Unit: 2 Central Force Motions | BSc Physics Notes | Kannur University | Semester 3

 

This unit covers central force motion, focusing on forces directed toward a fixed point, like gravitational and electrostatic forces. It includes analysis of planetary orbits, conservation laws, and derivation of Kepler’s Laws using Newtonian mechanics. Learn the physics behind orbital motion, effective potential, angular momentum, and the inverse-square law. Essential for BSc Physics students at Kannur University, this topic builds a strong foundation in classical mechanics and celestial dynamics. Gain insight into how bodies move under central forces and why elliptical orbits form. Perfect for exam preparation and competitive physics exams.

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Semester 3 | Kannur University | Notes | BSc Physics

Unit 1:  Non inertial Systems and Fictious Forces: Link

Unit 2:  Central Force Motions: Link

Unit 3:  Harmonic Oscillator: Link

Unit 4:  Waves: Link

Here’s a classic diagram illustrating the idea of central force motion—the force is always directed toward (or away from) a fixed center and typically depends only on the distance from that center.

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Unit II: Central Force Motions

BSc Physics (Kannur University – Semester 3)

1. Overview & Syllabus Highlights

• Duration: About 9 lecture hours, covering the following major topics:

  • Reduction to a one-body problem (using reduced mass)

  • General properties of central force motion

  • Motion confined to a plane

  • Conservation of energy and angular momentum

  • Law of equal areas (Kepler's 2nd law)

  • Energy diagrams and effective potential

  • Applications: planetary motion, Kepler’s laws, elliptical orbits, hyperbolic trajectories, satellite orbits
    (Kannur University)

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2. Core Concepts & Definitions

a) What Is a Central Force?

A central force is one that:

• Acts along the line joining the particle and a fixed center.

• Its magnitude depends only on the distance $r$ from the center.

Mathematically:

$$\mathbf{F}(\mathbf{r}) = f(r)\,\hat{\mathbf{r}}$$

with $\hat{\mathbf{r}} = \frac{\mathbf{r}}{|\mathbf{r}|}$.
(Scribd, Wikipedia)

Typical examples include gravitational and Coulomb forces—both conservative and spherically symmetric.
(Wikipedia, Vedantu)

b) Conservation Laws & Planar Motion

• Angular momentum $\mathbf{L} = \mathbf{r} \times m\dot{\mathbf{r}}$ is conserved since torque is zero. Motion is thus confined to a single plane.

• Mechanical energy (kinetic + potential) is conserved in conservative central fields.

• Equal area law: The radius vector sweeps equal areas in equal times—this is Kepler’s second law.
  (Wikipedia, Scribd, Vedantu)

c) Two-Body to One-Body Reduction

Using the concept of reduced mass $\mu = \frac{m_1 m_2}{m_1 + m_2}$, a two-body problem under mutual central force can be simplified to an effective one-body problem.
(physics.uwo.ca, opencw.aprende.org)

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3. Equations & Motion Analysis

a) Equations of Motion in Plane

In polar coordinates $(r, \theta)$, the radial and angular components of motion follow:

$$m(\ddot{r} - r\dot{\theta}^2) = f(r), \quad m(r\ddot{\theta} + 2\dot{r}\dot{\theta}) = 0$$

From the angular equation, $r^2 \dot{\theta} = \text{constant}$, expressing angular momentum conservation.
(Scribd)

b) Effective Potential & Radial Motion

Motion can be understood via an effective potential:

$$V_{\text{eff}}(r) = V(r) + \frac{l^2}{2\mu r^2}$$

where $l$ is angular momentum and $V(r)$ is the original potential.
This allows one to analyze turning points and radial behavior by visualizing energy diagrams.
(Wikipedia, thphys.nuim.ie)

c) Orbit Solutions (Inverse-Square Law)

For gravitational-like inverse-square forces ($V(r) \propto -1/r$), the orbits are conic sections:

$$r(\theta) = \frac{C}{1 + \varepsilon \cos\theta}$$

where $\varepsilon$ is eccentricity.

• $\varepsilon<1$: ellipse (bound)

• $\varepsilon=1$: parabola

• $\varepsilon>1$: hyperbola (unbound)
  (Wikipedia, Scribd)

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4. Applications & Examples

Scenario	Description
Planetary/Satellite Orbits	Elliptical orbits following Kepler’s laws; special case of inverse-square force.
Hyperbolic Trajectories	Scattering scenarios such as cometary fly-bys or particle collisions.
Bound and Unbound Motion	Determined by energy relative to effective potential—bound (elliptic) vs unbound.

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5. Sample Notes Outline

1. Introduction

   • Definition and characteristics of central forces

   • Reduction to one-body problem (reduced mass)

2. Fundamental Properties

   • Conservation of angular momentum

   • Planar motion

   • Conservation of energy

   • Equal areas law

3. Mathematical Treatment

   • Polar coordinates: radial & angular equations

   • Effective potential and radial motion analysis

   • Orbit classifications: elliptical, parabolic, hyperbolic

4. Specific Cases & Kepler's Laws

   • Inverse-square law derivations

   • Key features of elliptical orbits—semimajor axis, focus, eccentricity

5. Applications & Sample Problems

   • Satellite orbit parameters

   • Energy considerations in comet trajectories

   • Interpretation of effective potential diagrams

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6. Study Tips

• Visualize motion using diagrams (like the one above).

• Practice deriving angular momentum conservation and energy equations in polar coordinates.

• Sketch effective potential vs $r$ to identify bound vs unbound motion.

• Solve example problems: e.g., deriving orbit shape for given $E$ and $l$.

• Reinforce understanding of conic section orbits and Kepler’s laws.

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