Centripetal Acceleration and Centripetal Force

11 Key excerpts on "Centripetal Acceleration and Centripetal Force"

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  College Physics

  • 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.

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  Introductory Physics for the Life Sciences: Mechanics (Volume One)

  • David V. Guerra(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  12 Circular Motion and Centripetal Force

  DOI: 10.1201/9781003308065-12

  12.1 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

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  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  The direction of the centripetal acceleration vector always points towards the centre of the circle and continually changes as the object moves. a c = v 2 r (5.2) 5.3 Solve uniform circular motion dynamic problems. To produce a centripetal acceleration, a net force pointing towards the centre of the circle is required. This net force is called the centripetal force, and its magnitude F c is given by equation 5.3, where m and v are the mass and speed of the object, and r is the radius of the circle. The direction of the centripetal force vector, like that of the centripetal acceleration vector, always points towards the centre of the circle. F c = mv 2 r (5.3) 5.4 Solve problems involving banked curves. A vehicle can negotiate a circular turn without relying on static friction to provide the centripetal force, provided the turn is banked at an angle relative to the horizontal. The angle  at which a friction‐free curve must be banked is related to the speed v of the vehicle, the radius r of the curve, and the magnitude g of the acceleration due to gravity by equation 5.4. tan  = v 2 rg (5.4) 5.5 Analyse circular gravitational orbits. When a satellite orbits the earth, the gravitational force provides the centripetal force that keeps the satellite moving in a circular orbit. The speed v and period T of a satellite depend on the mass M E of the earth and the radius r of the orbit according to equations 5.5 and 5.6, where G is the universal gravitational constant. v = √ GM E r (5.5) T = 2r 3∕2 √ GM E (5.6) 5.6 Solve application problems involving gravity and uniform circular motion. The apparent weight of an object is the force that it exerts on a scale with which it is in contact. All objects, including people, on board an orbiting satellite are in free‐fall, since they experience negligible air resistance and they have an acceleration that is equal to the acceleration due to gravity.

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  College Physics

  • Paul Peter Urone, Roger Hinrichs(Authors)
  • 2012(Publication Date)
  • Openstax

    (Publisher)

  The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton’s second law of motion, net force is mass times acceleration: net F = ma . For uniform circular motion, the acceleration is the centripetal acceleration— a = a c . Thus, the magnitude of centripetal force F c is (6.23) F c = ma c . By using the expressions for centripetal acceleration a c from a c = v 2 r ; a c = rω 2 , we get two expressions for the centripetal force F c in terms of mass, velocity, angular velocity, and radius of curvature: (6.24) F c = m v 2 r ; F c = mrω 2 . You may use whichever expression for centripetal force is more convenient. Centripetal force F c is always perpendicular to the path and pointing to the center of curvature, because a c is perpendicular to the velocity and pointing to the center of curvature. Note that if you solve the first expression for r , you get (6.25) r = mv 2 F c . This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve. 208 Chapter 6 | Uniform Circular Motion and Gravitation This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Figure 6.11 The frictional force supplies the centripetal force and is numerically equal to it. Centripetal force is perpendicular to velocity and causes uniform circular motion. The larger the F c , the smaller the radius of curvature r and the sharper the curve. The second curve has the same v , but a larger F c produces a smaller r′ . Example 6.4 What Coefficient of Friction Do Car Tires Need on a Flat Curve? (a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s. (b) Assuming an unbanked curve, find the minimum static coefficient of friction, between the tires and the road, static friction being the reason that keeps the car from slipping (see Figure 6.12).

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  Physics for Scientists and Engineers

  Foundations and Connections, Extended Version with Modern Physics

  • Debora Katz(Author)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  WCN 02-300 6-6 Centripetal Force 169 All content on this page is © Cengage Learning. If the origin of a polar coordinate system is at the center of the circle, the centripetal force is written as F u c 5 2m v 2 r r ˆ (6.8) The centripetal force is not a new force. It is not generated by the circular motion of a particle; instead, it is a requirement of circular motion. Some physical force (or forces)—gravity, a spring force, the normal force, a tension force, static friction— must act on an object in uniform circular motion in such a way that the net force on the object is perpendicular to the velocity and points to the center of the circular path. Neither drag nor moving friction can generate a centripetal force because they are always directed opposite the velocity. In the case of uniform circular motion, the net force is the centripetal force, which is always perpendicular to the velocity. So, imagine that the source of the centripetal force were suddenly removed such that there was no net force exerted on the object. Then, ac- cording to Newton’s first law, the object would continue at the same speed but in a straight line tangent to the point where the object was when the force suddenly vanished. CONCEPT EXERCISE 6.11 The following objects are moving in uniform circular motion. Draw a free-body dia- gram for each object and identify the force responsible for the centripetal acceleration. Object 1. A person riding on the barrel-of-fun ride (Fig. 6.27, top) Object 2. The lead object in the laboratory set-up (Fig. 6.27, center) Object 3. A jogger running on a circular track (Fig. 6.27, bottom) Barrel-of-fun rider Lead object attached to spring Runner on track FIGURE 6.27 Problems that involve centripetal force are no different from other problems that require us to apply Newton’s second law. So, the strategy developed in Section 5-8 works here.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  The phrase “centripetal force” does not denote a new and separate force created by nature. The phrase merely labels the net force pointing toward the center of the circular path, and this net force is the vector sum of all the force components that point along the radial direction. Sometimes the centripetal force consists of a single force such as tension (see Example 5), friction (see Example 7), the normal force or a component thereof (see Examples 9 and 14), or the gravitational force (see Examples 10–12). However, there are circumstances when a number of different forces contribute simultaneously to the cen- tripetal force (see Section 5.7). In some cases, it is easy to identify the source of the centripetal force, as when a model airplane on a guideline flies in a horizontal circle. The only force pulling the plane inward is the tension in the line, so this force alone (or a component of it) is the centripe- tal force. Example 5 illustrates the fact that higher speeds require greater tensions. N S W E 1 2 A B C D E CYU FIGURE 5.2 CYU FIGURE 5.1 136 CHAPTER 5 Dynamics of Uniform Circular Motion Conceptual Example 6 deals with another case where it is easy to identify the source of the centripetal force. EX AMPLE 5 The Effect of Speed on Centripetal Force The model airplane in Figure 5.6 has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground. The path of the airplane and its guideline lie in the same horizontal plane, because the weight of the plane is balanced by the lift gener- ated by its wings. Find the tension in the guideline (length = 17 m) for speeds of 19 and 38 m/s. Reasoning Since the plane flies on a circular path, it experi- ences a centripetal acceleration that is directed toward the center of the circle. According to Newton’s second law of motion, this acceleration is produced by a net force that acts on the plane, and this net force is called the centripetal force.

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  College Physics, Global Edition

  • Raymond Serway, Chris Vuille(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  This is a geometric result relating the centripetal accelera- tion to the angular speed, but physically an acceleration can occur only if some force is present. For example, if a car travels in a circle on flat ground, the force of static friction between the tires and the ground provides the necessary centripetal force. Note that a c in Equations 7.13 and 7.17 represents only the magnitude of the cen- tripetal acceleration. The acceleration itself is always directed toward the center of rotation. O v i v f q u Top view v S r a b B A S S O r r Figure 7.6 (a) Circular motion of a car moving with constant speed. (b) As the car moves along the cir- cular path from Ⓐ to Ⓑ, the direc- tion of its velocity vector changes, so the car undergoes a centripetal acceleration. v -v i  u a b S S vf S v i v f q u B A S S O r r s Figure 7.7 (a) As the particle moves from Ⓐ to Ⓑ, the direction of its velocity vector changes from v S i to v S f . (b) The construction for determining the direction of the change in velocity D v S , which is toward the center of the circle. Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 200 TOPIC 7 | Rotational Motion and Gravitation Unless otherwise noted, all content on this page is © Cengage Learning. The foregoing derivations concern circular motion at constant speed. When an object moves in a circle but is speeding up or slowing down, a tangential component of acceleration, a t 5 r a, is also present. Because the tangential and centripetal com- ponents of acceleration are perpendicular to each other, we can find the magnitude of the total acceleration with the Pythagorean theorem: a 5 "a t 2 1 a c 2 [7.18] Quick Quiz 7.6 A racetrack is constructed such that two arcs of radius 80 m at Ⓐ and 40 m at Ⓑ are joined by two stretches of straight track as in Figure 7.8. In a particular trial run, a driver travels at a constant speed of 50 m/s for one complete lap.

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  Physics for Scientists and Engineers with Modern Physics

  • Raymond Serway, John Jewett(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  (b) Overhead view observed by someone in an inertial reference frame attached to the platform. The platform appears stationary, and the ground rotates counterclockwise. PITFALL PREVENTION 6.2 Centrifugal Force The commonly heard phrase “centrifugal force” is described as a force pulling outward on an object moving in a circular path. If you are feeling a “centrifugal force” on a rotating carnival ride, what is the other object with which you are interact- ing? You cannot identify another object because it is a fictitious force that occurs when you are in a noninertial reference frame. Example 6.7 Fictitious Forces in Circular Motion Consider the experiment described in the opening storyline: you are riding on the Mad Tea Party ride and holding your smartphone hanging from a string. Now suppose your friend stands on solid ground beside the ride watching you. You hold the upper end of the string above a point near the outer rim of the spinning tea cup. Both the inertial observer (your friend) and the noninertial observer (you) agree that the string makes an angle u with respect to the vertical. You claim that a force, which we know to be fictitious, causes the observed deviation of the string from the vertical. How is the magni- tude of this force related to the smartphone’s centripetal acceleration measured by the inertial observer? S O L U T I O N Conceptualize Place yourself in the role of each of the two observers. The inertial observer on the ground knows that the smartphone has a centripetal acceleration and that the deviation of the string is related to this acceleration. As the noninertial observer on the teacup, imagine that you ignore any effects of the spinning of the teacup, so you have no knowledge of any centripetal acceleration. Because you are unaware of this acceleration, you claim that a force is pushing sideways on the smart- phone to cause the deviation of the string from the vertical.

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  In the orbiting shuttle, however, you are floating around with no sensation of any force acting on you. Why this difference? The difference is due to the nature of the two centripetal forces. In the car, the centripetal force is the push on the part of your body touching the car wall. You can sense the compression on that part of your body. In the shuttle, the centripetal force is Earth’s gravitational pull on every atom of your body. Thus, there is no compression (or pull) on any one part of your body and no sensation of a force acting on you. (The sensation is said to be one of “weight- lessness,” but that description is tricky. The pull on you by Earth has certainly not disappeared and, in fact, is only a little less than it would be with you on the ground.) Another example of a centripetal force is shown in Fig. 6-8. There a hockey puck moves around in a circle at constant speed v while tied to a string looped around a central peg. This time the centripetal force is the radially inward pull on the puck from the string. Without that force, the puck would slide off in a straight line instead of moving in a circle. Note again that a centripetal force is not a new kind of force. The name merely indicates the direction of the force. It can, in fact, be a frictional force, a gravitational force, the force from a car wall or a string, or any other force. For any situation: Figure 6-8 An overhead view of a hockey puck moving with constant speed v in a circular path of radius R on a horizontal frictionless surface. The centripetal force on the puck is T → , the pull from the string, directed inward along the radial axis r extending through the puck. String Puck R v r T The puck moves in uniform circular motion only because of a toward-the- center force. From Newton’s second law and Eq. 6-17 (a = v 2 /R), we can write the magnitude F of a centripetal force (or a net centripetal force) as F = m v 2 R (magnitude of centripetal force).

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  Questioning the Universe

  Concepts in Physics

  • Ahren Sadoff(Author)
  • 2008(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  23 4 Forces 4.1 THE FUNDAMENTAL FORCES What is a force? One answer is that it is a push or a pull. A better answer, that we will find to be more useful, is that it is an interaction between two or more objects. For most of our discussion, two objects will suffice. Forces are no strangers to us since we interact with all sorts of things every day. Below is a list of forces I have compiled. Before reading my list, it would be instruc-tive for you to take out a piece of paper and make your own list. Hopefully you will come up with some not on my list. Gravity Electric Weak nuclear Strong nuclear Centrifugal Magnetic Centripetal Friction Wind force Contact force (between surfaces) Muscular force Chemical Atomic I am sure you have noticed that my list is arranged in columns or categories. Let us look at the last column first. Both items are, in fact, not forces at all, but adjec-tives describing the action of a particular force. A centrifugal force is any force that is directed outward from the center of a curve when an object is traveling in curved motion. Similarly, a centripetal force acts inward toward the center of the curve. Gravity is usually the force most people list first, as I have. It, of course, is very important to us since it keeps us bound to the earth and the earth to the sun. The second column contains many familiar forces under one heading. Why? Because all these seemingly different forces are all due to only one force. Electric and magnetic are not separate forces, but just different manifestations of what is known as the electromagnetic force (we will discuss this in more detail shortly). The force that holds the atom together is not some special new force, but is just due to the electri-cal attraction of the negatively charged electrons to the positively charged protons in the nucleus. Similarly, different atoms interact by the attraction or repulsion of the electrons and protons in one atom acting on the electrons and protons of another atom.

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  Reeds Vol 2: Applied Mechanics for Marine Engineers

  • Paul Anthony Russell(Author)
  • 2021(Publication Date)
  • Reeds

    (Publisher)

  NEWTON’S FIRST LAW OF MOTION states that every object will continue in a state of rest or in uniform motion, in a straight line, unless acted upon by an external, unbalanced force. However, if a stone is connected to the end of a piece of string and whirled around in a circular path, an inward pull must be continuously exerted to keep it travelling in a circle. The stone itself is exerting an outward radial pull, trying to get move away from the centre and continue on a straight path. If the speed is increased, the force becomes greater until the string reaches its breaking point and snaps; the stone then flies off in a straight line at a tangent to the circle determined by the moment at the time of failure of the string. This can also be demonstrated by swinging a weight in a circle on the end of a string. When up to speed, if the string is released, the weight flies off dragging the string along. The outward radial force created by a body travelling in a circular path, due to its natural tendency to stay in a straight line as stated in Newton’s first law, is termed the centrifugal force. The inward pull applied to counteract the centrifugal force and keep the object on its circular path is termed the centripetal force; it is equal in magnitude to the centrifugal force and opposite in direction, as stated by Newton’s third law of motion. Consider a body moving at a constant speed of v around a circle of radius r. Referring to Figure 5.1, at the instant it is passing point a its instantaneous velocity is v in the direction tangential to the circle at the point a; a little further around the circle it is passing point b and its velocity is now v tangential to the circle at point b. Although the speed is constant, the velocity has changed because there has been a change of direction. If the movement from a to b is through a small angle θ, then to find the change of velocity, the vector diagram of velocities is drawn (Figure 5.1).

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