowledge of physics to achieve your romantic goal. So you'll make a sharp turn. Which direction should you turn so as to make your friend slide closer to you? If the coefficient of static friction between your friend and the seat of the car is 0.40, and you drive at a constant speed of 18 m/s, what is the maximum radius you could make your turn and still have your friend slide your way?
   
   
6. Weight, Normal, Friction: During a freeway safety review, you are studying a piece of one road with a curve that is essentially 1/8 of a circle with a radius of 0.5 miles. The curve is banked so that the road makes an angle of 4 degrees to the horizontal throughout the curve. Your boss asks you to help determine the speed limit for a standard passenger car (about 2000 lbs) to complete the turn. You decide to start by considering the worst-case scenario, a slick, ice-covered road, and finding the constant speed a car must travel in order to maintain a horizontal path through the turn.
   
   
7. Weight, Tension: After watching the movie "Crocodile Dundee," you and some friends decide to make a communications device invented by the Australian Aborigines. It consists of a noise-maker swung in a vertical circle on the end of a string. Your design calls for a 400 gram noise-maker on a 60 cm string. You are worried about whether the string you have will be strong enough, so you decide to calculate the tension in the string when the device is swung with an acceleration which has a constant magnitude of 20 m/s2 . You and your friends can't agree whether the maximum tension will occur when the noise maker is at the highest point in the circle, at the lowest point in the circle, or is always the same. To settle the argument you decide to calculate the tension at the highest point and at the lowest point and compare them.
   
   
8. You are watching a TV news program when they switch to some scenes taken aboard the space shuttle which circles 500 miles above the Earth once every 95 minutes. To allow the audience to appreciate the distances involved, the announcer tells you that the radius of the Earth is about 4000 miles and the distance from the Earth to the Moon is about 250,000 miles. When an astronaut drops her pen it floats in front of her face. You immediately wonder how the acceleration of the dropped pen compares to the acceleration of a pen that you might drop here on the surface of the Earth.
   
   
9. Gravitational: You are still a consultant for the new Star Trek TV series. You were hired to make sure that any science on the show is correct. In this episode, the crew of the Enterprise discovers an abandoned space station in deep space far from any stars. This station, which was built by Earth in the 21st century, is a large wheel-like structure where people live and work in the rim. In order to create "artificial gravity," the space station rotates on its axis. The special effects department wants to know at what rate a space station 200 meters in diameter would have to rotate to create "gravity" equal to 0.7 that of Earth.
   
   
10. Gravitational: You did so well in your physics course that you decided to try to get a summer job working in a physics laboratory at the University. You got the job as a student lab assistant in a research group investigating the ozone depletion at the Earth's poles. This group is planning to put an atmospheric measuring device in a satellite which will pass over both poles. To collect samples of the upper atmosphere, the satellite will be in a circular orbit 200 miles above the surface of the Earth. To adjust the instruments for the proper data taking rate, you need to calculate how many times per day the device will sample the atmosphere over the South pole. Using the inside cover of your trusty Physics text you find that the radius of the Earth is 6.38 x 103 km, the mass of the Earth is 5.98 x 1024 kg, and the universal gravitational constant is 6.7 x 10-11 N m2/kg2.
    
    
11. Gravitational: You did so well in your physics course that you decided to try to get a summer job working in a physics laboratory at the University. You got the job as a student lab assistant in a research group investigating the ozone depletion at the Earth's poles. This group is planning to put an atmospheric measuring device in a satellite which will pass over both poles. To collect samples of the upper atmosphere, the satellite will be in a circular orbit 200 miles above the surface of the Earth where g is 95% of its value on the Earth's surface. To adjust the instruments for the proper data taking rate, you need to calculate how many times per day the device will sample the atmosphere over the South pole. Using the inside cover of your trusty Physics text you find that the radius of the Earth is 6.38 x 103 km and the mass of the Earth is 5.98 x 1024 kg.
    
    
12. Gravitational: You are reading a magazine article about pulsars. A few years ago, a satellite in orbit around the Earth detected X-rays coming from sources in outer space. The X-rays detected from one source, called Cygnus X-3, had an intensity which changed with a period of 4.8 hours. This type of astronomical object emitting periodic signals is called a pulsar. One popular theory holds that the pulsar is a normal star (similar to our Sun) which is in orbit around a much more massive neutron star. The period of the X-ray signal is then the period of the orbit. In this theory, the distance between the normal star and the neutron star is approximately the same as the distance between the Earth and our Sun. You realize that if this theory is correct, you can determine how much more massive the neutron star is than our Sun. All you need to do is first find the mass of the neutron star in terms of two unknowns, the universal gravitational constant G and the radius of the Earth's orbit. Then find the mass of our Sun in terms of the same two unknowns, G and the radius of the Earth's orbit. (The period of the Earth's orbit is 1 year). Then you can calculate how many times more massive the neutron star is than our Sun.
    

Radial Force Components

Note: Each problem begins with a list of forces necessary to solve the context-rich problem. These are for the benefit of the instructor. Delete the list before using the problems in your class.

13. Weight, Lift: You are reading an article about the aesthetics of airplane design. One example in the article is a beautiful new design for commercial airliners. You are worried that this light wing structure might not be strong enough to be safe. The article explains that an airplane can fly because the air exerts a force, called "lift," on the wings such that the lift is always perpendicular to the wing surface. For level flying, the wings are horizontal. To turn , the pilot "banks" the plane so that the wings are oriented at an angle to the horizontal. This causes the plane to have a trajectory which is a horizontal circle. The specifications of the 100 x 103 lb plane require that it be able to turn with a radius of 2.0 miles at a constant speed of 500 miles/hr. The article states that tests show that the new wing structure will support a force 4 times the lift necessary for level flight. Is the wing structure sufficiently strong for the plane to make this turn?
    
    
14. Weight, Lift: You are flying to Chicago when the pilot tells you that the plane can not land immediately because of airport delays and will have to circle the airport. This is standard operating procedure. She also tells you that the plane will maintain a speed of 400 mph at an altitude of 20,000 feet while traveling in a horizontal circle around the airport. To pass the time you decide to figure out how far you are from the airport. You notice that to circle, the pilot "banks" the plane so that the wings are oriented at 10o to the horizontal. An article in your in-flight magazine explains that an airplane can fly because the air exerts a force, called "lift," on the wings. The lift is always perpendicular to the wing surface. The magazine article gives the weight of the type of plane you are on as 100 x 103 pounds and the length of each wing as 150 feet. It gives no information on the thrust of the engines or the drag of the airframe.
    
    
15. Because of your physics background, you have been hired as a member of the team the state highway department has assigned to review the safety of Minnesota freeways. This week you are studying 35W which has a curve which is essentially 1/8 of a circle with a radius of 0.5 miles. The road has been designed with a banked curve so that the road makes an angle of 4û to the horizontal throughout the curve. To begin the study, the head of your department asks that you calculate the maximum speed for a standard passenger car (about 2000 lbs) to complete the turn while maintaining a horizontal path along the road. She asks that you first consider the case of a slick, ice covered road. When you have completed that calculation she wants you to do the case of a dry, clear road where the coefficient of kinetic friction is 0.70 and the coefficient of static friction is 0.80 between the tires and the road. This will give her team the two extremes of Minnesota driving conditions on which to base the analysis.
    
    
16. Tension, Weight: A neighbor's child wants to go to a neighborhood carnival to experience the wild rides. The neighbor is worried about safety because one of the rides looks dangerous. She knows that you have taken physics and so asks your advice. The ride in question has a 10-lb chair which hangs freely from a 30-ft long chain attached to a pivot on the top of a tall tower. When a child enters the ride, the chain is hanging straight down. The child is then attached to the chair with a seat belt and shoulder harness. When the ride starts up the chain rotates about the tower. Soon the chain reaches its maximum speed and remains rotating at that speed. It rotates about the tower once every 3.0 seconds. When you ask the operator, he says that the ride is perfectly safe. He demonstrates this by sitting in the stationary chair. The chain creaks but holds and he weighs 200 lbs. Has the operator shown that this ride safe for a 50-lb child?