Classical_Mechanics_lecture_slides_2021_compressed.pdf

SC 201-Physics I Classical Mechanics Dr. V. S. Gayathri [email protected] 25/08/21 [email protected] Dr VS Gayathri 1  Classical Mechanics Classi...

SC 201-Physics I Classical Mechanics Dr. V. S. Gayathri [email protected] 25/08/21 [email protected] Dr VS Gayathri 1  Classical Mechanics Classical mechanics is the branch of Physics which deals with the motion of physical bodies at macroscopic level. Classical mechanics is based on Newton’s laws of motion, so often called as Newtonian mechanics. To study the motion of a physical body, the most basic parameters are space and time. (both are continuous). So to describe the motion of a body , one has to specify its position in space as a function of time. This needs a suitable choice of a coordinate 25/08/21 Dr VS Gayathri system. 2  Frame of reference If we imagine a coordinate system attached to a rigid body and we describe the position of any particle w.r.t it, then such coordinate system is called as frame of reference (FOR). The simplest FOR is Cartesian coordinate system (x,y,z) P(x,y,z) Y "⃗ O X Z 25/08/21 Dr VS Gayathri 3 Types of frame of reference 1. Inertial frame of reference 2. Non-inertial frame of reference 1. Inertial frame of reference: i. Inertial frames of reference are the frames, in which law of inertia holds and other laws of physics are valid. ii. These frames are unaccelerated frames (at rest or moving with constant velocity) (acceleration of the frame, !" =0). iii. All frames which are moving with constant velocity w.r.t. an inertial frame are also inertial. 25/08/21 Dr VS Gayathri 4 2. Non-inertial frame of reference: i. If a frame is accelerated w.r.t an inertial frame (road, train moving with constant velocity), these frames are called non-inertial frames. acceleration of the frame, !" ≠ 0 ii. In non-inertial frames, newton’s laws of motion are not valid. iii. In non-inertial frames, %! ≠ & %! − & ≠ 0 = &) , Pseudo force Therefore, by adding an extra pseudo force with F, we are able to do all operations, 25/08/21 like inertial frames. Dr VS Gayathri 5  Various Coordinate systems To specify the position and motion of a body in specific frame of reference, we have various coordinate systems. i. Cartesian or rectangular coordinate system (x,y) ii. Plane polar coordinate system (r,θ) iii. Cylindrical coordinate system (ρ,ϕ,z) iv. Spherical polar coordinate system (r,θ, ϕ) Frequently used 2D coordinate system is Cartesian or rectangular coordinate system. 25/08/21 Dr VS Gayathri 6  Cartesian or rectangular coordinate system To specify the position of any point in space , we need two perpendicular coordinates X-axis and Y-axis. P(x,y) We represent the position of the point P Y-axis as P(x,y) in CC. y It is a 2D coordinate system suitable for straight line motion. x X-axis 25/08/21 Dr VS Gayathri 7  Plane polar coordinates To describe circular motion, we need plane polar coordinate system. This two-dimensional coordinate system is based on the three dimensional cylindrical coordinate system. In this coordinate system, position of particle P is represented by a ordered pair (r, θ), where r—radial distance from origin /pole. P(r,θ) θ---angle from a fixed direction(x-axis). "⃗ The coordinates r and θ are called plane θ polar coordinates, as motion is restricted Pole in x-y plane here. 25/08/21 Dr VS Gayathri 8 Comparison of constant coordinate lines for the Cartesian coordinate system and for the plane polar coordinate system: 25/08/21 θ is +ve if we move counter clockwise from x-axis. Dr VS Gayathri 9 Relation Between Cartesian coordinates(CC) and plane polar coordinates(PPC). In Cartesian coordinates, position vector r is given as "=xî+yĵ ⃗ while in polar coordinates, we have Here, "̂ is radial unit vector and "=r ⃗ r̂ is given as "̂ =cos θ ı̂ + sin θ ȷ̂ P(x,y) =r (cos θ ı̂ + sin θ ȷ̂ ) P(r,θ) On comparing above equations, we have, xî+yĵ=cos θ ı̂ + sin θ ȷ̂ "⃗ - = /012 3 y 4 = /256 3 On squaring &adding and then taking ratio, we have θ / = √-8 + 48 Pole x ;< 4 3 = 9:6 - 25/08/21 Dr VS Gayathri 10  Unit Vectors in Cartesian coordinates 25/08/21 Dr VS Gayathri 11  Unit Vectors in Polar coordinates In CC, there are base unit vectors "̂ and #̂ where "̂ indicates the direction of increasing x and #̂ indicates the direction of increasing y. In the same way, in PPC also, we have two base unit vectors, $̂ and &% that points in the direction of increasing r and increasing θ. The directions of $̂ and &% vary with $'̂ position, whereas "̂ and #̂ have fixed directions. 25/08/21 Dr VS Gayathri 12 Unit vectors represention Properties of unit vectors of PPC 1. "̂ = %$ =1 2. "̂. %$ =0 25/08/21 Dr VS Gayathri 13  ()* - (. !"#$% &' ",( (+ (+ : =4 ℎ7?4 /0 123345406276208 5̂ 701 ;, Standard Notation 15 1C5 = 5̇ = 5̈ 16 16 C 1; 1C; = ;̇ C = ;̈ 16 16 ()* - (. =.̇. - ̇* = −.) (+ (+ 25/08/21 Dr VS Gayathri 14  Motion in Plane Polar Coordinates 25/08/21 Dr VS Gayathri 15  Velocity in plane polar coordinates The position vector "⃗ in polar coordinate is given by "=" ⃗ "̂ The velocity in PPC is given by &"⃗ $⃗ = &' ( =(* ""̂ () ()̂ = (* "̂ + " (* ="̇ "̂ + "-̇ -. $⃗ = "̇ "̂ + "-̇ -/ ="0&102 $32451'6 + '07837'102 $32451'6 Therefore, 25/08/21 the velocity in PPC is a combination Dr VS Gayathri of radial and tangential motion. 16 Case 1: Radial velocity (θ = constant, r varies). ̇ 0, If, θ is a constant, "= and, $⃗ = %& ̇% This implies one-dimensional motion in a fixed radial direction. Case 2: Tangential velocity (r = constant, θ varies). In this case $⃗ = '"̇ "( Since r is fixed, the motion lies on the arc of a circle(in the tangential direction). 25/08/21 Dr VS Gayathri 17  Acceleration in plane polar coordinates #$ Acceleration in plane polar coordinates, "= ⃗ #% = radial acceleration + tangential acceleration 25/08/21 Dr VS Gayathri 18  "⃗ = $̈ $̂ + $'̈ '( − $'̇ +$̂ + 2$̇ '̇ '( the Coriolis acceleration the centripetal acceleration radially inward the acceleration that arises from the changing tangential speed. the acceleration due to a change in radial speed. 25/08/21 Dr VS Gayathri 19  The Coriolis acceleration we discussed here, is a real acceleration that is present whenever r and θ both change with time. Half of the Coriolis acceleration is due to the change in direction of the radial velocity, dvr/dt = vr ".̇ The other half arises, due to tangential speed vθ = r".̇ If r changes by Δr, then vθ changes by Δvθ = Δr",̇ and the contribution to the tangential acceleration is therefore #̇ ",̇ the other half of the Coriolis acceleration. 25/08/21 Dr VS Gayathri 20  Newton’s law in plane polar coordinates "⃗ = $%⃗ In radial direction, In tangential direction, Newton’s law in polar coordinates do not follow its Cartesian form as, It means, the form of Newton’s law is different in different coordinate systems. 25/08/21 Dr VS Gayathri 21  Classical Mechanics Dynamics of system of particles 25/08/21 Dr. V.S. Gayathri Dr VS Gayathri 22 Ø Earlier, we have studied, the motion of a single particle which is represented as a point mass, m. Ø Now, we will extend our study to the dynamics of a system of particles. What is a system of particles? System of particles or extended objects : A group of inter-related particles. m1 m1 m m3 mn m2 m5 m4 single particle 25/08/21 Dr VS Gayathri System of particles 23  System of Particles System of particles or extended objects : A group of inter-related particles. v How to characterize the system of particles? 25/08/21 Dr VS Gayathri Fig:Fundamentals of Physics, Resnick24et al.,  Center of Mass Definition: The center of mass of a system of particles is the point that moves as though: v all of the mass were concentrated there. v all external forces were acting at that point. v For a two particle system along x-axis: !"#$ = $% $&% ''$$( &( (Mass−weighted mean of !* & !, ) % ( If -* = -,, &% ' &( !"#$ = , (geometrical center) 25/08/21 Dr VS Gayathri 25  Center of Mass v Generalizing to n-particles(along x-axis), Total mass of the system ! = #$ + #& + ⋯ + #( The location of center of mass is ,-.- / ,0.0 /⋯…/,2.2 )*+, = ,- / ,0 /…./,2 $ ( )*+, = ∑67$ #6 )6 4 25/08/21 Dr VS Gayathri 26  Center of Mass For particles distributed in 2D: y If the two particles are not along x-axis, 12 = !3 54 + )3 0̂ 16 = !7 54 + )7 0̂ $% &% ' $( &( !"#$ = $% ' $( $% +% ' $( +( )"#$ = $% ' $( x , = !"#$.̂ + )"#$ 0̂ 25/08/21 Dr VS Gayathri 27  Center of Mass y v 3-particle system: $% &% ' $( &( '$) &) !"#$ = x $% ' $( '$) $% ,% ' $( ,( '$) ,) *"#$ = $% ' $( '$) - = !"#$ /̂ + *"#$ 1̂ 25/08/21 Dr VSRef: Classical Gayathri Dynamics of Par3cles & Systems, Marion et28 al.,  Center of Mass v For an n-particle system(in 3D): & $' (' !"#$ = ) & $ ' +' *"#$ = ) & $ ' -' ,"#$ = ) The position vector of the center of mass is:. = !"#$ 0̂ + *"#$ 2̂ + ,"#$ 43 & $ ' 5' R= ) 3 (position vector of the 6 78 particle: 59 = !9 0̂ +*9 2̂ + ,9 4) 25/08/21 Dr VSRef: Classical Gayathri Dynamics of Par3cles & Systems, Marion et29 al.,  Center of Mass Note: v The position of center of mass of a system of particles is independent of origin. v If origin is chosen at the center of mass, ! "# $# R= % =0, Σ *+ ,+ = *-,- + */,/ + ⋯ + *1 ,1 = 0 ∴ 34* 56 *5*7893 = 0 Moment of masses about the origin is zero. 25/08/21 Dr VS Gayathri 30  Center of Mass of a System of Particles In the con)nuum limit, divide the body into N mass elements. If !" is the posi)on of the #$% element, and &" is its mass, then. 1 ' = + &" !" * ",- In the limit of N → ∞,. 1 ' = lim + &" !" 5→6 * ",- - = ∫! 9& , (9& ∶ 9;