Central Force Conservative M Sudhadhar Lecturer in Physics

Central Force - Conservative M. Sudhadhar Lecturer in Physics GDC, Tuni Commissionerate of Collegiate Education

Central force • A force acting on a body - Whose magnitude depends on the distance of the body from a fixed point - Direction is towards or away from a fixed point Then the force is called central force Commissionerate of Collegiate Education

and Direction is away from or towards O Thus a Central force is denoted O as Commissionerate of Collegiate Education A r θ F

Examples Gravitational Force of Attraction between Sun and Planets Sun Planet It is directed towards the Sun Commissionerate of Collegiate Education

Electrostatic Force of Attraction between Nucleus and Electron It is directed towards the Nucleus Commissionerate of Collegiate Education

Restoring Force acting on a Simple harmonic oscillator It is directed towards the Mean position O Commissionerate of Collegiate Education

Conservative Force Ø Work done by the force is independent of the path followed between any two positions Ø Simply work done is not a path function Ø In other words work done around a closed path subjected to conservative force is zero. Commissionerate of Collegiate Education

Proof • • • Initial Position = A Final Position = B Path I : A P P’ B Path II : A Q Q’ B Work done for Path I : W 1 Work done for path II : W 2 B P’ P Path I Q’ A Commissionerate of Collegiate Education Path II Q

For conservative nature we have to prove W 1 = W 2 P’ Draw to arcs of radii r and r + dr with r+dr center as A Forces acting on the particle at P & Q be F 1, F 2 B P Path I F 1 F 2 r A Commissionerate of Collegiate Education Q’ Path II Q

Displacement from P to P’ : dr 1 from Q to Q’ : dr 2 Angle between F 1 & dr 1 : θ 1 F 2 & dr 2 : θ 2 B P’ dr 1 P θ 1 Path I F 1 r+dr F 2 r A Commissionerate of Collegiate Education Q θ 2 dr 2 Q’ Path II

Work done from P to P’ along Path I : dw 1 B P’ dr 1 P r+dr θ 1 Path I F 1 F 2 as r A Commissionerate of Collegiate Education Q θ 2 dr 2 Q’ Path II

Work done from Q to Q’ along Path I : dw 2 B P’ dr 1 P r+dr θ 1 Path I F 1 F 2 as r A Commissionerate of Collegiate Education Q θ 2 dr 2 Q’ Path II

As P and Q are at same distance from A and F is the central force so F 1 = F 2 = F r+dr B P’ dr 1 P θ 1 Path I F 1 F 2 For the whole path we get by integrating r A Commissionerate of Collegiate Education Q θ 2 dr 2 Q’ Path II

The work done along Path I i. e. APP’B : W 1 B P’ dr 1 P r+dr The work done along Path II i. e. AQQ’B : W 2 θ 1 Path I F 1 F 2 r A Commissionerate of Collegiate Education Q θ 2 dr 2 Q’ Path II

Thus W 1 = W 2 For any path between A & B the work done is same. Hence Work done is the independent of the path followed. The Work done is not a path function Commissionerate of Collegiate Education

Let the body is moved along a closed path i. e. from A to B and B to A. The work done along Path I i. e. APP’B : W 1 B P’ P Path I Q’ A Q Commissionerate of Collegiate Education Path II

The work done along Path II i. e. BQ’QA : W 2 B P’ P Path I Q’ A Q Commissionerate of Collegiate Education Path II

B P’ Total work done around the closed path is or P Path I Q’ A Q Thus Central Force is Conservative Commissionerate of Collegiate Education Path II