Physics
Centripetal Force
Centripetal force is the force that keeps an object moving in a circular path. It acts towards the center of the circle and is necessary to maintain the object's circular motion. This force is proportional to the mass of the object and the square of its velocity, and inversely proportional to the radius of the circular path.
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12 Key excerpts on "Centripetal Force"
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
In the case of a car moving on a flat circular track, the force is friction between the car and track. A satellite in circular orbit around Earth has a centripetal acceleration due to the gravitational force between the satellite and Earth. Some books use the term “Centripetal Force,” which can give the mistaken impression that it is a new force of nature. This is not the case: The adjective “cen- tripetal” in “Centripetal Force” simply means that the force in question acts toward a center. The force of tension in the string of a yo-yo whirling in a vertical circle is an example of a Centripetal Force, as is the force of gravity on a satellite circling the Earth. Consider a puck of mass m that is tied to a string of length m that is tied to a string of length m r and is being whirled r and is being whirled r at constant speed in a horizontal circular path, as illustrated in Figure 7.9. Its weight is supported by a frictionless table. Why does the puck move in a circle? Because of its inertia, the tendency of the puck is to move in a straight line; however, the string prevents motion along a straight line by exerting a radial force on the puck—a ten- sion force—that makes it follow the circular path. The tension T S S is directed along the string toward the center of the circle, as shown in the figure. In general, converting Newton’s second law to polar coordinates yields an equa- a- a tion relating the net Centripetal Force, F c c , which is the sum of the radial compo- nents of all forces acting on a given object, to the centripetal acceleration. The magnitude of the of the net Centripetal Force equals the mass times the magnitude of the Centripetal Force equals the mass times the magnitude of the centripetal acceleration: F c c 5 ma c 5 m v 2 r [7.19] A net force causing a centripetal acceleration acts toward the center of the circu- lar path and effects a change in the direction of the velocity vector.
No longer available |Learn morePhysics 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.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
The force of tension in the string of a yo-yo whirling in a vertical circle is an example of a Centripetal Force, as is the force of gravity on a satellite circling the Earth. Consider a puck of mass m that is tied to a string of length r and is being whirled at constant speed in a horizontal circular path, as illustrated in Figure 7.9. Its weight is supported by a frictionless table. Why does the puck move in a circle? Because of its inertia, the tendency of the puck is to move in a straight line; however, the string prevents motion along a straight line by exerting a radial force on the puck—a ten- sion force—that makes it follow the circular path. The tension T S is directed along the string toward the center of the circle, as shown in the figure. In general, converting Newton’s second law to polar coordinates yields an equa- tion relating the net Centripetal Force, F c , which is the sum of the radial compo- nents of all forces acting on a given object, to the centripetal acceleration. The magnitude of the net Centripetal Force equals the mass times the magnitude of the centripetal acceleration: F c 5 ma c 5 m v 2 r [7.19] A net force causing a centripetal acceleration acts toward the center of the circu- lar path and effects a change in the direction of the velocity vector. If that force should vanish, the object would immediately leave its circular path and move along a straight line tangent to the circle at the point where the force vanished. Centrifugal (‘center-fleeing’) forces also exist, such as the force between two particles with the same sign charge (see Topic 15). The normal force that pre- vents an object from falling toward the center of the Earth is another example of Tip 7.2 Centripetal Force Is a Type of Force, Not a Force in Itself! “Centripetal Force” is a classifica- tion that includes forces acting toward a central point, like the horizontal component of the string tension on a tetherball or gravity on a satellite.
- 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- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(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).
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)
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 = 2r 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.
- 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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 History of Centrifugal and Centripetal Forces In physics, the history of centrifugal and Centripetal Forces illustrates a long and complex evolution of thought about the nature of forces, relativity, and the nature of physical laws. Huygens, Leibniz, Newton, and Hooke Early scientific ideas about centrifugal force were based upon intuitive perception, and circular motion was considered somehow more natural than straight line motion. According to Domenico Meli: For Huygens and Newton centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently represented. Hence it is not located in nature, but is the result of a choice by the observer. In the first case a mathematical formulation mirrors centrifugal force; in the second it creates it. Christiaan Huygens coined the term centrifugal force in his 1659 De Vi Centrifiga and wrote of it in his 1673 Horologium Oscillatorium on pendulums. Isaac Newton coined the term Centripetal Force ( vis centripita ) in his discussions of gravity in his 1684 De Motu Corporum . Gottfried Leibniz as part of his solar vortex theory conceived of centrifugal force as a real outward force which is induced by the circulation of the body upon which the force acts. An inverse cube law centrifugal force appears in an equation representing planetary orbits, including non-circular ones, as Leibniz described in his 1689 Tentamen de motuum coelestium causis . Leibniz's equation is still used today to solve planetary orbital problems, although his solar vortex theory is no longer used as its basis. Leibniz produced an equation for planetary orbits in which the centrifugal force appeared as an outward inverse cube law force in the radial direction: .
eBook - PDFQuestioning 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.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity v and has magnitude F c = ma c , which can also be expressed as F c = m v 2 r or F c = mrω 2 , ⎫ ⎭ ⎬ ⎪ ⎪ 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force • Rotating and accelerated frames of reference are non-inertial. • Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames. 6.5 Newton’s Universal Law of Gravitation • Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this is 228 Chapter 6 | Uniform Circular Motion and Gravitation This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 F = G mM r 2 , where F is the magnitude of the gravitational force. G is the gravitational constant, given by G = 6.674×10 –11 N ⋅ m 2 /kg 2 . • Newton’s law of gravitation applies universally. 6.6 Satellites and Kepler’s Laws: An Argument for Simplicity • Kepler’s laws are stated for a small mass m orbiting a larger mass M in near-isolation. Kepler’s laws of planetary motion are then as follows: Kepler’s first law The orbit of each planet about the Sun is an ellipse with the Sun at one focus. Kepler’s second law Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times. Kepler’s third law The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun: T 1 2 T 2 2 = r 1 3 r 2 3 , where T is the period (time for one orbit) and r is the average radius of the orbit. • The period and radius of a satellite’s orbit about a larger body M are related by T 2 = 4π 2 GM r 3 or r 3 T 2 = G 4π 2 M.
- 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).
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
155 © 2010 Taylor & Francis Group, LLC Motion Under a Central Force A central force is a force whose line of action passes through a single point or center (fixed or in motion with constant velocity) and whose magnitude depends only on the distance from the center. Forces, such as gravity and electrostatic force, are all central forces. Perhaps the first example of central force motion to be recognized was that of the planets about the sun. In old quantum theory, Bohr’s hydrogen atom was described in terms of a classical two-body central picture. Certain two-body nuclear interactions, such as the scattering of alpha particles by nuclei, undoubtedly have a central character. 6.1 TWO-BODY PROBLEM AND REDUCED MASS Consider a conservative system of two particles of mass m 1 and m 2 . We shall limit ourselves to the case where the only forces acting are equal and opposite, directed along the line connecting the masses. Such a system has six degrees of freedom and, hence, six independent generalized coordinates. We can select these to be the three components of arrowrightnosp r 1 and the three components of arrowrightnosp r 2 . Alternatively, we can choose the three components of the position vector arrowrightnosp R of the center of mass (CM) of the system and the three components of arrowrightnosp arrowrightnosp arrowrightnosp r r r ( ) = -1 2 . Components arrowrightnosp ′ r 1 and arrowrightnosp ′ r 2 are position vec-tors of m 1 and m 2 with respect to the CM. The relationships among the various position vectors are illustrated in Figure 6.1. The position vector arrowrightnosp R of the center mass is given by arrowrightnosp arrowrightnosp R m r m r m m = + + 1 1 2 2 1 2 (6.1) and so in the CM system, m r m r 1 1 2 2 0 arrowrightnosp arrowrightnosp ′+ ′ = from which we have arrowrightnosp arrowrightnosp ′ = -′ r m m r 2 1 2 1 ( ) / .
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