Equilibrium in rigid body systems video introduction

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# Equilibrium in rigid body systems video introduction

Equilibriumin physicsthe condition of a system when neither its state of motion nor its internal energy state tends to change with time. A simple mechanical body is said to be in equilibrium if it experiences neither linear acceleration nor angular acceleration; unless it is disturbed by an outside force, it will continue in that condition indefinitely. For a single particle, equilibrium arises if the vector sum of all forces acting upon the particle is zero. A rigid body by definition distinguished from a particle in having the property of extension is considered to be in equilibrium if, in addition to the states listed for the particle above, the vector sum of all torques acting on the body equals zero so that its state of rotational motion remains constant.

An equilibrium is said to be stable if small, externally induced displacements from that state produce forces that tend to oppose the displacement and return the body or particle to the equilibrium state. Examples include a weight suspended by a spring or a brick lying on a level surface. An equilibrium is unstable if the least departure produces forces that tend to increase the displacement.

An example is a ball bearing balanced on the edge of a razor blade. In thermodynamics the concept of equilibrium is extended to include possible changes in the internal state of a system, as characterized by its temperature, pressuredensity, and any other quantities needed to specify its state completely. At strict thermodynamic equilibrium, the temperature of the system is uniform otherwise heat would flowand any gradients in state functions such as pressure or density are balanced by external forces so that they remain constant.

For example, the equilibrium pressure at the bottom of a column of air is higher than at the top because of the force of gravityand density gradients in a centrifuge are balanced by the centrifugal force. It is also useful to consider quasi-equilibrium processes where, for example, temperature gradients are allowed if the rate of heat flow is too slow to be significant adiabatic processesbut the system is otherwise in local thermodynamic equilibrium. For example, the adiabatic expansion of a rising column of air accounts for the decrease of atmospheric temperature with altitude.

Article Media. Info Print Cite. Submit Feedback. Thank you for your feedback. Equilibrium physics. See Article History. Alternative Titles: mechanical equilibrium, static equilibrium. This article was most recently revised and updated by William L. HoschAssociate Editor. Learn More in these related Britannica articles:.Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law kinetics or their derivative form Lagrangian mechanics.

The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.

If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement. Determine the resultant force and torque at a reference point Rto obtain. For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation.

Use the center of mass C as the reference point, so these equations for Newton's laws simplify to become. Several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections.

The first attempt to represent an orientation is attributed to Leonhard Euler. He imagined three reference frames that could rotate one around the other, and realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space using two rotations to fix the vertical axis and another to fix the other two axes. The values of these three rotations are called Euler angles.

These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane.

In aerospace engineering they are usually referred to as Euler angles. Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis Euler's rotation theorem. Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle.

Therefore, any orientation can be represented by a rotation vector also called Euler vector that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector.

A similar method, called axis-angle representationdescribes a rotation or orientation using a unit vector aligned with the rotation axis, and a separate value to indicate the angle see figure. With the introduction of matrices the Euler theorems were rewritten. The rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix.

The above-mentioned Euler vector is the eigenvector of a rotation matrix a rotation matrix has a unique real eigenvalue.

The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe. Orientation may be visualized by attaching a basis of tangent vectors to an object. The direction in which each vector points determines its orientation. Another way to describe rotations is using rotation quaternionsalso called versors. They are equivalent to rotation matrices and rotation vectors.

With respect to rotation vectors, they can be more easily converted to and from matrices.Arrange the comparative stabilities of the three objects shown in the figure in terms of work and potential energy. The figure shows a dish kept at an orientation with a marble placed on it. If the marble in figure is displaced slightly, it would rattle about for a while and come to rest in its original place. This is an example of :.

The rock has a mass of 1 kg. What is the mass of the measuring stick if it is balanced by a support force at the one-quarter mark?

The stick is balanced by the 1 kg rock that is one-quarter of its length to the left and by its own weigh which is at the Centre of Mass, one-quarter of its length to the right. Therefore, the mass of the stick must be one kilogram, 1 kg. A meter rule is supported at its centre.

It is balanced by two weights, A and B.

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If A and B are placed at a distance 30 cm, and 60 cm from the centre of scale, find the weight of B. The weight of A is 50N.

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Determine the force that must be applied to end of the class 1 lever shown below to lift the lb load. Five forces are separately applied to a flat object lying on a table of negligible friction as shown in figure. The force which will not cause the object to rotate about the centre of the object is.

Class 11 chapter 7 - Rotational Motion 03 - Rotational Equilibrium IIT JEE / NEET - Torque Problem -

There are two blocks of mass 3 kg and 5 kg hanging from the ends of a rod of negligible mass. The rod is marked in eight equal parts as shown.

## 2-7 Equilibrium of a System of Rigid Bodies

At which of the points indicated should a string be attached if a rod is to remain horizontal when suspended from the string? Login New User. Sign Up. Forgot Password? New User? Continue with Google Continue with Facebook. Gender Male Female. Create Account. Already Have an Account?In order to make a curve with a bicycle or a motorcycle there must be sufficient friction between the tires and the road, because the frictional force provides the centripetal force that makes the bike follow the curved trajectory.

But the frictional force also makes the rider and the bike rotate sideways in the direction opposite to the center of the curve; to counteract that rotation the rider tilts with the bike, making their weights produce a tendency to rotate in the opposite direction.

The tendency to rotate produced by a force is called moment and will be studied in this chapter. The high speeds in motorcycle races imply larger tilt angles.

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To make the motrocycle tilt more, the rider first turns the wheel in the opposite sense of the curve and moves his body away from the motorcycle in the direction of the center of the curve. The vectors introduced in chapter 2 are free vectors, which are considered equal if they have the same magnitude, direction and sense, regardless of the point of the space where they are. In the case of forces, it is not enough to know their magnitude, direction and sense.

For example, when a force is applied to a door to close it, besides the magnitude, direction and sense of the force, the point at which that force is applied will also be important. The further away from the hinges the force is applied, the easier it is to close the door. The force required to close the door would be very high if it was applied at a point near the hinges.

Suppose we want to move a chair to another place, lifting it with just one hand. As the chair moves up, it will rotate until it reaches an equilibrium position as shown on the right-hand side of the figure, where the weight and the force applied by the hand are in the same vertical line.

If we could apply the upward force in a point in the same vertical line as the center of gravity C, the chair would go straight up without rotating. We conclude that to predict the effect of a force it is necessary to know its magnitude, direction, sense and also its line of actionwhich is the straight line passing through the point where the force is applied, in the direction of the force.

A force produces different effects if it is moved to a different line of action, even if its magnitude, direction and sense are kept equal.

These kind of vectors, with a specific line of action, are called sliding vectors. The point where they are applied is not important, as long as it is in their line of action.

The chair would rotate in the same direction, with the same angular acceleration. Whenever it was necessary to add forces in chapter 4we assumed that they could be moved freely and added together as free vectors, using the parallelogram rule.

In the following sections it is shown that this sum of forces as if they were free vectors is not wrong, as long as the rotation effect introduced when moving a force to a different line of action is also taken into account.

The translation of a body is determined by the resultant force, which can be obtained adding all external forces as free vectors.In this course, you will learn the conditions under which an object or a structure subjected to time-invariant static forces is in equilibrium - i. This course is suitable for learners with interest in different Engineering disciplines such as civil engineering, architecture, mechanical engineering, aerospace. Non engineering disciplines may also find the course very useful, from archaeologist who are concerned about the stability of their excavation sites to dentists interested in understanding the forces transmitted through dental bridges, to orthopedic surgeons concerned about the forces transmitted through the spine, or a hip or knee joint.

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The content will be primarily delivered using light board. Katafygiotis is going to write and sketch with color markers directly on the board while facing you. You will have an exciting and interactive learning experience online! Awesome the way they teach the concept is great and I completely satisfied by taking this course thankyou Thank you very much.

We will study the effect of a system of forces and couples acting on a rigid body and will state the conditions of equilibrium for particles, rigid bodies and assemblies of rigid bodies. The number of independent number of equations that are necessary and sufficient for satisfying equilibrium will be clearly stated.

Loupe Copy. Mastering Statics. Enroll for Free. From the lesson. Equivalent Systems of Forces and Equilibrium. Taught By. Lambros Katafygiotis Professor. Try the Course for Free.

Explore our Catalog Join for free and get personalized recommendations, updates and offers. Get Started. All rights reserved.The concept of equilibrium is introduced to describe a body which is stationary or which is moving with a constant velocity. A body under such a state is acted upon by balanced forces and balanced couples only.

There is no unbalanced force or unbalanced couple acting on it. The concept must be really understood by every student. The size of a particle is very small compared to the size of the system being analysed. Rigid body. A body is formed by a group of particles. The size of a body affects the results of any mechanical analysis on it. Most bodies encountered in engineering work can be considered rigid from the mechanical analysis point of view becase the deformations that take place within these bodies under the action of loads can be neglected when compared to other effects produced by the loads. Springs undergo deformations that cannot be neglected when acted upon by forces or moments.

For the analyses ini this book, only the effects of the deformations of springs on a rigid body interacting with the springs are considered but the springs themselves will not be analysed as a body. In general, a force acting on a particle tends to cause the particle to translate.

Also, a force on on a body not only tends to cause the body to translate as in the case of the particle but also tends to cause the body to rotate about any axis which does not intersect with or is not parallel to the line of action of the force. To see what actually happens to any particular part of a structure, that part has to be isolated from the other parts of the body.

A mechanical system is defined as a body system that can be isolated from other bodies. The system can be formed by a single body, part of a body, or a group of connected bodies. The bodies forming the system can either be rigid or non-rigid.

A mechanical system can be solid, fluid, or even a combination of solild and fluid. The isolation of a mechanical system is achieved by cutting and isolating the system from its surroundings. The isolation enables us to see the interactions between the isolated part and the other parts. The part which has been cut imaginarilyforms a free body. A diagram which portrays the free body, complete with the system of external forces acting on it due to its interaction with the parts which have been removed, is called the free-body diagram FBD of the isolated part.

The FBD of of a body system shows all loads acting on the external boundary of the isolated body. Assume that an analysis is to be carried-out on the whole structure of the arm when it is carrying a load as shown, where the weight of the component members of the arm can be nglected compared to the weight of the load.

Assume also that all joints of the arm do not prevent rotation around the respective joints, i. Because the direction and the sense of every reactive force are not known, the direction nad sense shall be assumed. The arm can be isolated from the body of the lift truck at point A where it is pinned to the body of the lorry and at point C where it is acted upon by the active forceof the hydraulic piston rod.

The isolated arm is shown in Figure 3. Figure 3. If what is to be analysed is the load container only, the FBD shown in Figure 3. Please note that, in the FBDs shown, the direction and sense of all the reactions are drawn arbitarily because they are assumed to be unknown.

We will learn later how to determine the direction of some types of reactivce forces through observation.The Mechanics Map is an open textbook for engineering statics and dynamics containing written explanations, video lectures, worked examples, and homework problems. All content is licensed under a creative commons share-alike license, so feel free to use, share, or remix the content.

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Newtonian Mechanics Video Introduction 1.

### 2-7 Equilibrium of a System of Rigid Bodies

Statically Equivalent Systems Video Introduction 4. Engineering Structures Video Introduction 5. Friction and Friction Applications Video Introduction 6. Tipping 6. Particle Kinematics Video Introduction 7.

### 5. Dynamics of rigid bodies

Work and Energy in Particles Video Introduction 9. Impulse Momentum Methods Video Introduction Work and Energy Methods Video Introduction Welcome to the Mechanics Map Digital Textbook: The Mechanics Map is an open textbook for engineering statics and dynamics containing written explanations, video lectures, worked examples, and homework problems.

Mechanics Basics: 1. Rigid Body Kinematics: Newton's Second Law for Rigid Bodies: Vector and Matrix Math: A1. Moment Integrals: A2. Video Introduction. 