CLASSICAL MECHANICS NOTES PART - 2

coriolis force

 the Coriolis force is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise (or counterclockwise) rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect.

The effect of the Coriolis force is an apparent deflection of the path of an object that moves within a rotating coordinate system. The object does not actually deviate from its path, but it appears to do so because of the motion of the coordinate system.

FOR VIDEO DEMONSTRATION OF CORIOLIS FORCE CLICK HERE

CORIOLIS FORCE FORMULA :-2m(ω⃗ × v⃗ )

FOR THE COMPLETE DERIVATION OF CORIOLIS FORCE CLICK HERE

MECHANICS OF SYSTEM OF PARTICLES

FOR COMPLETE THEORY AND DERIVATIONS FOLLOW THIS PDF

WE WILL HAVE A OVERLOOK AT DEFINATIONS & FORMULAE OF 

THIS CONTENT:-

                               

                          

                         

Lagrangian and Hamiltonian mechanics

Holonomic constraints :- 

                                        For a system of particles with positions given by ri(t) for i = 1, . . . , N, constraints that can be expressed in the form g(r1, . . . , rN , t) = 0, are said to be holonomic. Note they only involve the configuration coordinates.

Degrees of freedom :-

                                      The dimension of the configuration space is called the number of degrees of freedom. Thus we can transform from the ‘old’ coordinates r1, . . . , rN to new generalized coordinates q1, . . . , qn where n = 3N − m:

                                                   r1 = r1(q1, . . . , qn, t),

                                                    .

                                                    .

                                                    .

                                                   rN = rN (q1, . . . , qn, t)

D’Alembert’s principle:-

                                          The principle states that the sum of the differences between the forces acting on a system of massive particles and the time derivatives of the momenta of the system itself projected onto any virtual displacement consistent with the constraints of the system is zero.^{[clarification needed]} Thus, in mathematical notation, d'Alembert's principle is written as follows,

                                      

where :

${\displaystyle i}$	is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,
${\displaystyle \mathbf {F} _{i}}$	is the total applied force (excluding constraint forces) on the ${\displaystyle i}$-th particle,
${\displaystyle m_{i}\scriptstyle }$	is the mass of the ${\displaystyle i}$-th particle,
${\displaystyle \mathbf {v} _{i}}$	is the velocity of the ${\displaystyle i}$-th particle,
${\displaystyle \delta \mathbf {r} _{i}}$	is the virtual displacement of the ${\displaystyle i}$-th particle, consistent with the constraints.

 Hamilton’s principle:-

 It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it :

                                                                                                                     H=T+V

 Constraint:-

                     In classical mechanics, a constraint on a system is a parameter that the system must obey.

Lagrange principle:-

                                    A variational integral principle in the dynamics of holonomic systems restricted by ideal stationary constraints and occurring under the action of potential forces that do not explicitly depend on time.

THE LAGRANGIAN EQUATION CAN BE WRITTEN AS:-

                                                                                            L = T - V 

FOR COMPLETE INFORMATION HAMILTONIAN AND LAGRANGIAN OPERATORS CLICK HERE