Physics
Circular Motion and Gravitation
Circular motion refers to the movement of an object along a circular path, experiencing a centripetal force that keeps it in orbit. Gravitation is the force of attraction between objects with mass, such as planets and stars, and is responsible for keeping celestial bodies in orbit around each other. These concepts are fundamental to understanding the motion of objects in space.
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11 Key excerpts on "Circular Motion and Gravitation"
eBook - PDFThe Mechanical Universe
Mechanics and Heat, Advanced Edition
- Steven C. Frautschi, Richard P. Olenick, Tom M. Apostol, David L. Goodstein(Authors)
- 2008(Publication Date)
- Cambridge University Press(Publisher)
The new description begins with the mathematics of uniform circular motion, studied in Section 5.8 . In uniform circular motion we found that the body is not only constantly changing its position as it goes around, but is also constantly changing its velocity, which means that it has an acceleration. We showed that the acceleration always points toward the center, giving rise to the name centripetal, and has magnitude v 2 /r; 7.5 CIRCULAR ORBITS 155 -v 2 a = r. (7.23) According to Newton's second law, if a body has a centripetal acceleration there must be a force inducing the acceleration. In the case of orbits it's the force of gravity that supplies the centripetal acceleration. The force of gravity on the moon is F = Y—^t-(7.24) and it causes the moon to have an acceleration a = ¥/M M . (7.25) Combining Eqs. (7.24) and (7.25) , we obtain the moon's acceleration due to gravity, a = — ^ -E * . (7.26) The acceleration is centripetal - radially inward toward the center of the earth - and the orbit of the moon is very nearly a circle. Comparing Eqs. (7.26) and (7.23) , and canceling a common factor of r, we find that the requirement that gravity supply the centripetal acceleration for uniform circular motion of speed v is satisfied if . ( 7 . 2 7 ) At precisely this speed, the force of the earth's gravity makes the moon fall just the right amount to stay in its circular orbit. This is true not only of the moon but also for any satellite of any planet, including artificial ones. It is the basic mechanism of the solar system. Figure 7.11 Quantities pertaining to the moon's orbital motion. Example 4 Assuming no atmosphere, what horizontal speed would have to be imparted to a golf ball in order for it to orbit the earth at its surface? 156 UNIVERSAL GRAVITATION AND CIRCULAR MOTION Taking the radius of the orbit as the radius of the earth and using Eq.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
This geocentric model, which can be made progressively more accurate by adding more circles, is purely descriptive, containing no hints as to what are the causes of these motions. (b) The Copernican model has the Sun at the center of the solar system. It is fully explained by a small number of laws of physics, including Newton’s universal law of gravitation. Glossary ω , the rate of change of the angle with which an object moves on a circular path Δs , the distance traveled by an object along a circular path the curve in a road that is sloping in a manner that helps a vehicle negotiate the curve the point where the entire mass of an object can be thought to be concentrated a fictitious force that tends to throw an object off when the object is rotating in a non-inertial frame of reference the acceleration of an object moving in a circle, directed toward the center any net force causing uniform circular motion the fictitious force causing the apparent deflection of moving objects when viewed in a rotating frame of reference a force having no physical origin a proportionality factor used in the equation for Newton’s universal law of gravitation; it is a universal constant—that is, it is thought to be the same everywhere in the universe the angle at which a car can turn safely on a steep curve, which is in proportion to the ideal speed the sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction the maximum safe speed at which a vehicle can turn on a curve without the aid of friction between the tire and the road an environment in which the apparent net acceleration of a body is small compared with that produced by Earth at its surface 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 an accelerated frame of reference Chapter 6 | Uniform Circular Motion and Gravitation 227
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
WCN 02-200-202 202 TOPIC 7 | Rotational Motion and Gravitation Unless otherwise noted, all content on this page is © Cengage Learning. APPLYING PHYSICS 7.2 ARTI F I C I AL G R AV ITY a centrifugal force. Sometimes an insufficient centripetal force is mistaken for the presence of a centrifugal force (see section 7.4.2 Fictitious Forces, page 206). A radial force is a vector and has a direction. The second law for uniform cir- r- r cular motion involves forces that are directed either towards the center of a circle or away from it. A force acting towards the center of the circle is by convention negative. Examples include the gravity force on a satellite or the string tension of a whirling yo-yo. A force acting away from the center of the circle is positive. Exam- ples include the normal force on a car traveling over the circular crest of a hill or the force of repulsion between like electric charges. Similarly, the centripetal accel- eration is negative because it acts towards the center of the circle. Newton’s second law for uniform circular motion, written as a vector, therefore reads 2 m v 2 r 5 a F r r [7.20] where the forces F r r are the radial forces acting on the mass r are the radial forces acting on the mass r m, positive if the force is away from the center of the circle, negative if the force is towards the center of the circle. Centripetal, or center-seeking, forces have negative radial components, whereas centrifugal, or center-fleeing, forces have positive radial components. Newton’s second law for uniform circular motion is illustrated in Applying Physics 7.2 and Examples 7.6–7.8. Astronauts spending lengthy periods of time in space experi- ence a number of negative effects due to weightlessness, such as weakening of muscle tissue and loss of calcium in bones. These effects may make it very difficult for them to return to their usual environment on Earth.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
(b) When the object moves along the circle from O to P, the radius r traces out the same angle u. Here, the sector COP has been rotated clockwise by 908 relative to its orientation in Figure 5.2. 110 Chapter 5 | Dynamics of Uniform Circular Motion As Example 2 discusses, the object in Figure 5.4 would travel on a tangent line if it were released from its circular path suddenly at point O. It would move in a straight line to point A in the time it would have taken to travel on the circle to point P. It is as if, in the process of remaining on the circle, the object drops through the distance AP, and AP is directed toward the center of the circle in the limit that the angle u is small. Thus, the object in uniform circular motion accelerates toward the center of the circle at every moment. Since the word “centripetal” means “moving toward a center,” the acceleration is called centripetal acceleration. 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 airplane, 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 airplane 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. Answer (a) is correct. In the absence of a net force, the plane or any object would continue to move at a constant speed along a straight line in the direction it had at the time of release, con- sistent with Newton’s first law.
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.
- Jerry D. Wilson, Anthony J. Buffa, Bo Lou(Authors)
- 2022(Publication Date)
- CRC Press(Publisher)
We are used to feeling N upward on our feet to balance gravity. Rotation at the proper speed would simulate normal gravity. To an outside observer, a dropped ball would follow a tangential straight-line path, as shown. (b) A colonist on board the space colony would observe the ball to fall downward as in a normal gravitational situation. 184 College Physics Essentials • A centripetal force, F c , (the net force directed toward the center of a circle) is a requirement for circular motion, the magnitude of which is F ma mv r c c = = 2 (7.11) • Angular acceleration ( α ) is the time rate of change of angular velocity and is related to the tangential accelera- tion ( a t ) in magnitude by a r t = α (7.13) • According to Newton’s law of gravitation, every par- ticle attracts every other particle in the universe with a force that is proportional to the masses of both particles and inversely proportional to the square of the distance between them: F Gm m r G g 2 2 N m /kg = = × ⋅ - 1 2 2 11 6 67 10 . (7.14) • Acceleration due to gravity at an altitude h above the Earth’s surface: a GM R h g E E = + ( ) 2 (7.17) • Gravitational potential energy of a system of two masses: U Gm m r =- 1 2 (7.18) • Kepler’s first law (law of orbits): Planets move in elliptical orbits, with the Sun at one of the focal points. • Kepler’s second law (law of areas): A line from the Sun to a planet sweeps out equal areas in equal lengths of time. • Kepler’s third law (law of periods): T Kr 2 3 = (7.22) (K depends on the mass of the object orbited; for objects orbiting the Sun, K = 2.97 × 10 −19 s 2 /m 3 .) End of Chapter Questions and Exercises Multiple Choice Questions 7.1 Angular Measure 1. The radian unit is a ratio of (a) degree/time, (b) length, (c) length/length, (d) length/time. 2. For the polar coordinates of a particle traveling in a cir- cle, the variables are (a) both r and θ, (b) only r , (c) only θ, (d) none of the preceding.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
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. Gravity is the centripetal force that keeps the planets in orbit around the Sun and the Moon around the Earth. Newton was able to demonstrate mathematically that the centripe- tal acceleration of the Moon toward the Earth conformed to the inverse square law. The Moon's period around the Earth with respect to the distant stars is 27.3 days (2.36 X 10 6 s). This period is known as the sidereal month. It is shorter than the lunar month of 29.5 days due to the fact that the lunar month is mea- sured with respect to the Earth, which itself is moving around the Sun. Assuming the Moon orbits the Earth in uniform circular motion at a radius (distance between centers of Earth Rotational Motion 53 and Moon) of approximately 3.90 X 10 8 m, the magnitude of the angular velocity of the Moon around the Earth is and the angular acceleration of the Moon toward the Earth is This value can be compared to the value of the Earth's gravitation at the Moon's loca- tion calculated using the inverse square law. The distance between the Earth and Moon is approximately 60 times the radius of the Earth. The value of g at the Earth's surface is 9.8 m/s 2 . According to the inverse square law, the gravitational force is inversely pro- portion to the square of the distance: G <* l/d 2 . This means the value of g at a distance of 60 times the Earth radius should be about This value is identical (accounting for round- ing error and using approximate values in the calculation) to the calculated centrip- etal acceleration. Therefore, the centripetal acceleration of the Moon toward the Earth is provided by gravity.
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- 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.
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)
Applying Newton’s second law shows that the net force exerted on the planet is gravity and that it equals the centripetal force. Substitute Equation 6.7, F c 5 m 1v 2 / r 2 , for F C and Equation 7.4, F G 5 G 1m 1 m 2 / r 2 2 , for F G . a F r 5 M P a c F G 5 F C G M } M P r 2 5 M P v 2 r The planet mass M P cancels out of this equation, and so does one power of r. v 2 5 G M } r (7.7) We find the planet’s speed from Equation 4.30. v 5 2pr T (4.30) Square that speed and set it equal to Equation 7.7. v 2 5 4p 2 r 2 T 2 5 GM } r Rearrange this equation to arrive at Kepler’s third law for a planet in a circular orbit around the Sun. T 2 5 a 4p 2 GM } b r 3 (7.5) COMMENTS A more general derivation yields Equation 7.6 which holds when the orbit is ellipti- cal with semimajor axis a and an object of mass M is at one focus. If G and M are expressed in SI units, then T and a are in seconds and meters, respectively. T 2 5 a 4p 2 GM b a 3 (7.6) GENERAL FORM OF KEPLER’S THIRD LAW ▲ Special Case r ˆ F G u a c u FIGURE 7.16 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 7-3 Newton’s Law of Universal Gravity 195 Gravitational and Inertial Mass It may seem obvious that the gravitational force between any two objects depends on the objects’ masses, but there is no theoretical basis for this. That is, no one knows why the gravitational force depends on inertia (mass) and not some other property of the objects, such as volume, density, or composition. The mass in the law of universal gravity (Eq. 7.4) is referred to as the object’s gravitational mass. It is the property of the particles that creates a gravitational force between them. As we know, the mass in Newton’s second law ( g F u tot 5 ma u ) is the inertial mass of an object. Experimental evidence supports the idea that the gravita- tional mass of any object equals its inertial mass.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Forces do not always cause motion, however. For example, when you are sitting, a gravitational force acts on your body and yet you remain stationary. As a second example, you can push (in other words, exert a force) on a large boulder and not be able to move it. What force (if any) causes the Moon to orbit the Earth? Newton answered this and related questions by stating that forces are what cause any change in the velocity of an object. The Moon’s velocity changes in direction as it moves in a nearly circular orbit around the Earth. This change in velocity is caused by the gravitational force exerted by the Earth on the Moon. When a coiled spring is pulled, as in Figure 5.1a, the spring stretches. When a stationary cart is pulled, as in Figure 5.1b, the cart moves. When a football is kicked, as in Figure 5.1c, it is both deformed and set in motion. These situations are all examples of a class of forces called contact forces. That is, they involve physical contact between two objects. Other examples of contact forces are the force exerted by gas molecules on the walls of a container and the force exerted by your feet on the floor. Another class of forces, known as field forces, does not involve physical contact between two objects. These forces act through empty space. The gravitational force of attraction between two objects with mass, illustrated in Figure 5.1d, is an exam- ple of this class of force. The gravitational force keeps objects bound to the Earth and the planets in orbit around the Sun. Another common field force is the electric force that one electric charge exerts on another (Fig. 5.1e), such as the attractive electric force between an electron and a proton that form a hydrogen atom. A third example of a field force is the force a bar magnet exerts on a piece of iron (Fig. 5.1f). The distinction between contact forces and field forces is not as sharp as you may have been led to believe by the previous discussion.
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