Force as a Vector
In the context of mathematics, force as a vector refers to the representation of force as a quantity with both magnitude and direction. This concept is essential for understanding how forces act on objects in different directions and can be combined using vector addition. By treating force as a vector, mathematical models can accurately describe the complex interactions of forces in various physical systems.
8 Key excerpts on "Force as a Vector"
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...A force is a vector quantity and in this chapter, the resolution of forces is introduced. Resolving forces is very important in structures, where the principle is used to determine the strength of roof trusses, bridges, cranes, etc. The resolution of forces is also used in studying the motion of vehicles and other particles in dynamics, and in the case of the navigation of ships, aircraft, etc., the vectors take the form of displacements, velocities and accelerations. This chapter gives a sound introduction to the manipulation and use of scalars and vectors, by both graphical and analytical methods. Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved. 20.2 Scalar and vector quantities Quantities used in engineering and science can be divided into two groups: (a) Scalar quantities have a size (or magnitude) only and need no other information to specify them. Thus, 10 centimetres, 50 seconds, 7 litres and 3 kilograms are all examples of scalar quantities. (b) Vector quantities have both a size or magnitude and a direction, called the line of action of the quantity. Thus, a velocity of 50 kilometres per hour due east, an acceleration of 9.81 metres per second squared vertically downwards and a force of 15 newtons at an angle of 30 degrees are all examples of vector quantities. 20.3 Centre of gravity and equilibrium The centre of gravity of an object is a point where the resultant gravitational force acting on the body may be taken to act. For objects of uniform thickness lying in a horizontal plane, the centre of gravity is vertically in line with the point of balance of the object...
eBook - ePubBasic Engineering Mechanics Explained, Volume 1
Principles and Static Forces
- Gregory Pastoll, Gregory Pastoll(Authors)
- 2019(Publication Date)
- Gregory Pastoll(Publisher)
...and vectors Any phenomenon such as length, weight, and time, that can be measured, is referred to as a ‘quantity’. A quantity can be either a scalar or a vector. Scalars: These are quantities that can be measured on some single scale of magnitude. For example: mass, length, volume, density, elasticity, and temperature. Vectors: These are all quantities that need to be specified in terms of both magnitude and direction, in order to be properly defined. For example: displacement, velocity, acceleration, force, and stress. Suppose you are told about a force with a magnitude of 86 N. This number alone cannot completely define the force, because a downward force of 86 N is very different from a sideways or an upward force of the same value. Likewise, a displacement of 24 m to the left of a reference point is completely different from a displacement of 24 m to the right of that point. Any given vector acts in a specific direction, which has to be specified, in order to define the vector. All vectors can be indicated graphically by drawing an arrow showing the direction in which they have an effect. The magnitude of the vector is indicated by the length of the arrow, drawn to some chosen scale. Adding scalars In cases where it is meaningful to add scalars, they can be added arithmetically, because they don’t have different effects in different directions. If a bucket contains 3 kg of sand, and you add another 4kg of sand, you have altogether 7 kg of sand...
eBook - ePubStructure for Architects
A Primer
- Ramsey Dabby, Ashwani Bedi(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
...CHAPTER 7 Working with Forces 7.1 Forces, Vectors, and Lines of Action From Chapter 5 we have a basic understanding of a force. A force is also a vector quantity, meaning that it has magnitude (e.g., pounds, kips) and direction (e.g., up, down, left, right). Because it is a vector, a force can be represented graphically by being drawn to size at a particular scale and in a particular direction (bearing). The infinite imaginary line passing through a vector force is called its line of action (Figure 7.1). Figure 7.1 Force as a Vector Quantity. A 10 lb force with a line of action having a bearing from (0, 0) of N 60 degrees E. Principle of Transmissibility The principle of transmissibility states that a given force can be applied on a body (the point of application) anywhere along the force's line of action without causing any external effect on that body. Note external effect! In fact, changing the point of application can have a marked change on the internal effect on that body. Let's use the following example of a load on a truss to see the difference between the external and internal effect of a point of application. EXAMPLE 7a: Principle of Transmissibility A 4 kip downward vertical force P is applied at point M (the point of application) on the top chord of a truss (Figure 7.2). What are reactions R 1 and R 2 ? Figure 7.2 Point of Application of Force on Top Chord Using the equilibrium equations, R 1 and R 2 are readily determined to be 3 kips and 1 kip, respectively. P is now applied along the same line of action at point N on the bottom chord of the truss (Figure 7.3)...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Chapter 2 Statics 2.1 Introduction Statics is concerned with the equilibrium of bodies under the action of forces. Thus, this chapter deals with situations where forces are in equilibrium and the determination of resultant forces and their moments. 2.1.1 Scalar and vector quantities Scalar quantities can be fully defined by just a number; mass is an example of a scalar quantity. To specify a scalar quantity all we need to do is give a single number to represent its size. Quantities for which both the size and direction have to be specified are termed vectors. For example, force is a vector and if we want to know the effect of, say, a 100 N force then we need to know in which direction the force is acting. To specify a vector quantity we need to indicate both size and direction. To represent vector quantities on a diagram we use arrows. The length of the arrow is chosen according to some scale to represent the size of the vector and the direction of the arrow, with reference to some reference direction, the direction of the vector, e.g. Figure 2.1 to represent a force of 300 N acting in a north-east direction from A to B. Figure 2.1 A vector In order to clearly indicate in texts when we are referring to a vector quantity, rather than a scalar or just the size of a vector quantity, it is common practice to use a bold letter such as a, or when hand-written by underlining the symbol a. When we are referring to the vector acting in directly the opposite direction to a, we would use - a, with the minus sign being used to indicate that it is in the opposite direction to a. If we want to just refer to the size of a we write a. 2.1.2 Internal and external forces The term external forces is used for the forces applied to an object from outside (by some other object). The term internal forces is used for the forces induced in the object to counteract the externally applied forces (Figure 2.2). To illustrate this, consider a strip of rubber being stretched...
eBook - ePubBiomechanics of Human Motion
Applications in the Martial Arts, Second Edition
- Emeric Arus, Ph.D.(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...The kilogram (kg) is an SI base unit of mass. From here results that a mass of 1 kg = 9.8 m/s 2. A man of 70 kg mass × 9.8 m/s 2 will have 686 Newton force. 8.2 FORCES Recall that kinetics deal with motion that includes the forces that cause the motion. The physical property of the force, perhaps, is the most important between other physical properties, such as momentum, impulse, energy, velocity, and so on, which are related to objects in static positions or dynamic motions, including humans and animals alike. Force is described as the effect one object has on another, which can alter the state of a matter by pushing, pulling, twisting, sliding, and so on, and this effect of the force (F) is the product of a mass and its acceleration F = m ⋅ a. There are many different kinds of forces such as internal forces; for example, in biomechanics, a muscle contraction produces internal forces that move a segment of the body. External forces represent actions on the system of the human body such as a pushing, pulling, or twisting force or even push of the wind. The gravity and inertial forces have been described in Chapter 6. Force is a vector quantity that includes magnitude and direction. The force of action usually includes a point and a line of application. The force vector is represented by an arrow, usually a straight line from the origin of the force that shows the direction of the force and the length of the arrow represents the magnitude of the force. Other vector quantities are velocity, acceleration, momentum, torque, and impulse. In the human body, the force is represented by the skeletal muscle. A specification that is important to be mentioned here is that muscles can execute only pulling actions. To better understand the pulling actions of the muscles, here is the explanation: When you pull a cart, you say there is a pulling action of the biceps muscle. But when you push a table, you probably will say there is a pushing action...
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...In order to move the sledge from rest the parent has to apply a force to the rope to overcome the resistance due to friction. Once the friction is overcome and the sledge is moving, any further increase in the force will merely accelerate the sledge. If two children are sitting on the sledge, the parent has to pull twice as hard to move the sledge. This shows that the greater the load, the greater the force needed to overcome friction. A basic frictional system can be idealized as in Figure 4.8, where N = force applied due to the weight of the block and is always at right angles to the friction plane and F r = friction force (resisting force due to friction), as explained later in section 6.2. From these observations the following equation can be deduced and is developed in more detail in Chapter 6 : F r = μ N (6.2) Fig. 4.8 Basic friction system. 4.3 Vector Representation of a Force (Two-Dimensional) A force has the following characteristics: magnitude direction or line of action point of application. In Figure 4.9(a) the force in the rope, which suspends the block, is exerting a force F in a vertical direction. This force F can be represented by a vector since a vector requires both magnitude and direction for it to be valid. Suppose the block weighs 50 N. The force in the rope will be 50 N at all points along its length and can be depicted by a line whose scale length represents the 50 N. An arrowhead shows the direction in which the force is applied, as shown in Figure 4.9(b). Since forces always act in pairs, a force and its reaction, there is a force W acting downwards, of 50 N, due to the weight of the block. It is represented by a vector pointing downwards. There is also a reactive force F, supporting the weight and directed upwards. The representing vector should also follow that direction. It is said to show the sense of the force...
eBook - ePub- Paul McMullin, Jonathan Price, Paul W. McMullin, Jonathan S. Price(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...This is because gravitational force on each object is proportional to its mass. Third Law: for every action, there is an equal and opposite reaction : Objects are in static equilibrium when an applied force is opposed by an equal and opposite force. This occurs when all the applied forces on an object, including self-weight, sum to zero. For example, a book resting on your table (Figure 5.2) has a weight measured in lb f or N. The table pushes back up with an equal force (normal or perpendicular to the book), therefore the book is in equilibrium. If you were to place too much weight on the table, it would collapse and therefore it would not be in static equilibrium. Newton’s Third Law simply states that the sum of all forces on an object must total zero, see Equation (5.1). Without question, Newton’s First and Third Laws provide the foundation upon which modern structural analysis techniques are built. To apply these laws, we assume three basic conditions apply. The object must: be motionless (or “static”) be perfectly rigid (i.e. no significant change in shape) not rotate under the action of the applied forces. Figure 5.2 Book supported by a table 5.3 Vectors A force vector has magnitude, line of action, and direction (see Figure 5.3). Direction is important, as is the magnitude. Also, direction determines the magnitude of each component (x, y, and z). Three entities are needed to graphically describe a force vector: Magnitude (a scalar quantity, pounds or kips or N or kN: 1 k = 1,000 pounds, 1 kN = 1,000 N). Line of action or direction. Sense. When represented graphically, vectors are drawn as arrows with these three entities shown: magnitude is a numerical annotation, line of action is a straight line, and the arrowhead represents the sense of the vector. Figure 5.3 Force vector terminology 5.3.1 Force Vectors The pull of gravity on an object’s mass is an example of a force vector. Resistance to gravitational forces can also be described as vectors...
eBook - ePub- Alan Hendrickson(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...If it is not, the problem’s givens will have to change to reduce this speed. Torque The rotational equivalent of force is torque. Torque is a twisting force, the action a screwdriver applies to a woodscrew, or a wrench applies to a nut. In machinery, motor’s and gear reducer’s shafts provide torque to drive their loads, and both roller chain sprockets and cable drums convert easily between a torque on a shaft and a force (in the form of chain or cable tension). Torque results from force acting at some angle and at some distance from an axis of rotation. Mathematically it is defined as: T = r × F T = r F sin θ Where T, T = torque, vector and scalar respectively (ft-lb, in-lb, Nm) r, r = radius, or the distance from axis of rotation to the point where force is applied (ft, in, m) F, F = force (lb, N) θ = the angle between the vectors r and F (deg, or rad) Vectors appear here briefly because they are an essential part of the definition of torque, but they will soon vanish again because the movements of scenery we need to describe are generally simple enough as to be done without the full vector treatment. The bold × indicates a “cross product”, an operation between two vectors with a vector result that follows the “right hand rule” illustrated in Figure 7.5. Basically, the mathematics has to differentiate between the clockwise and counterclockwise twists that are the directions of torque, and this is accomplished with the direction convention that is a part of the cross product. If, as was done in the linear section, we confine all our problems to one dimension, or one axis of rotation, the direction of torque can only take on two directions, easily represented mathematically as + and – signs. Figure 7.5 The right hand rule helps to determine the direction of T The scalar component of torque, that is its magnitude or size, involves a sine term to account for the effect that the angle between radius and force has on the value of torque...
