Lecture On Theoretical Mechanics - 1

PBC Lecture Notes Series in Physics: Classical Mechanics -I Prepared by Dr. Abhijit Kar Gupta, e-mail: [email protected]

Lecture 1

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Particle Dynamics: inertial frame, linear and angular momentum The mathematical description of mechanics of a particle starts from its position vector. We can derive all other quantities like displacement, velocity, acceleration, force etc. from that. In Cartesian coordinate system: Position vector, r = iˆx + ˆjy + kˆz .

Velocity, v =

d r ˆ dx ˆ dy ˆ dz , =i + j +k dt dt dt dt

d 2r ˆ d 2x ˆ d 2 y ˆ d 2z =i 2 + j 2 +k 2 . Acceleration, a = dt 2 dt dt dt From Newton’s 2nd law of motion force is given by

F=

dp dv =m = ma , where the momentum p = mv . dt dt

dv = 0 since m ≠ 0 . dt ∴ v = const. This means the body is moving with a uniform velocity or the body is at rest

When the applied force on a particle F = 0 , we have

( v = 0). Thus we arrive at Newton’s 1st law of motion. Example of Force: Weight of a body is a force exerted by the Earth on it. F = m g , where g is the acceleration due to gravity. Reference frame:

To describe motion, we need a coordinate system. Position vector is defined with respect to that. The coordinate system should be fixed in a frame, we call this reference frame. For example, we are standing on earth and this can be our reference frame. When we are riding on a bus and we want to describe motion happening inside the bus, the bus is our reference frame. The reference frame, if it is in absolute rest or in uniform motion with respect to another frame at rest, we call it inertial frame. Suppose, we have two frames S and S ′ where S ′ is moving with a uniform speed v along some direction (say, x-direction) with respect to S . We can write at time the following at any time, t x ′ = x − vt y′ = y z′ = z The time in two frames are equal, t ′ = t .

PBC Lecture Notes Series in Physics: Classical Mechanics -I Prepared by Dr. Abhijit Kar Gupta, e-mail: [email protected]

Lecture 1

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The above relationship between the two sets of coordinates, in the two frames are called Galilean transformation.

Y′

Y

vt

x

•P

x′

X Z

X′

The coordinates of the point P is ( x, y, z ) and ( x ′, y ′, z ′) in two coordinate systems S and S′.

Z′

We can conclude from the above equations that d 2 x′ d 2 x d 2 y ′ d 2 y d 2 z′ d 2 z = = = 2 . , and dt 2 dt 2 dt 2 dt 2 dt 2 dt nd Therefore, Newton’s 2 law of motion,  d 2 x′  d 2x d 2 y′ d2y d 2 z′  d 2z  F = m iˆ 2 + ˆj 2 + kˆ 2  = m iˆ 2 + ˆj 2 + kˆ 2  remains invariant under dt dt dt  dt   dt  dt such transformation. Newton’s 2nd law of motion is called law of inertia. Thus we say, the frame of reference which is either at absolute rest or in uniform motion relative to any other frame at rest, the law of inertia holds good in it. Such a frame is thus called inertial reference frame. Different kinds of motion:

A particle can have two types of motion: (i) translational or linear and (ii) rotational or angular. Any kind of motion of a particle or of a body can be either of the above or a combination of the two. For linear motion, we define linear momentum, p = mv (1) dp dv =m = ma = F (Force). dt dt For angular motion, a similar physical quantity is angular momentum, L = r × p Differentiating (2),

Differentiating above, we get

( )

dL d dr dp = × p+r× r× p = . dt dt dt dt

(2)

PBC Lecture Notes Series in Physics: Classical Mechanics -I Prepared by Dr. Abhijit Kar Gupta, e-mail: [email protected] But

∴

Lecture 1

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( )

dr × p = v × mv = m v × v = 0 . dt

dL dp dL = r× = r × F , where we call Γ = (Torque). dt dt dt Note:

Angular momentum, L = r × p = moment of momentum Torque, Γ = r × F = moment of force

dL = 0, we have L = const. When there is no torque applied on a particle, dt its angular momentum remains conserved.

Now if Γ =

Example: Planets moving around the Sun in closed orbits where angular momentum of a planet remain conserved. (We shall prove this in the chapter of central force motion later.) -------------------------