Physics
Applications of Circular Motion
Circular motion has various applications in physics, including the motion of planets around the sun, the motion of electrons around the nucleus of an atom, and the motion of a car around a curve. Understanding circular motion is important in fields such as engineering, astronomy, and particle physics.
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12 Key excerpts on "Applications of Circular Motion"
eBook - ePub- Jerry Marion(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
5CIRCULAR MOTION, TORQUE, AND ANGULAR MOMENTUM
Publisher Summary
This chapter presents Newtonian dynamics related to rotation and translation. It discusses circular motion of an object. It describes the general properties of motion in a circle, including the idea of centripetal acceleration. One of the important cases of motion in two dimensions is that of circular motion. An object that moves in a circular path with constant speed undergoes uniform circular motion. An object undergoing uniform circular motion moves with constant speed, but the velocity vector continually changes direction. The chapter discusses centripetal acceleration and uniform circular motion. It explains the use of torque in statics problems. The angular momentum is equal to the product of the linear momentum and the radius of the path. The chapter presents the law of conservation of angular momentum. If no net external torque acts on an object, the angular momentum of the object will remain constant. The chapter explains magnitude and direction of angular momentum. Any change in the state of rotation of an object involves an angular acceleration.We now extend the discussion of Newtonian dynamics to include rotation as well as translation . To simplify matters, we restrict our attention to circular motion. We begin by considering the general properties of motion in a circle, including the idea of centripetal acceleration .5-1 CIRCULAR MOTION
The Period
One of the important cases of motion in two dimensions is that of circular motion. An object that moves in a circular path with constant speed undergoes uniform circular motion . If the constant speed is v and if the radius of the circle is r
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
5.2 Dynamics of Uniform Circular Motion For objects undergoing uniform circular motion, the acceleration is centripetal accel- eration, in which case the net force is toward the center of the circular path. This net force is called the centripetal force and its magnitude is denoted F c . According to Newton’s second law, then F m v r c 2 = (5.2.1) The centripetal force is generally a combination of several forces whose overall effect is a net force pointing toward the center of the circular path along which the object is traveling. When using Equation 5.2.1, you must choose the positive direction to be toward the center of the circular path. 5.3 Applications The centripetal force that keeps an object moving along a circular path may be provided by the tension in a string, friction, or the normal force during a banked turn. When solving a problem involving forces, always start with a free-body diagram. The centripetal force is the sum of the forces (or components of forces) that are either toward or away from the center of the circular path, with the positive direction taken to be toward the center of the circular path. (5.1.1) The centripetal acceleration of an object in terms of its speed and the radius of its circular path (5.2.1) The centripetal force on an object in terms of its mass, it’s speed, and the radius of its circular path KEY EQUATIONS a v r c 2 = F m v r c 2 = CHAPTER 5 PROBLEMS * Number of asterisks indicates level of problem difficulty. 5.1 Uniform Circular Motion and Centripetal Acceleration *Problem 5.1.1. Objects sitting on the Earth’s surface move in circu- lar paths about the Earth’s axis of rotation. (a) What is the speed of an object at the equator due to the Earth’s rotation? (b) What is the magnitude of the centripetal acceleration of an object at the equator? **Problem 5.1.2. The city of Beijing, China, is at latitude 39.9° N, meaning that a line drawn from the center of the Earth to Beijing makes a 39.9º angle with the equatorial plane.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Recall that Newton’s first law tells us that motion is along a straight line at constant speed unless there is a net external force. We will therefore study not only motion along curves, but also the forces that cause it, including gravitational forces. In some ways, this chapter is a continuation of Dynamics: Newton's Laws of Motion as we study more applications of Newton’s laws of motion. This chapter deals with the simplest form of curved motion, uniform circular motion, motion in a circular path at constant speed. Studying this topic illustrates most concepts associated with rotational motion and leads to the study of many new topics we group under the name rotation. Pure rotational motion occurs when points in an object move in circular paths centered on one point. Pure translational motion is motion with no rotation. Some motion combines both types, such as a rotating hockey puck moving along ice. 6.1 Rotation Angle and Angular Velocity In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion. Rotation Angle When objects rotate about some axis—for example, when the CD (compact disc) in Figure 6.2 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance.
- David V. Guerra(Author)
- 2023(Publication Date)
- CRC Press(Publisher)
12 Circular Motion and Centripetal Force
DOI: 10.1201/9781003308065-1212.1 Introduction
Until this point in the volume, the motion of the objects studied has been predominantly in one dimension, so the change in the velocity of an object was focused on the change in the magnitude of the velocity vector. In this chapter, the change in velocity of an object in uniform circular motion is all about the change in the direction of the object’s velocity. Therefore, this motion, which is common in nature, requires its own analysis. First, by studying the change in the direction of the velocity vectors of an object moving in a circle at a constant speed the centripetal acceleration is derived. From the acceleration the associated net force, known as the centripetal force, is explained. Then, a series of examples employed in these concepts are provided in which the forces of tension, friction, gravity, electrostatics, and magnetism are involved.- Chapter question: A centrifuge is a device that separates solutions, like blood, into its different constituents by spinning the solution at high speeds. The solution is poured into test tubes, loaded into the centrifuge, and spun at a high rate until the constituents of the solution are separated. As a centrifuge spins faster, heavier particles in the solution move away from the center of the circle, toward the bottom of the test tube. In the case of blood, the denser red blood cells move to the outside of the circle with the largest radius r, as shown in Figure 12.1 , which is often referred to as the bottom of the tube, the white cells and platelets move to the center of the tube, and the blood plasma moves to the inside, which is the top of the tube.
FIGURE 12.1
eBook - PDF- Philip Dyke, Roger Whitworth(Authors)
- 2017(Publication Date)
- Red Globe Press(Publisher)
CHAPTER 6 Circular Motion 6.1 Introduction We may not realise it yet, but the effects of circular motion are a common daily experience. The motion of record players, the act of cornering in a car or on a bicycle, and the behaviour of washing in a spin dryer are common examples, and there are plenty more. Circular motion is an example of two-dimensional motion. Previously, it has helped to analyse motion in two perpendicular directions, and a Cartesian coordinate system has proved ideal for this purpose. In Cartesians, the position of a body is represented parametrically, in terms of the time t , in the form x x t , y y t ± see for example the case of projectile motion in Chapter 5. For the analysis of motion in a circle, it is more convenient to use polar coordinates ( r , ). The advantages of using such a coordinate system will become obvious in the next section. 6.2 Polar coordinates The particle P shown in Figure 6.1 is describing a circle with centre O in a plane. The radius of the circle is a metres. At a particular instant, the position of P is given in polar coordinates by ( r a , 0 radians) and at some subsequent time, its position has moved to ( r a , = 3 radians). The dis-tance between the two positions is the most direct distance, which using trigonometry, is 2 a sin = 6 . However, a more significant measure of the motion is the distance that P has travelled (that is, along the arc of the circle between the two points). This is given by the expression a = 3 metres. More generally, if the angular coordinates of two points on a circle, of radius r metres, differ by an angle of radians, then the distance along the circular arc between them is r metres. In Figure 6.2, a particle has travelled from a point A to point B , on a circle of radius r . The angle subtended at the centre of the circle by the arc AB is radians. The angle radians is defined as the angular displacement experienced by the particle moving from A to B .
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Circular motion Whirlpools and storms his circling arm invest With all the might of gravitation blest. Alexander Pope These pictures show some objects which move in circular paths. What other examples can you think of? The answers to these questions lie in the nature of circular motion. Even if an object is moving at constant speed in a circle, its velocity keeps changing because its direction of motion keeps changing. Consequently the object is accelerating and so, according to Newton’s first law, there must be a force acting on it. The force required to keep an object moving in a circle can be provided in many ways. Without the earth’s gravitational force, the moon would move off at constant speed in a straight line into space. The wire attached to the athlete’s hammer provides a tension force which keeps the ball moving in a circle. When the athlete lets go, the ball flies off at a tangent because the tension has disappeared. Although it would be sensible for the pilot to be strapped in, no upward force is necessary to stop him falling out of the plane because his weight contributes to the force required for motion in a circle. In this chapter, these effects are explained. 12 Q UESTION 12.1 What makes objects move in circles? Why does the moon circle the earth? What happens to the ‘hammer’ when the athlete lets it go? Does the pilot of the plane need to be strapped into his seat at the top of a loop in order not to fall out? 12.1 Notation To describe circular motion (or indeed any other topic) mathematically you need a suitable notation. It will be helpful in this chapter to use the notation (attributed to Newton) for differentiation with respect to time in which, for example, d d s t is written as s . , and d d 2 t 2 as .. . Figure 12.1 shows a particle P moving round the circumference of a circle of radius r , centre O. At time t , the position vector OP ⎯→ of the particle makes an angle (in radians) with the fixed direction OA ⎯→ .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 5 Dynamics of uniform circular motion LEARNING OBJECTIVES After reading this module, you should be able to: 5.1 define uniform circular motion 5.2 solve uniform circular motion kinematic problems 5.3 solve uniform circular motion dynamic problems 5.4 solve problems involving banked curves 5.5 analyse circular gravitational orbits 5.6 solve application problems involving gravity and uniform circular motion 5.7 analyse vertical circular motion. INTRODUCTION The iconic giant face welcoming visitors to Sydney’s Luna Park has changed over the years but one attraction of this harbourside fun park has been a mainstay. The Rotor ride spins a cylindrical room at a dizzying 30 revolutions per minute, pinning riders to its curved walls through the forces of friction and centripetal motion. Rider’s faith in the physics of circular motion is put to the test as the floor drops away, leaving them suspended in mid-air by nothing more than the clothes on their back. In this chapter we will examine the features of circular motion and the physics of following a curved path. 1 5.1 Uniform circular motion LEARNING OBJECTIVE 5.1 Define uniform circular motion. There are many examples of motion on a circular path. Of the many possibilities, we single out those that satisfy the following definition. Definition of uniform circular motion Uniform circular motion is the motion of an object travelling at a constant (uniform) speed on a circular path. As an example of uniform circular motion, figure 5.1 shows a model aeroplane on a guideline. The speed of the plane is the magnitude of the velocity vector v, and since the speed is constant, the vectors in the drawing have the same magnitude at all points on the circle. FIGURE 5.1 The motion of a model aeroplane flying at a constant speed on a horizontal circular path is an example of uniform circular motion.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
WCN 02-300 Unless otherwise noted, all content on this page is © Cengage Learning. 224 IN THE STUDY OF LINEAR MOTION, objects were treated as point particles without structure. It didn’t matter where a force was applied, only whether it was applied or not. The reality is that the point of application of a force does matter. In American football, for example, if the ball carrier is tackled near his midriff, he might carry the tackler several yards before falling. If tackled well below the waistline, however, his center of mass rotates toward the ground, and he can be brought down immediately. Tennis provides another good exam- ple. If a tennis ball is struck with a strong horizontal force acting through its center of mass, it may travel a long distance before hitting the ground, far out- of - bounds. Instead, the same force applied in an upward, glancing stroke will impart topspin to the ball, which can cause it to land in the opponent’s court. The concepts of rotational equilibrium and rotational dynamics are also important in other disciplines. For example, students of architecture benefit from understanding the forces that act on buildings, and biology students should understand the forces at work in muscles and on bones and joints. These forces create torques, which tell us how the forces affect an object’s equilibrium and rate of rotation. We will find that an object remains in a state of uniform rotational motion unless acted on by a net torque. That principle is the equivalent of Newton’s first law. Further, the angular acceleration of an object is proportional to the net torque acting on it, which is the analog of Newton’s second law. A net torque acting on an object causes a change in its rotational energy. Finally, torques applied to an object through a given time interval can change the object’s angular momentum.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202 Unless otherwise noted, all content on this page is © Cengage Learning. | Problems 217 The purpose of their rotation is to simulate gravity for the inhabitants. Explain the concept behind this proposal. 8. Describe the path of a moving object in the event that the object’ s acceleration is constant in magnitude at all times and (a) perpendicular to its velocity; (b) parallel to its velocity. 9. A pail of water can be whirled in a vertical circular path such that no water is spilled. Why does the water remain in the pail, even when the pail is upside down above your head? 10. A car of mass m follows a truck of mass 2 m follows a truck of mass 2 m m around a circular m around a circular m turn. Both vehicles move at speed v. (a) What is the ratio of the truck’s net centripetal force to the car’s net centripetal force? (b) At what new speed v truck truck will the net centripetal k will the net centripetal k force acting on the truck equal the net centripetal force act- t- t ing on the car still moving at the original speed v? 11. Is it possible for a car to move in a circular path in such a way that it has a tangential acceleration but no centripetal acceleration? 12. A child is practicing for a BMX race. His speed remains con- stant as he goes counterclockwise around a level track with two nearly straight sections and two nearly semicircular sec- tions, as shown in the aerial view of Figure CQ7.12. (a) What are the directions of his velocity at points A, B, and C ? For each point, choose one: north, south, east, west, or nonexistent. (b) What are the directions of his acceleration at points A, B, and C ? 13. An object executes circular motion with constant speed when- ever a net force of constant magnitude acts perpendicular to the velocity.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
52 Rotational Motion acceleration, a c , is directed everywhere per- pendicular to the tangential velocity vector and toward the center of circular motion. The magnitude of the centripetal accel- eration is equal to the tangential speed (v) squared divided by the radius: Since rco is equal to v, it can be substituted into the above equation to give a - rco 2 . The centripetal acceleration is produced by a force directed toward the center of circular motion termed the centripetal force. If F c represents centripetal force and m the mass of an object in uniform circular motion, F. is found by applying Newton's second law: Centripetal is a general term that identi- fies the force as directed toward the center of a circle. The actual source of the centripetal force can be gravity, friction, electromag- netic, tension, or various other forces. When a car rounds a corner, the centripetal force is supplied by friction between the tires and the road. The equation for centripetal force shows that the centripetal force is directly proportional to the square of the tangen- tial speed and inversely proportional to the radius. This means that as the car travels faster and the turn gets tighter, a greater cen- tripetal force is required to make the turn. There is a limit to how much frictional force can be supplied by the tires, dictated by the coefficient of friction between the road and the rubber tire. If the road is wet or icy, the frictional force decreases, and driving too fast or trying to make a tight turn results in lost of control and sliding off the road. One way to supply a greater centripetal force and allow turns at higher speeds is to bank or ele- vate the roadway toward the outside. In this manner, both friction and the normal force of the roadway on the car supply the cen- tripetal force. Speedways, such as Daytona, use this principle to enable cars to maintain high speeds around the turns. The racetrack essentially pushes the car around the corner.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Problems 123 PROBLEMS Section 5.1 Uniform Circular Motion, Section 5.2 Centripetal Acceleration 1. A child sitting on the edge of a merry-go-round is moving at a speed of 1.2 m/s. If the merry-go-round has a diameter of 2.1 m, find the centripetal acceleration of the child. 2. Speedboat A negotiates a curve whose radius is 80 m. Speedboat B negotiates a curve whose radius is 240 m. Each boat experiences the same centripetal acceleration. What is the ratio v A /v B of the speeds of the boats? 3. How long does it take a plane, traveling at a constant speed of 150 m/s, to fly once around a circle whose radius is 3430 m? 4. Venus orbits the sun in about 225 days. If the average distance from the sun is about 108 million km, calculate the approximate centripetal acceleration in m/s 2 . 5. Computer-controlled display screens provide drivers of modern race cars with a variety of information about how their cars are performing. For instance, as a car is going through a turn, a speed of 109 m/s and cen- tripetal acceleration of 3.00 g (three times the acceleration due to gravity) are displayed. Determine the radius of the turn (in meters). *6. A centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance, to cause the more dense red blood cells to settle through the less dense blood serum and collect at the bot- tom of the container. Suppose the centripetal acceleration of the sample is 6.25 3 10 3 times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 5.00 cm from the axis of rotation? *7. The earth rotates once per day about an axis passing through the north and south poles, an axis that is perpendicular to the plane of the equator.
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
But, what is its origin? It originates from the source of this force. In fact, there are numerous interesting 80 Essential Physics © 2010 Taylor & Francis Group, LLC situations that demonstrate the origin and role of these forces. Figure 4.7 is a typical diagram that shows an object of mass m circulating in uniform horizontal circular motion. The centripetal force F c in the diagram is the net force acting on the object and keeps it circulating at a fixed distance r from the center of the circle. EXAMPLE 4.10 A small block of mass m = 321 g is attached firmly through a locked hook to a string of length 0.450 m. The mass then is set into a horizontal circular motion by a boy whirling the block at a constant speed of 1.20 m/s. a. Determine the centripetal acceleration of the circulating mass. b. Determine the centripetal force acting on the circulating mass. c. What is the source of the centripetal force acting on the circulating mass? S OLUT ION a. From Equation 3.11 the centripetal acceleration is a v r c = 2 . Thus, a / / c 2 (1.20 m s) 0.450 m 3.20 m s = = 2 . b. The centripetal force is F net,c = ma c = (0.321 kg)(3.20 m/s 2 ) = 1.03 N. c. The centripetal force in this situation is the tension, which the boy is exerting on the string as he is whirling it. A NALYSIS 1. The above value for the tension on the mass is reasonably feasible for the boy to exert. If he intends to go to much higher speeds, he has to apply more force through his grip on the string. F c C F c F c a c a c a c m m m m FIGURE 4.7 An object of mass m circulating in a uniform horizontal circular motion. The net centripetal force F c acting on the object is the centripetal force. 81 Newton’s Laws: Implications and Applications © 2010 Taylor & Francis Group, LLC 2. From Newton’s third law, the mass will react on the boy’s hand and acts on it with a force equal to the tension the boy is exerting on the mass via the string.
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