Syllabus (MATHEMATICS)

Course Type: MAJ-6

Semester: 5

Course Code: BMTMMAJ06T

Course Title: Dynamics of a Particle and System of Particles

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: After successful completion of the course, students will be able to CO1: get a basic knowledge of forces and moments that covers concepts of particle dynamics. CO2: understand the co

Syllabus:

Unit -1: Dynamics of a Particle [Credit: 4, 60L]

Kinematics

1. Expressions for velocity & acceleration for

(i) Motion in a straight line;

(ii) Motion in a plane;

(a) Cartesian co-ordinates, (b) Polar co-ordinates, (c) Intrinsic co-ordinates

Kinetics

2. Newton’s laws of motion, Equation of motion of a particle moving under the action of given external forces.

(a) Motion of a particle in a straight line under the action of forces μxⁿ, n = 0, ± 1, n = -2 (μ>0 or < 0) with physical interpretation,

(b) Simple harmonic motion and elementary problems,

(c) The S.H.M. of a particle attached to one end of an elastic string, the other end being fixed,

(d) Harmonic oscillator, effect of a disturbing force, linearly damped harmonic motion and forced oscillation with or without damping,

(e) Vertical motion under gravity when resistance varies as some integral power of velocity, terminal velocity.

(f) Works, power and energy, Conservation laws: conservation of linear momentum, angular momentum and total energy for conservative system of forces.

3. Impulse of force, Impulsive forces, change of momentum under impulsive forces, Examples, Collision of two smooth elastic bodies, Newton’s experimental law of impact, Direct and oblique impacts of (i) Sphere on a fixed horizontal plane, (ii) Two smooth spheres, Loss of Kinetic Energy and Impulse

4. Motion in plane/ two dimensions:

(a) Motion of a particle moving on a plane referred to a set of rectangular axes, Angular velocity and acceleration, Circular motion,

(b) Trajectories in a medium with the

(i) Motion of a projectile under gravity in free space;

(ii) Motion of a projectile under gravity with air resistance proportional to velocity, square of the velocity;

(c) Motion of a particle moving on a plane referred to polar co-ordinate system, radial and transverse Accelerations.

(d) Central forces and central Orbits: Motion under a central force, basic properties and differential equation of the path under given forces and velocity of projection, Apses, Time to describe a given arc of an orbit, Law of force when the center of force and the central orbit are known. Special study of the following problems; to find the central force for the following orbits:

(i) A central conic with the force directed towards the focus;

(ii) Equiangular spiral under a force to the pole;

(iii) Circular orbit under a force towards a point on the circumference.

To determine the nature of the orbit and of motion for different velocity of projection under a force per unit mass equal to –

(i) μ / (dist)² towards a fixed point ;

(ii) under a repulsive force μ / (dist)² away from a fixed point .

(e) Circular orbit under any law of force μf(r) with the centre of the circle as the centre of force, stability analysis of a circular orbit under a force μ f (r) towards the center. Particular case μf(r) =1/rⁿ.

(f) Kepler’s laws of planetary motion from the equation of motion of a central orbit under inverse square law, Modification of Kepler’s third law from consideration of motion of a system of two particles under mutual attractions according to Newton’s law of gravitational attraction, Escape velocity.

(g) Constrained Motion: Motion of a particle along a smooth curve, Examples of motion under gravity along a smooth vertical circular curve, smooth vertical cycloidal arc (cycloidal pendulum), Motion of a particle along a rough curve (circle, cycloid) & in a resisting medium.

Unit -2: Dynamics of System of Particles [Credit: 1, 15L]

Fundamental concepts: Centre of mass, linear momentum, angular momentum, kinetic energy, work done by a field of force, conservative system of forces – potential and potential energy, internal potential energy, total energy.

The following results to be deduced in connection with the motion of system of particles:

(i) Centre of mass moves as if the total external force were acting on the entire mass of the system concentrated at the centre of mass (examples of exploding shell, jet and rocket propulsion).

(ii) The total angular momentum of the system about a point is the angular momentum of the system concentrated at the centre of mass, plus the angular momentum for motion about the center.

(iii) Similar theorem as in (ii) for kinetic energy.

An idea of constraints that may limit the motion of the system, definition of rigid bodies, D’Alembert’s principle, principle of virtual work for equilibrium of a connected system.

Reading References:

1. Loney, S. L., An Elementary Treatise on the Dynamics of particle and of Rigid Bodies, Loney Press.
2. Terence Tao, Analysis II, Hindustan Book Agency, 2006.
3. Ganguly and Saha; Analytical Dynamics of Particles; New Central Book Agency.
4. Dutta and Jana; Dynamics of a Particle; Shreedhar Prakashani.
5. A. S. Ramsey; Dynamics (Part 1); CBS Publisher.
6. J. L. Synge & B. A. Griffith ; Principles of Mechanics; McGraw Hill Book Company Inc.
7. Chorlton; A Text Book of Dynamics; D. Van Nostrand Company Ltd.
8. Mollah, S. A., Advanced Dynamics of Particles, Books & Allied.
9. H. Goldstein; Classical Mechanics; Addison-Wesley.
10. Ghosh, Chakraborty; Advanced Analytical Dynamics; U.N. Dhur Publishing House.
11. R. G. Takwale, P. S. Puranik; Introduction to Classical Mechanics; McGraw Hill Education.