Central Forces

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Date: 13.05.2020 E-CONTENT SUBJECT – PHYSICS CLASS - B.Sc. ( Hons.) Part I Dr. Debarati Ghosh Department of Physics, T. P. S. College, Patna. CENTRAL FORCES � � � � � � Fig. 1 Suppose that a force acting on a particle of mass in is such that [see Fig. 1]: (a) it is always directed from � toward or away from a fixed point O , (b) its magnitude depends only on the distance r from O . Then we call the force a central force or central force field with O as the center of force. Thus F is a central force if and only if (1) 1 r where r = is a unit vector in the direction of r . 1 r The central force is one of attraction toward O or repulsion from O according as f (r)<0 or f (r)>0 respectively. SOME IMPORTANT PROPERTIES OF CENTRAL FORCE FIELDS If a particle moves in a central force field, then the following properties are valid. Property 1: The path or orbit of the particle must be a plane curve, i.e. the particle moves in a plane. Property 2: The angular momentum of the particle is conserved, i.e. is constant. Property 3: The particle moves in such a way that the position vector or radius vector drawn from O to the particle sweeps out equal areas in equal times. In other words, the time rate of change in area (i.e. the areal velocity) is constant. This is sometimes called the law of areas. Property 4: A central force is conservative in nature. Prove that if a particle moves in a central force field, then its path must be a plane curve i.e. the particle moves in a plane. Let F=f (r)r1 be the central force field. Then (1) Since r1 is a unit vector in the direction of the position vector r . Since d v F=m , this can be written dt 2 (2) or (3) Integrating, we find (4) where h is a constant vector. Multiplying both sides of equation (4) by r • , (5) using the fact that Thus r is perpendicular to the constant vector h , and so the motion takes place in a plane. We shall assume that this plane is taken to be the xy plane whose origin is at the center of force. Prove that for a particle moving in a central force field the angular momentum is conserved. Let F=f (r)r1 be the central force field. Then (1) Since r1 is a unit vector in the direction of the position vector r . Since d v F=m , this can be written dt (2) 3 or (3) Integrating, we find (4) where h is a constant vector. Multiplying both sides of equation (4) by m , we have m(r×v)=mh =constant (5) Since the left side of expression (5) is the angular momentum, it follows that the angular momentum is conserved, i.e. is always constant in magnitude and direction. 4.

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Central Forces Notes
Central Forces Corinne A. Manogue with Tevian Dray, Kenneth S. Krane, Jason Janesky Department of Physics Oregon State University Corvallis, OR 97331, USA [email protected] c 2007 Corinne Manogue March 1999, last revised February 2009 Abstract The two body problem is treated classically. The reduced mass is used to reduce the two body problem to an equivalent one body prob- lem. Conservation of angular momentum is derived and exploited to simplify the problem. Spherical coordinates are chosen to respect this symmetry. The equations of motion are obtained in two different ways: using Newton’s second law, and using energy conservation. Kepler’s laws are derived. The concept of an effective potential is introduced. The equations of motion are solved for the orbits in the case that the force obeys an inverse square law. The equations of motion are also solved, up to quadrature (i.e. in terms of definite integrals) and numerical integration is used to explore the solutions. 1 2 INTRODUCTION In the Central Forces paradigm, we will examine a mathematically tractable and physically useful problem - that of two bodies interacting with each other through a force that has two characteristics: (a) it depends only on the separation between the two bodies, and (b) it points along the line con- necting the two bodies. Such a force is called a central force. Perhaps the 1 most common examples of this type of force are those that follow the r2 behavior, specifically the Newtonian gravitational force between two point masses or spherically symmetric bodies and the Coulomb force between two point or spherically symmetric electric charges.
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Central Forces
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08. Central Force Motion I Gerhard Müller University of Rhode Island, [email protected] Creative Commons License
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