Honors Assignment - Circular and Periodic Motion
Objectives/HW:
|
|
The student will be able to: |
HW: |
|
1 |
Define and calculate period and frequency. |
|
|
2 |
Apply the concepts of position, distance, displacement, speed, velocity, acceleration, and force to circular motion. |
|
|
State and apply the relation between speed,
radius, and period for uniform circular motion. |
|
|
|
State and apply the relation between speed,
radius, and centripetal acceleration and force for uniform circular motion. |
||
|
Distinguish and explain the concepts of
centripetal vs. centrifugal force. |
||
|
State and apply |
||
|
Define and calculate the gravitational field, g, and solve related problems
including orbital motion. |
||
|
State and apply the properties of Simple
Harmonic Motion and solve related problems involving Hooke’s Law, spring
constant, and period. |
||
|
State and apply the relation between length,
period, and g for a pendulum. |
Homework Problems
1. (a) Is it possible to go around any type of curve with zero acceleration? Explain. (b) Is it possible to go around any type of curve with a constant acceleration? Explain. (c) If an object goes around a circular curve with a constant speed, will the acceleration be a constant or changing vector quantity? Explain.
2. It takes a 615 kg car 14.3 s to travel at a uniform speed around a circular racetrack of 50.0 m radius. (a) What is the speed of the car? (b) What is the acceleration of the car? (c) What amount of centripetal force must the track exert on the tires to keep the car moving in the circle?
3. An athlete whirls a 7.00 kg hammer tied to the end of a 1.3 m chain in a horizontal circle. The hammer revolves at a frequency of 1.5 Hz. (a) What is the speed of the hammer? (b) What is the centripetal acceleration of the hammer? (c) Ignoring the effect of gravity, what is the tension in the chain,?
4. Sue whirls a yo-yo in a horizontal circle. The yo-yo has a mass of 0.20 kg and is attached to a string 0.80 m long. Ignore the effect of gravity. (a) If the tension in the string is 5.0 N what will be the resulting speed of the yo-yo? (b) Determine the resulting period of the yo-yo’s revolution.
5. A coin is placed on a stereo record revolving at 33.3 rpm. (a) In what direction must the coin accelerate to remain on the record? (b) Determine the acceleration rate of the coin when it is placed 5.0, 10, and 15 cm outward from the center of the record. (c) What type of force causes this acceleration? (d) At which of these three radii would the coin be most likely to fly off? Why?
6. A 50.0 gram mass is placed on a level rotating platter 20.0 cm from its center. The coefficient of starting friction is 0.60. Determine the maximum frequency with which the platter can rotate without the mass sliding off its surface.
7. According to a recent edition of the Guinness Book of World Records the highest rotary speed ever attained was 2010 m/s. This speed occurred at one end of a rod 15.3 cm long that rotated about the other end. (a) What was the centripetal acceleration of the free end of the rod? (b) If a 1.00 gram object were attached to the end how much force would be needed to hold it there while rotating at the given rate? (c) What was the frequency of rotation of the rod in revolutions per minute?
8. In a type of amusement park ride sometimes called the “Gravitron”, the people are spun around inside a large cylinder. If the rate of rotation is great enough the rider may “stick” to the inside wall of the cylinder with their feet not touching the floor. Suppose the radius of the cylinder is 2.0 m and that it rotates with a frequency of 1.1 Hz. (a) Determine the speed of a rider. (b) Determine the centripetal acceleration of a rider. (c) The force causing this acceleration is what type and exerted by what? (d) Determine the coefficient of friction that is necessary to keep a rider from sliding down the wall. See p. 141, Fig. 5-36, for a picture.
9. An early major objection to the idea that the Earth is spinning on its axis was that Earth would turn so fast at the equator that people would be thrown off into space. Show the error in this logic by solving the following problems for a 97.0 kg person standing on the equator where the Earth’s radius is 6378 km. Such a person moves in a circular path due to the Earth’s rotation about its own axis. (a) Determine the centripetal force necessary to keep this person moving in such a fashion. (b) Determine the pull of gravity acting on the person. (c) Determine the normal force acting on the person. (Note: the person is not “at rest” unless the Earth’s surface is the frame of reference. For this problem, let the Earth’s axis be the frame of reference. The person is in motion around the axis even though he is not moving relative to the surface of the Earth.)
10. An automobile wheel and tire is balanced by a 20.0 g piece of lead located on the rim of a 40.6 cm diameter wheel. As the wheel is balanced horizontally on the machine at the tire store it rotates clockwise with a constant frequency of 15.0 Hz. (a) Find the period of the motion. For the instant at which the lead reaches the eastern most part of its circular path find: (b) its velocity, (c) its acceleration, and (d) the net force upon it. (e) What distance does the lead travel in 1.00 minute?
11. Choose and solve one of the following problems from your book: pp. 139 – 144: 5, 8, 10, 12, 13, 17, 19, 46, 63, 73
12. A car magazine reports that a 1100 kg car has skid pad results of 0.80 g. This is an indication of the maximum centripetal acceleration possible before the car slides. (a) What is the maximum speed at which this car can go around a level curve with radius 30.0 m? (b) What is the minimum radius of curvature the car can go around at a highway speed of 30.0 m/s? (c) What magnitude of friction is on the car in either of these turns? (d) What coefficient of friction is necessary to achieve these turns?
13. What is the maximum speed at which a car can travel around a level circular track of radius 80.0 m if the coefficient of static friction between the tire and road is 0.30? (Note: you do not need the car’s mass to get the answer. The answer would be the same for any mass car.)
14. In what direction relative to the velocity of an object must a force be applied in order to cause the object to move uniformly in a circle? Explain.
15. While driving your car,
if you go around a curve rapidly, you will feel as if you are being pushed away
from the center point of the curve. Physicists often refer to this as a
fictitious force because there is no actual force pushing you away from the
center. Instead there is a real force pulling you toward
the center of the curve. Explain using
16. FHS science department owns centrifuges that operate at 3400 rpm. A particle at the bottom of a test tube in this centrifuge is 10 cm away from the axis of rotation. From the reference frame of the test tube the particles in the tube experience a centrifugal force equal to some multiple of their weight (i.e. a certain number of “g”s). Determine the centrifugal force of these centrifuges in g’s.
17. The Moon and the Earth are attracted to each other by gravitational force. Does the more massive Earth attract the Moon with a greater force than the Moon attracts the Earth?
18. Imagine going straight down into a deep cavern in the Earth. As you go down in this cavern you are getting closer to the center of the Earth. Will the force of Earth’s gravity acting on you increase, decrease, or stay the same? Explain. (Hint: Remember all mass attracts all other mass!)
19. Tom has a mass of 70.0
kg and sally has a mass of 50.0 kg. Tom and Sally are standing 20.0 m
apart on the dance floor. Is the handsome young Tom attracted to the
pretty young Sally? According to
20. Two bowling balls each have a mass of 6.8 kg. They are located next to one another with their centers 21.8 cm apart. What amount of gravitational force does one exert on the other?
21. The gravitational force between two electrons 1.00 m apart is 5.42 ´ 10-71 N. Determine the mass of an electron.
22. Two spherical balls are placed so their centers are 2.6 m apart. The force between the two balls is 2.75 ´ 10-12 N. What is the mass of each ball if one ball is twice the mass of the other?
23. Use appropriate information from the given Solar System Data Table to determine which object pulls more on the Moon – the Earth or the Sun. Which exerts the greater gravitational pull and how many times greater than the other? Assume the distance from the Sun to the Moon is essentially the same as the distance from the Sun to the Earth (which it is).
24. A spacecraft traveling from Earth to the Moon reaches a point where the pull of the Moon’s gravity is equal to that of the Earth’s. Before this point the speed of the spacecraft is decreasing and after this point the speed is increasing (a little like coasting up a hill and then back down the other side). Determine the precise location of this point.
25. The asteroid Ceres has a mass 7.0 ´ 1020 kg and a radius of 500 km. (a) What is g on its surface? (b) How much would a 85 kg astronaut weigh on the surface of Ceres?
26. A proposed space station shaped like a huge bicycle wheel with diameter of 625 m is to spin at a rate to produce "artificial gravity" for the occupants in the "rim" of the station. What should be the period of the station's rotation?
27. Suppose the Moon ceased to revolve around Earth. (a) Using information from the Solar System Data Table, find the acceleration each body would have toward the other. (b) As the Moon and Earth got closer together would their rates of acceleration decrease, increase, or remain constant? Explain. (c) The Moon and the Earth do accelerate toward one another at the amounts you found and yet the distance between them does not change! Explain.
28. It is often said the astronauts are "weightless" in orbit. Indeed the amount of gravity will decrease as the astronauts are moved farther from earth. Does it diminish to the extent that it can be dismissed as insignificant? Compare for an astronaut with mass 75.0 kg. (a) Find their weight on earth. (b) Find the magnitude of the force of gravity upon them while aboard the shuttle in orbit 525 km above the earth. (c) The astronauts are not really “weightless” if by that we mean there is no gravity. So why do they appear to “float” around inside the space shuttle? Explain using physics concepts.
29. How would you answer the question, “What keeps a satellite up?” Or in other words if gravity pulls it downward why doesn’t it fall to the Earth?
30. Suppose an engineer at NASA is designing a space probe that will orbit the planet Mercury at an altitude of 400 km. In order to calculate the required orbital parameters such as speed, period, etc. what other numerical information will be needed?
31. A geosynchronous satellite appears to remain over one spot on Earth. A geosynchronous satellite has an orbital radius of 4.23 ´ 107 m. (a) Calculate its speed in orbit. (b) Calculate its orbital period and show that it is indeed 24 hours.
32. On July 19, 1969, Apollo 11’s orbit around the Moon was adjusted to an average altitude of 111 km. (a) At that altitude how many minutes did it take to orbit once? (b) At what speed did it orbit the Moon?
33. Determine the mass of the Sun based on its effect on the earth. Earth orbits the Sun at a distance of 150 Gm and completes its circular orbit once every year. (Even though you could just look up the mass of the Sun in a table somewhere it is only by this type of calculation that anyone was ever able to determine this value!)
34. Mimas, a moon of Saturn, has an orbital radius of 187 Mm and an orbital period of 23 hours. Use this information to calculate the mass of Saturn.
35. The space shuttle typically orbits earth (radius 6378 km) in a circular path at an altitude of 525 km. (a) Find the speed of the shuttle. (b) Find the time in minutes to complete one orbit.
36. Choose and solve one of the following problems from your book, pp. 139 – 144: 31, 36, 45, 49, 50, 66, 67, 72, 72, 77, 78.
37. For a satellite to orbit uniformly about the Earth, what must be true of its speed at orbits of greater and greater radius? To help answer this question derive a single equation that gives the speed of a satellite, v, in terms of the gravitational constant, G, the mass of the planet, M, and the radius of the orbit, r.
38. A 500 gram mass attached to a horizontal spring slides back and forth on a horizontal frictionless surface. The spring constant is 12.0 N/m and the amplitude of the motion is 20.0 cm. (a) Determine the period of the motion. (b) Determine the maximum magnitude of acceleration for this oscillating mass.
39. A spring is attached to a support and hangs vertically. A mass of 1.5 kg is hung from the spring. When the mass hangs at rest, the spring is stretched 18.0 cm from its initial length (i.e. the elongation is 18.0 cm). The mass is pulled downward so that the spring’s elongation is 24.0 cm and then the mass is released from rest. (a) Determine the spring constant, k. (b) Determine the period of the mass’s motion. (c) What is the acceleration of the mass at a point in its motion when the spring’s elongation is 16.0 cm?
40. In a classroom
demonstration a Homer doll is attached to the end of a horizontal strip of wood
and allowed to oscillate back and forth in a horizontal plane. Homer has
a mass of 203 grams and the period of oscillation is 1.2 seconds. (a)
Determine the force constant for the strip of wood, assuming it behaves like a
spring. (b) Determine the period of oscillation if Homer is removed and a
70.0 g Gumby is instead attached to the same wood strip.
41. A certain pendulum has a length of 10.0 cm. (a) Determine its period on earth. (b) Determine its period on the moon where g = 1.62 m/s2.
42. Suppose you wanted to improvise a stopwatch by making a pendulum that has a period of exactly 2.00 second (so that you could time something by counting the swings forward and back). What should be the length of such a pendulum?
43. An old clock that uses a pendulum to regulate time is taken up in an aircraft flying at an altitude of 15.0 km. If the pendulum in the clock is 20.0 cm long, by how much will the clock lose or gain time every hour (assuming it works perfectly at sea level)?
0.10
0.80
Sun pulls 2.3 times the Earth
9.01 ´ 10-31 kg
0.37 kg, 0.75 kg
5.6 ´ 1026 kg
1.99 ´ 1030 kg
66.6 ms
0.635 s
0.70 s
0.852 s
1.1 s
1.28 s
1.56 s
slow, by 8.50 s every hour
35.5 s
95.1 minutes
120 minutes
24.0 h
0.86 Hz
126,000 rpm
15 cm
99.3 cm
110 m
1.15 km
346 Mm from Earth, 38.3 Mm from Moon
4.5 m/s
12 m/s
14 m/s
15 m/s
15 m/s
19.1 m/s, 270.0°
22.0 m/s
1600 m/s
3070 m/s
7600 m/s (= 17000 mph!)
32 mm/s2 toward moon
2.6 mm/s2 toward earth
0.19 m/s2
0.61, 1.2, 1.8 m/s2
1.09 m/s2, 270.0°
4.80 m/s2
9.65 m/s2 toward center
95 m/s2
120 m/s2
1800 m/s2, 180.0°
1300 g
2.64 ´ 107 m/s2
0.584 nN
65 nN toward the other ball
3.27 N
16 N
36.0 N, 180.0°
627 N
735 N
810 N
948 N, 90°
951 N
5940 N
8.6 kN
26.4 kN toward the center
5.6 N/m
81.7 N/m