Equation of Motion
The equation of motion is a mathematical expression that describes the relationship between an object's motion and the forces acting upon it. It is commonly used in physics and engineering to predict the behavior of objects in motion, such as projectiles, vehicles, and machinery. The equation of motion is derived from Newton's laws of motion and can be used to solve for various parameters like velocity, acceleration, and displacement.
4 Key excerpts on "Equation of Motion"
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...Rockets, for instance, have a fairly large rate of change of mass as fuel is consumed during takeoff and must be considered. The majority of mechanical systems on earth are composed of rigid or flexible bodies that do not suffer a mass change during their motions. For this reason, Equation 2.2 is most often used in the form, (2.3) where is defined to be the acceleration of the particle. (2.4) Both the velocity and the acceleration of the particle relate its motion to an inertial reference frame and they are termed the absolute velocity and absolute acceleration respectively. 2.4 Deriving Equations of Motion for Particles Newton's laws provide a very convenient method for deriving the equations of motion of simple systems. Equations of motion are the differential equations that, when solved, can be used to predict the response of the system to a set of applied forces. The procedure for deriving the equations has only four steps and, if they are followed, the derivations are very straightforward. The steps are, Kinematics – choose the coordinates (also known as degrees of freedom) to be used to describe the motion and derive expressions for the absolute velocities and accelerations of the masses under consideration. Use the methods of Chapter 1. Free body diagrams (FBDs) – sketch the masses under consideration as if they are in space with no forces acting on them. Then add to the sketches all of the externally applied forces acting on the masses. Also add the internal forces of interaction between the masses, being sure that they act in equal and opposite pairs as stipulated by Newton's third law. The FBDs should show the positive sense of the accelerations derived in step 1. Force balance equations (Newton's second law) – using the FBDs, equate the vector sum of forces on each body to its mass multiplied by its vector acceleration. Manipulate and solve the equations – the first three steps will lead to a set of equations with a set of unknowns...
eBook - ePubAdvanced Mechanical Vibrations
Physics, Mathematics and Applications
- Paolo Luciano Gatti(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...Chapter 2 Formulating the equations of motion 2.1 Introduction In order to describe the motion of a physical object or system, we must specify its position in space and time with respect to some observer or frame of reference. In this regard, physics teaches us that not all observers are on an equal footing because for a special class of them, called inertial observers, the laws of motion have a particularly simple form. More specifically, if one introduces the convenient concept of material particle – that is, a body whose physical dimension can be neglected in the description of its motion and whose position in space at time t is given by the vector r (t), then for all inertial observers, the particle’s Equation of Motion is given by Newton’s second law F = m a, where F is the vector sum of all the forces applied to the particle, a (t) = d 2 r / d t 2 (often also denoted by r ¨ (t)) is the particle acceleration and m, which here we assume to be a constant, is its (inertial) mass. This equation, together with the first law : ‘ a body at rest or in uniform rectilinear motion remains in that state unless acted upon by a force ’ and the third law : ‘ for every action there is an equal and opposite reaction ’ – both of which, like the second law, hold for inertial observers – is the core of Newtonian mechanics, which, as we mentioned at the beginning of the book, is one of the pillars of classical physics. Also, note that we spoke of all inertial observers because the class of inertial observers is potentially unlimited in number. In fact, any observer at rest or in uniform rectilinear motion with respect to an inertial observer is an inertial observer himself. Remark 2.1 In accordance with the knowledge of his time, Newton regarded the concepts of space and time intervals as absolute, which is to say that they are the same in all frames of reference...
eBook - ePub- Clifford Matthews(Author)
- 2011(Publication Date)
- Wiley-Blackwell(Publisher)
...Section 5 Motion 5.1 Making Sense of Equilibrium The concept of equilibrium lies behind many types of engineering analyses and design. 5.1.1 Definitions Formally An object is in a state of equilibrium when the forces acting on it are such as to leave it in its state of rest or uniform motion in a straight line. Practically The most useful interpretation is that an object is in equilibrium when the forces acting on it are producing no tendency for the object to move. Figure 5.1 Figure 5.1 shows the difference between equilibrium and non-equilibrium. 5.1.2 How is It Used? The concept of equilibrium is used to analyse engineering structures and components. By isolating a part of a structure (a joint or a member) which is in a state of equilibrium, this enables a ‘free body diagram’ to be drawn. This aids in the analysis of the stresses (and the resulting strains) in the structure. When co-planar forces acting at a point are in equilibrium, the vector diagram closes. 5.2 Motion Equations 5.2.1 Uniformly Accelerated Motion Bodies under uniformally accelerated motion follow the general equations given here. 5.2.2 Angular Motion 5.2.3 General Motion of a Particle in a Plane First law Everybody will remain at rest or continue in uniform motion in a straight line until acted upon by an external force. Second law When an external force is applied to a body of constant mass it produces an acceleration which is directly proportional to the force. i.e. Force (F) = mass (m) × acceleration (a) Third law Every action produces an equal and opposite reaction. 5.3.1 Comparisons: Rotational and Translational Motion 5.4 Simple Harmonic Motion (SHM) A particle moves with SHM when it has constant angular velocity (ω)...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Chapter 6 Dynamics 6.1 Introduction Dynamics is the study of objects in motion. This chapter thus follows on from Chapter 4, where the terms and equations used in describing linear motion were introduced. Here we now consider the forces responsible for motion and deal with the forces and linear momentum involved with linear motion. 6.2 Newton's laws The fundamental laws involved with the study of dynamics are Newton’s laws of motion. These can be stated as: Law 1 A body will continue in a state of rest or uniform motion in a straight line unless it is compelled to change that state by an externally applied force. Thus if an object is at rest then, unless a resultant force is applied to it there will be no motion. If an object is moving with a constant velocity then it will keep on in this motion until some externally applied force causes it to change its direction of motion and/or the magnitude of its velocity. Law 2 The rate of change of momentum of a body is proportional to the applied external force and takes place in the direction of action of that force. Momentum is defined as being the product of mass m and velocity v of a body. It is a vector quantity with the basic unit of kg m/s. The second law can thus be written as: Force is proportional to the rate of change of momentum (mv) The unit of force, the newton (N), is defined so that when the mass is in kg and the velocity in m/s, the force is in N and so: force = rate of change of momentum (mv) If mass m is constant, the above expression becomes: force = m × rate of change of v = ma where the rate of change of velocity is the acceleration a...
