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

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� �

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� � 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.

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