Physics 1 Assignment - Periodic Motion

AP Physics 1 Assignment - Circular and Periodic Motion

Reading: Chapter 6, sections 16.1 – 16.4, Open Stax College Physics;

Chapter 5, Sections 10.1 – 10.4, 10.6, Etkina et. al.

Objectives/HW:

	The student will be able to:	HW:
1	Define and calculate period and frequency.	1 – 14
2	Apply the concepts of position, distance, displacement, speed, velocity, acceleration, and force to circular motion.
3	State and apply the relation between speed, radius, and period for uniform circular motion.
4	State and apply the relation between speed, radius, and centripetal acceleration and force for uniform circular motion.
5	Distinguish and explain the concepts of centripetal vs. centrifugal force.	15 –16
6	State and apply Newton's Law of Universal Gravitation.	17 – 28
7	Define and calculate the gravitational field, g, and solve related problems including orbital motion.	29 – 37
8	State and apply the properties of Simple Harmonic Motion and solve related problems involving Hooke’s Law, spring constant, and period.	38 – 40
9	State and apply the relation between length, period, and g for a pendulum.	41 – 43

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 level circular path 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 as it moves in the circle?

3. 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?

4. Sue whirls a yo-yo above her head 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 45 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 is the “centripetal force” (causing this acceleration)? (d) Determine the minimum value of the static coefficient of friction for which the coin would not slide at any of these locations. Explain the reasoning behind this result.

6. The static coefficient of friction μ_s applies to an object of mass m placed on a platter that revolves about a vertical axis at frequency f. Derive an expression for the maximum distance r from the center of the platter at which the object can remain without sliding off the platter.

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.

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. Use the formulas a = v²/r and v = 2πr/T to derive the following formulas: (a) centripetal acceleration in terms of v and T (but not r). (b) centripetal acceleration in terms of r and T (but not v).

11. The “Ring of Fire” is a popular fair ride in which riders complete a vertical circle of radius 8.0 m, going upside down in the process. Suppose a rider of mass 80.0 kg is seated on the ride and reaches a speed of 10.0 m/s at the top of the ride and 15.0 m/s at the bottom of the ride. The centripetal force follows the formulas for uniform circular motion at these two points even though the speed changes. (a) Find the range of normal force (seat acting on the rider) at these speeds. (b) What speed at the top of the loop would make the normal force zero? What does the person experience in that case?

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. A car of mass m performs a U-turn of radius r on a level pavement for which the coefficient of static friction is μ. Determine the minimum amount of time t in which this can be accomplished without sliding.

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 using Newton’s Laws of Motion.

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 Newton’s Laws.

16. FHS science department has 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. If all mass attracts all other mass why doesn’t everything in the universe collapse into one big wad of stuff? Or is it just a matter of time before this happens? Justify using physics.

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 Newton, Yes! Determine the amount of gravitational pull that attracts Tom toward Sally.

20. Two bowling balls each of mass 6.8 kg and diameter 21.8 cm are at rest and nearly touching. The gap between the two balls is 0.2 cm. (a) What amount of gravitational force does one exert on the other? (b) If the two balls were completely free to move, how much time (at most) would it take for gravity to “close the gap” and cause the balls to touch? (And why is this “at most”?)

21. A space explorer standing on an alien world hangs a 500.0 gram object from a spring scale and its weight there is found to be 3.50 newtons. Given the radius of the world is 6100 km, determine its mass.

22. Two spherical balls are placed so their centers are 2.6 m apart. The gravitational force of the larger ball attracting the smaller ball is 2.75 × 10⁻¹² N. What is the mass of each ball if the larger 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. i.e. 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 (true).

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 9.393 × 10²⁰ kg and a radius of 473 km (as determined in 2015 by the DAWN spacecraft). (a) What is g on its surface? (b) How much would a 85 kg astronaut weigh on the surface of Ceres?

26. Determine the gravitational field strength at the surface of each of the following as a multiple of Earth’s g: (a) a planet with twice the radius but mass equal to Earth’s, (b) a star with 350 thousand times the mass and 100 times the diameter as Earth, (c) a neutron star with 500 thousand times the mass and 1 one thousandth the diameter of Earth, and (d) a gas giant planet with 10 times the radius but 1 fifth the density of Earth.

27. A proposed space station shaped like a huge bicycle wheel with diameter of 625 m is to spin at a certain rate to produce "artificial gravity" for the occupants in the "rim" of the station. (a) What should be the period of the station's rotation? (b) Describe a specific observation or experiment that a person could perform on the space station in order to distinguish the fake gravity and a real planetary gravitational field.

28. 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) If the distance were halved, by what factor would each acceleration value change? (c) The Moon and the Earth do accelerate toward one another at the amounts you found in part (a) and yet the distance between them does not change! Explain.

29. 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 his weight on earth. (b) Find the magnitude of the force of gravity upon him while aboard the space station in orbit 400 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 station? Explain using physics concepts.

30. 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? Explain in physics terms.

31. 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?

32. 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!)

33. A moon moving with speed v completes its orbit of a certain planet with a period T. (a) Derive the mass M of the planet in terms of v, T, and appropriate physical constants. (b) Explain why the mass of the moon cannot be determined using v and T.

34. A geosynchronous satellite appears to “hover” stationary directly above a particular point on the Earth’s equator. In order to achieve this the period of the orbit must match the sidereal period of Earth’s rotation, which is 23 hours 56 minutes. Solve for the altitude and speed of such a satellite.

35. 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.

36. Observations of spiral galaxy M33 show that material near its edge orbits at a speed roughly 4 times greater than expected based on the mass of the stars that are visible in the galaxy. Physicists conclude this is caused by mass that is not visible, which is called “dark matter”. What fraction of M33’s total mass is dark matter? Hint: use result of previous problem.

37. The International Space Station (ISS), mass 419000 kg, orbits Earth at altitude 390 km. However, over the course of several months its altitude will “decay” and decrease by 10 km due to drag with Earth’s atmosphere. Rocket thrusters firing continuously for 2.0 orbits are used to boost the ISS back to the correct orbit. (a) Determine the “delta-v” (change in speed) associated with an increase in altitude of 10 km (assuming circular orbits at each altitude). (b) Determine the amount of thrust necessary for the boost – you can treat the situation like pushing an object up a ramp that is 10 km high and two orbits long (ignore drag).

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. The position of a 300.0 g mass attached to a certain spring is given by x = 0.20 sin(5.0t), where x is in meters and t is in seconds. (To graph or calculate use radian mode.) The 5.0 in the equation represents the “angular frequency” and, with units, is: ω = 5.0 radians/second. (Angular frequency satisfies either of these equations: ω = 2πf or ω = 2π/T.) (a) Determine the spring constant k. (b) Determine the maximum amount of force acting on the 300 g mass. (c) Determine a new position function if the original mass is replaced by a 600.0 g object (attached to the same spring) and released from a position of x = 0.30 m at t = 0.0. (Hint: using a cosine function is convenient this time.)

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. Because it exhibits simple harmonic motion like a mass on a spring, it is possible to determine the value of k (the “spring constant”) for a pendulum. (a) Derive a formula for k in terms of pendulum mass m, length L, and appropriate constants. (b) Use your formula to determine the value of k for a pendulum with mass 100.0 g and length 20.0 cm. (c) Use this value of k to determine the acceleration of this pendulum when the mass is 1.50 cm, 180.0° from equilibrium.

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