Physics
Circular Motion
Circular motion refers to the movement of an object along a circular path. It involves a constant change in direction, even if the speed remains constant. The centripetal force, directed towards the center of the circle, is responsible for keeping the object in circular motion.
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10 Key excerpts on "Circular Motion"
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
The extremely interesting features and properties of Circular Motion or near-Circular Motion (electrons in their atomic orbits, planets in their orbits, cars driven on flat curved roads, ramps, etc.) warrants special attention and extended treatment. The simplest Circular Motion is known as uniform Circular Motion, where an object rotates in a fixed circle with a constant speed. The conditions that make this rotation possible will be discussed in the next chapter. This motion is introduced with the general definition of the average acceleration that an object is subjected to during a time interval Δ t. Assumed to be constant, this was stated for one- and two-dimensional motion in the general form a v = ∆ ∆ t . (3.10) Since v is a vector quantity that has a magnitude and direction, the above definition may involve a change in the magnitude of the object’s velocity, v , a change in the direction of its velocity, or both. Accordingly, an object executing motion that is not along a straight line is expected to have an acceleration even if it is moving with a velocity of constant magnitude because this velocity is constantly changing in direction. This is the essence of uniform Circular Motion, in which an object is rotating along a circular path with a velocity v , also known as the tangential velocity, which is of a fixed magnitude but constantly changing in direction. The notion for v is tangential because the direction of this velocity is along the tangent to the circular path. Thus, the object experiences an acceleration resulting from the continuous change in the direction of v , along the circular path. This acceleration is always directed toward the center and is thus called centripetal acceleration. To illustrate this problem, consider an object rotating on a circle with a constant speed, v. Take two positions, P 1 and P 2 along the object’s path on the circle (Figure 3.3a).
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
130 Motion Along a Circular Path 5 In the sport of Grand Prix motorcycle racing, motorcycles compete on closed-loop, paved tracks. Although the length and shape of the track varies from venue to venue, all of them have a number of turns. The sharpness of a turn is quantified by its radius of curvature (defined as the radius of the circle that would be formed by continuing the turn until it closed onto itself). These turns, taken at high speeds, are a dramatic illustration of motion along a circular path. Although the speed of the motorcycles is approximately constant through a turn, there is plenty of acceleration due to the turning. Acceleration due to the changes in direction is called centripetal acceleration. David Acosta Allely/Shutterstock Uniform Circular Motion and Centripetal Acceleration | 131 5.1 Determine the magnitude and direction of the centripetal acceleration of an object in Circular Motion. An object moving in a circular path at constant speed is said to be executing uniform Circular Motion. Animated Figure 5.1.1 shows a rubber puck of mass m, viewed from above, moving on a horizontal ice surface that we will assume is frictionless. The puck is connected to a string, which constrains the puck to move along a circular path of radius r. The speed v of the puck is constant, but the velocity v is continuously changing. It is the direction of the velocity that is changing, and a changing velocity means that there is accel- eration. We can determine the direction of the acceleration with the help of Newton’s second law. At the end of the animation, a view of the puck at point P is shown, which is a free-body diagram of the puck as viewed from the level of the ice. The three forces acting on the puck are the tension in the string, the normal force, and the weight of the puck. The puck remains on the surface of the ice, so a y = 0.
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- 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.
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)
( b ) ( a ) P O C r r at time t t v v at time t 0 v θ θ Δ Δ υ CHAPTER 5 Dynamics of uniform Circular Motion 121 Centripetal acceleration is a vector quantity and, therefore, has a direction as well as a magnitude. The direction is towards the centre of the circle, and conceptual example 2 helps us to set the stage for explaining this important fact. CONCEPTUAL EXAMPLE 2 Which way will the object go? In figure 5.4 an object, such as a model aeroplane on a guideline, is in uniform Circular Motion. The object is symbolised by a dot (•), and at point O it is released suddenly from its circular path. For instance, suppose that the guideline for a model plane is cut suddenly. Does the object move (a) along the straight tangent line between points O and A or (b) along the circular arc between points O and P? FIGURE 5.4 If an object (•) moving on a circular path were released from its path at point O, it would move along the straight tangent line OA in the absence of a net force. C O P A v θ Reasoning Newton’s first law of motion (see section 4.2) guides our reasoning. This law states that an object continues in a state of rest or in a state of motion at a constant velocity (i.e. at a constant speed along a straight line) unless compelled to change that state by a net force. When an object is suddenly released from its circular path, there is no longer a net force being applied to the object. In the case of the model aeroplane, the guideline cannot apply a force, since it is cut. Gravity certainly acts on the plane, but the wings provide a lift force that balances the weight of the plane. Answer (b) is incorrect. An object such as a model aeroplane will remain on a circular path only if a net force keeps it there. Since there is no net force, it cannot travel on the circular arc.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
Many of the concepts describing translational motion have anal- ogous counterparts for Circular Motion. These will be developed in this chapter. The concepts in this chapter apply to both rota- tional and revolutionary motion. Rotation involves Circular Motion around an axis that passes through the object itself, for example, a rotating top. Revolution involves Circular Motion around an axis outside of the object. The Earth rotates on its axis and revolves around the Sun. In describing translational motion, a simple Cartesian coordinate system was 50 Rotational Motion used to reference position. This system can also be used for Circular Motion. When using the Cartesian coordinate system for Circular Motion, the position of an object can be referenced to the origin and the posi- tive x axis (Figure 4.1). Angles are measured counterclockwise starting from the positive x axis. The angular displacement is the angle swept out during a specific time period. For any two points along the circumference, the angular displacement is equal to A9. When starting from the positive x axis, where 0 is defined as zero, the angular displacement is just 0. The distance an object moves in one revolution around the circle is the circum- ference of the circle, 2irr. The time it takes for an object to make one revolution around the circle is called the period, symbolized by T. The tangential or linear speed of an object moving in a circular path of radius r around the origin is equal to the change in distance along the circumference divided by the change in time, As/At (if referenced from the positive x axis, the distance is just s). Since an object will travel a distance / equal to the circumference in one period, the tangential speed can be found by dividing the circumference by the period: 2rrr tangential speed = ~^r When the tangential speed is constant, the motion is described as uniform Circular Motion. The tangential velocity, v, is the vector analogy to tangential speed.
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- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
131 CHAPTER 5 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 Analyze circular gravitational orbits. 5.6 Solve application problems involving gravity and uniform Circular Motion. 5.7 Analyze vertical Circular Motion. Dynamics of Uniform Circular Motion Crew members are refueling a V-22 Osprey helicopter. The green lights near the tips of its rotating blades experience a net force and acceleration that points toward the center of the circle. We will now see how and why this net force and acceleration arise. Chief Petty Officer Joe Kane (U.S. Navy), Image taken from https:// commons.wikimedia.org/wiki/ File:20080406165033%21V-22_Osprey_ refueling_edit1.jpg 5.1 Uniform Circular Motion There are many examples of motion on a circular path. Of the many pos- sibilities, we single out those that satisfy the following definition: DEFINITION OF UNIFORM Circular Motion Uniform Circular Motion is the motion of an object traveling at a constant (uniform) speed on a circular path. As an example of uniform Circular Motion, Figure 5.1 shows a model airplane 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 draw- ing have the same magnitude at all points on the circle. Sometimes it is more convenient to describe uniform Circular Motion by specifying the period of the motion, rather than the speed. The period T is the time required to travel once around the circle—that is, to make one complete revolution. There is a relationship between υ υ υ υ FIGURE 5.1 The motion of a model airplane flying at a constant speed on a horizontal circular path is an example of uniform Circular Motion.
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- 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.
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