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EA3_particledynamics - EA3 Systems Dynamics II Dynamics of...

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EA3: Systems Dynamics Dynamics of Particles Sridhar Krishnaswamy 1 II Dynamics of Particles II. 1. Motion of a particle: In EA2, you encountered the basic concepts of mechanics. You learnt how to solve for the motion of rigid particles under the action of external forces. Let us quickly recapitulate Newton's laws of motion. First, the linear momentum p of a particle is defined as mass m times its velocity v : p = m v . * Then, Newton's laws can be stated as follows. First Law: The linear momentum of a particle is conserved (does not change) unless the particle is acted upon by external forces. This is equivalent to saying that a particle will remain in its state of rest or uniform motion unless acted upon by external forces. Second Law: The time rate of change of linear momentum of a particle is equal to the net external force acting on it. That is: F = d p dt = d ( m v ) dt = m d v dt neglecting relativity } = m a (2.1) A particle acted upon by an external force will therefore accelerate in the direction of the force with a magnitude proportional to the magnitude of the applied load. Third Law: If a body A exerts an action (force) on a body B, the body B exerts a reaction (force) on body A of equal magnitude and opposite direction. Remarks: (i) Recall that Newton's law (2.1) is valid in any reference frame as long as it is inertial (that is it is not accelerating, and this includes rotation). (ii) We can use (2.1) in Cartesian, polar or any other coordinate system. We just need to remember that we express both the force and the acceleration vectors in the same system. * Bold letters will be used to denote vectors. (In class, we use under-tilde to denote vectors.}
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EA3: Systems Dynamics Dynamics of Particles Sridhar Krishnaswamy 2 Cartesian coordinates: F x i + F y j + F z k = m a x i + a y j + a z k { } = m d v x dt i + d v y dt j + d v z dt k = m d 2 r x dt 2 i + d 2 r y dt 2 j + d 2 r z dt 2 k (2.2) where i , j , k are unit vectors along the x,y and z-directions; F is the force vector, a is the acceleration vector, and r is the position vector. Normal and Tangential components: F t e t + F n e n = m a t e t + a n e n { } = m dv dt e t + v 2 e n (2.3) Where e t and e n are unit vectors tangential and normal to the path, and ρ is the radius of curvature of the path. (iii) In general, if we know the external forces acting on an object, we can use the above to compute the trajectory of the object. We will work out a few examples later. II. 2. Motion of Systems of Particles : It turns out that Newton's law F =m a holds for a system of particles, and indeed even for bodies of finite size (not just particles) provided we reinterpret its meaning somewhat: ie, if we consider in ' F ' the net external forces acting on the system , and if 'm' denotes the total mass, and ' a ' refers to the acceleration of a point called the center of mass of the system of particles. To see this, consider a collection of N particles as our system. Then, let the ith particle have a mass m i , and be located at position ri with respect to some chosen coordinate system. Let us consider the forces on the ith particle in two parts:
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