Classical Mechanics

Classical Mechanics

Module 1

Basics

 $F_i=F_i^e\ \ +\sum_{j=1}^NF_{ij}$  

• According to Newton's second law,  $F_i=\frac{\text{d}P_i}{\text{d}t}=m_i\frac{\text{d}v_i}{\text{d}t}=m\frac{\text{d}^2r_i}{\text{d}t^2}$  

• Sum is taken over all the particles, equation (1) becomes,  $\frac{\text{d}^2}{\text{d}t^2}\sum_i^{ }m_ir_i=\sum_i^{ }F_i^e\ +\ \sum_i^{ }\sum_j^{ }F_{ij}$  

According to Newton's third law, any two particles of the system exert equal and opposite forces on each other, 
 $F_{ij}\ +F_{ji}=0$  

• Second sum represents the internal forces in pairs, consequently sum vanishes. Thus, 

•   $\text{}F^e=\frac{\text{d}^2}{\text{d}t^2}\sum_i^{ }m_ir_i$ 

Centre of the mass R of the system is,

 $R=\frac{\Sigma_im_ir_i}{\text{M}}$  

• Then,       $F^e=M\frac{\text{d}^2R}{\text{d}t^2}=Ma_{cm}$  

• Thus, acceleration of centre of mass is due to only the external forces.

Center of mass-frame of reference

• Inertial frame attached with the centre of mass of an isolated system. 

• C.M. remains at rest (v=0) and total linear momentum is always zero.

• C-frame: zero-momentum frame.

Relation between rectangular (x,y) and polar coordinates

 $\left(r,\theta\right)$  

•  $x=r\cos\theta$   $y=r\sin\theta$  

•  $r=\sqrt{x^2+y^2}$  

•  $\theta=\tan_{ }^{-1}\left(\frac{y}{x}\right)$  

Relation between cylindrical and Cartesian coordinates,

•  $x=\rho\cos\theta$   $y=\rho\sin\theta$   $z=z$  

•  $\rho=\sqrt{x^2+y^2}$   $\theta=\tan^{-1}\ \frac{y}{x}=\sin^{-1}\ \frac{y}{\rho}$  

Relation between Spherical and Cartesian coordinates,

•  $x=r\sin\theta\cos\phi$   $y=r\sin\theta\sin\phi$   $z=r\cos\theta$  

•  $r=\left(x^2+y^2+z^2\right)^{\frac{1}{2}}$   $\theta=\tan_{ }^{-1}\ \frac{\sqrt{x^2+y^2}}{z}$  

•  $\phi=\tan^{-1}\ \frac{y}{x}$  

General relationships for spherical and Cartesian coordinates is,

•  $r=r\left(r,\theta,\phi,t\right)$  

•  $r=r\left(q_{1,\ }q_{2,}q_3,t\right)$  if coordinates are represented by  $q_{1,}q_{2,}q_3$   

Degrees of freedom

• Minimum no: of independent variables or coordinates to specify the position of a dynamical system.

• Example: Three coordinates are required to specify the motion of a particle moving freely in space.

• For an N particle system: 3N degres of freedom.

Constraints

• Limitations on the motion of a system.

• Motion is called constrained motion.

Holonomic Constraint

•  $f\left(r_{1,}r_{2,............,}t\right)$  

  Constraints are expressed in the form of equations.

• Example: simple pendulum with rigid support.

• Distance between position vectors  of i and j particles is,  $\left|r_i-r_j\right|=C_{ij}$  

Non-holonomic constraint

• Corresponds to non-integrable differential equations of constraints.

•  $\left|r\right|\ge a$  or  $r-a\ge0$  

  Equation represents the motion of a particle placed on the surface of a sphere of radius a.

• Example: Gas molecules in a container are constrained to move inside it.

Rheonomous Constraint

• Constraint equation contain time as an explicit variable.

Sceleronomous

• Constraints are not explicitly dependent on time.

Conservative

• Total mechanical energy of a system is conserved during the constrained motion.

• Constraint forces do not do any work.

Dissipative

• Constraint forces do work & total mechanical energy is not conserved.

• Time-dependent (rheonomic) constraints are generally dissipative.

Multiple Choice

Name the type of constraint involved in the motion of a body on an inclined plane under gravity.

• Rheonomic

• Sceleronomic

• Dissipative

• Holonomic