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.
6 Key excerpts on "Circular Motion and Gravitation"
eBook - ePub- Alan Hendrickson(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...Constant velocity implies a constant speed and a constant direction—in other words perfectly straight line motion. Whenever rotation occurs, all points on a spinning object, except those directly on the axis of rotation, constantly change direction. Change, be it in the speed of an object or in the direction of its travel, requires an acceleration. Centripetal acceleration is the name given to the always-directed-inward acceleration that must occur during rotation to keep an object spinning in a circle (see Figure 7.4c). Mathematically, centripetal acceleration equals Figure 7.4 Illustration of the terms involved in: a. Angular and linear displacement, b. Angular and linear speed and acceleration, c. Tangential and centripetal acceleration. (In both b. and c. rotational speed and acceleration are indicated in an easily understood way. The actual vectors for rotational velocity and acceleration would be shown by arrows along the axis of rotation, straight out of the page.) a c e n t r i p e t a l = v 2 r = r ω 2 Newton’s second law states that force is required to accelerate mass, and so a centripetal force can be defined. F c e n t r i p e t a l = m a c e n t r i p e t a l = m r ω 2 If you were to tie a weight to a string and spin it around above your head, the tension in the line would equal F centripetal. The cohesive forces within the steel of a shaft supply F centripetal, keeping the shaft from flying apart as it spins (though there are limits to the speed it can withstand)...
eBook - ePubNewton's Philosophy of Nature
Selections from His Writings
- Sir Isaac Newton, H. S. Thayer, H. S. Thayer(Authors)
- 2012(Publication Date)
- Dover Publications(Publisher)
...This is the nature of gravity upon earth; let us now see what it is in the heavens.That every body continues in its state either of rest or of moving uniformly in a right line, unless so far as it is compelled to change that state by external force, is a law of Nature universally received by all philosophers. But it follows from this that bodies which move in curved lines, and are therefore continually bent from the right lines that are tangents to their orbits, are retained in their curvilinear paths by some force continually acting. Since, then, the planets move in curvilinear orbits, there must be some force operating by the incessant actions of which they are continually made to deflect from the tangents.Now it is evident from mathematical reasoning, and rigorously demonstrated, that all bodies that move in any curved line described in a plane and which, by a radius drawn to any point, whether at rest or moved in any manner, describe areas about that point proportional to the times are urged by forces directed toward that point. This must therefore be granted. Since, then, all astronomers agree that the primary planets describe about the sun and the secondary about the primary areas proportional to the times, it follows that the forces by which they are continually turned aside from the rectilinear tangents and made to revolve in curvilinear orbits are directed toward the bodies that are placed in the centers of the orbits. This force may therefore not improperly be called centripetal in respect of the revolving body, and in respect of the central body attractive, from whatever cause it may be imagined to arise.Moreover, it must be granted, as being mathematically demonstrated, that if several bodies revolve with an equable motion in concentric circles and the squares of the periodic times are as the cubes of the distances from the common center, the centripetal forces will be inversely as the squares of the distances...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Chapter 23 Circular motion 23.1 Introduction This chapter is concerned with motion in a circle. Velocity is a vector quantity and has both size and direction. Thus, there is a constant velocity if equal distances are covered in the same straight line in equal intervals of time. Acceleration is the rate of change of velocity. Thus, there will be an acceleration if a velocity changes either as a result of equal distances in the same straight line not being covered in equal times or, if equal distances are being covered they are not in the same straight line, i.e. there is a change in direction. This is what happens with circular motion where we can have equal distances round the circumference of the circular path covered in equal times, i.e. constant speed, but the velocity is changing because the direction is continually changing and so there is an acceleration. 23.2 Centripetal acceleration Figure 23.1 Motion in a circle Consider a point object of mass m rotating with a constant speed in a circular path of radius r (Figure 23.1(a)). At point A the velocity will be v in the direction indicated. At B, a time t later, the velocity will have the same size but be in a different direction. If the direction has changed by the angle θ in time t, then the amount by which the velocity has changed can be obtained by resolving the velocity at B into two components, one in the same direction as the velocity at A of v cos θ and the other at right angles to it of v sin θ, i.e. along the radial direction AC. The acceleration in the direction of the velocity at A is thus (v cos θ 2 v)/ t = (cos θ - 1)v/t. The acceleration at an instant, rather than the average over the time t, is obtained by considering the value the average value tends to as we make t small and hence θ small. So, since cos θ tends to 1 as θ tends to 0, (1 - cos θ) tends to 0. Hence, the instantaneous acceleration in the direction of the velocity at A tends to 0...
eBook - ePub- William Bolton(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...It drives, via a belt drive, another pulley. What diameter will this need to be if it is required to rotate a shaft at 2.5 rev/s? 4.4.4 Combined linear and angular motion Consider objects which have both a linear motion and angular motion, e.g. a rolling wheel. For a wheel of radius r which is rolling, without slip, along a straight path (Figure 4.24), when the wheel rotates and rolls its centre moves from C to C′ then a point on its rim moves from O to O′. The distance CC′ equals OO′ But OO′ = r θ. Thus: Figure 4.24 Rolling wheel horizontal distance moved by the wheel x = r π [31] If this movement occurs in a time t, then differentiating equation [ 31 ]: and thus: horizontal velocity v x = r ω [32] where ω is the angular velocity of the wheel. Differentiating equation [ 32 ] gives: and thus: horizontal acceleration = ra [33] where a is the angular acceleration. 4.5 Force and linear motion In considering the effects of force on the motion of a body we use Newton’s laws. Newton’s laws can be expressed as: First law A body continues in its state of rest or uniform motion in a straight line unless acted on by a force. Second law The rate of change of momentum of a body is proportional to the applied force and takes place in the direction of the force. Third law When a body A exerts a force on a body B, B exerts an equal and opposite force on A (this is often expressed as: to every action there is an opposite and equal reaction). Thus the first law indicates that if we have an object moving with a constant velocity there can be no resultant force acting on it. If there is a resultant force, then the second law indicates that there will be a change in momentum with: [34] For a mass m which does not change with time: d v /d t is the acceleration. The units are chosen so that the constant of proportionality is 1 and thus the second law can be expressed as: F = ma [35] If a body of mass m is allowed to freely fall then it would fall with the acceleration due to gravity g...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 19 Force, mass and acceleration Why it is important to understand: Force, mass and acceleration When an object is pushed or pulled, a force is applied to the object. The effects of pushing or pulling an object are to cause changes in the motion and shape of the object. If a change occurs in the motion of the object then the object accelerates. Thus, acceleration results from a force being applied to an object. If a force is applied to an object and it does not move, then the object changes shape. Usually the change in shape is so small that it cannot be detected by just watching the object. However, when very sensitive measuring instruments are used, very small changes in dimensions can be detected. A force of attraction exists between all objects. If a person is taken as one object and the Earth as a second object, a force of attraction exists between the person and the Earth. This force is called the gravitational force and is the force that gives a person a certain weight when standing on the Earth’s surface. It is also this force that gives freely falling objects a constant acceleration in the absence of other forces. This chapter defines force and acceleration, states Newton’s three laws of motion and defines moment of inertia, all demonstrated via practical everyday situations. At the end of this chapter, you should be able to: define force and state its unit appreciate ‘gravitational force’ state Newton’s three laws of motion perform calculations involving force F = ma define ‘centripetal acceleration’ perform calculations involving centripetal force = mv 2 r define ‘mass moment of inertia’ Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved. 19.1 Introduction When an object is pushed or pulled, a force is applied to the object...
eBook - ePubBasic Engineering Mechanics Explained, Volume 1
Principles and Static Forces
- Gregory Pastoll, Gregory Pastoll(Authors)
- 2019(Publication Date)
- Gregory Pastoll(Publisher)
...To understand this, we need to understand the connection between weight and gravity. Weight, Newton’s Law of Gravitation, and gravitational acceleration Isaac Newton, following the work of Robert Hooke, confirmed that a gravitational force exists between any two objects in the universe. Simply due to the fact that the objects have mass, they attract each other. This means that if two rocks are drifting in space, near one another, they will each exert a pull on the other, the result of which is that they will gradually accelerate towards one another. The more mass the two objects possess, the greater is this force. The further apart they are, the smaller is this force. Newton’s Law of Gravitation tell us that the magnitude of the gravitational force, F, depends on these variables, in the following relationship: F = (Gm 1 m 2)/r 2 Where G is the universal gravitation constant, m 1 and m 2 are the masses of the two objects, and r is the distance between their centres of gravity. This law applies to all bodies, not only to the gravitational force between a given object and the Earth. Every object that possesses mass attracts every other object with a force of attraction given by the above equation. It might be hard to believe that two pencils lying on your desk exert a gravitational pull on one another, but they do. The magnitude of this pull is insignificant, because both of their masses are vastly smaller than the mass of the earth. The magnitude of the gravitational force of attraction between two small objects was first determined in a famous experiment by the British scientist Henry Cavendish in 1797. A torsion wire suspended from an overhead beam supported a horizontal rod, on the ends of which were mounted two small lead spheres. When the rod had settled into an equilibrium position, two larger spheres were brought from a distance, to a position on the circle of movement close to the small ones, and equidistant from them...
