Classical Mechanics (part 1)

7/29/2019 Classical Mechanics (part 1)

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Classical Mechanics (Newtonian physics)

praveen kumar reddy.A (M.sc Physics)


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Newtonian physics


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What is the main theme to study Mechanics ?

Central concept in Mechanics is to understand

Motion.


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Types of motion


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Calculus

Newton discovered Calculus ( a new branch of mathematics) to describemotion.


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Kinematics

First step in classical mechanics is to

describe motion in terms ofspace and

time while ignoring agents that caused

motion. Portion of that classical

mechanics is called Kinematics.


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Motion in one dimension

Now we will concentrate on the motion of a body which is moving linearly

along only on one dimension (may be along X axis, Y axis , Z axis)


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Particle model

In study of translatory motion , we use particle model. a particle

is a point-like objectthat is, an object with mass but having infinitesimal

size.


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Parameters to describe motion

Position

Velocity / Speed

Acceleration


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Position

The motion of a particle is completely known if the particles position in

space is known at all times.

A particles position is the location of the particle with respect to a

chosen reference point that we can consider to be the origin of a

coordinate system.


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Position

Position time graph


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Distance vs displacement

Distance is the path traversed

by the particle.

Displacement is the difference between

initial and final position.

Distance is a scalar quantity , whiledisplacement is a vector quantity.


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Average Velocity or Average speed

The average velocity vxof a particle is defined as the particles displacement x

divided by the time interval during which that displacement occurs:

Units of velocity are meter per second.

Note : here velocity is displacement/time , while Speed is distance/time


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Instantaneous velocity or speed

Often we need to know the velocity of a particle at a particular instant in time,

rather than the average velocity over a finite time interval.

the instantaneous velocity vx

equals the limiting value of the ratiox/tast approaches zero

In calculus notation it is written as

The instantaneous velocity can

be positive, negative, or zero.

When the slope of the

positiontime graph is positive


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Average acceleration

The average acceleration ax of the particle is defined as the change in velocity

vx divided by the time intervalt during which that change occurs:

Units of acceleration are meter per second square.


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Instantaneous acceleration

In some situations, the value of the average acceleration may be different over

different time intervals. It is therefore useful to define the instantaneous

acceleration as the limit of the average acceleration ast approaches zero.


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When the objects velocity and acceleration are in

the same direction, the object is speeding up.

On the other hand, when the objects velocity and

acceleration are in opposite directions, the object is

slowing down.


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Now we will understand a simple type

of motion - A body moving in onedimension with constant acceleration.


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body moving with constant acceleration

Position time graph Velocity time graph Acceleration time graph


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Equations of motion - applicable only to body moving with constantacceleration :


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Freely Falling objects (Example of motion of body with constantacceleration)

Bodies whoose radius is very small compared to radius of

earth , which are falling very near to earth is approximated as

a freely falling body. They move with constant acceleration

(that is acceleration due to gravity)

Freely falling bodies should fall only under the influence of

gravity , not with any other force.


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Freely falling objects - Galileo

It is well known that, in the absence of air resistance, all

objects dropped near the Earths surface fall toward the Earth

with the same constant acceleration under the influence of

the Earths gravity.


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Motion of a body in two dimensions


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Position vector

Let us extend this idea to motion in the xy

plane. We begin by describing the position

of a particle by its position vectorr, drawn

from the origin of some coordinate system

to the particle located in the xy plane,


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Position , velocity and acceleration

Difference between final position vector and initial position vector

average velocity of a particle during the time intervalt as the displacementof the particle divided by the time interval

The instantaneous velocity v is defined as the limit of the average velocity


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Position , Velocity and acceleration

the average acceleration of a particle

The instantaneous acceleration a is defined as


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It is important to recognize that various changes can occur when aparticle accelerates.

First, the magnitude of the velocity vector (the speed) may change

with time as in straight-line (one-dimensional) motion.

Second, the direction of the velocity vector may change with time even ifits magnitude (speed) remains constant, as in curved-path(two-dimensional) motion.

Finally, both the magnitude and the direction of the velocity vector may

change simultaneously.


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Two dimensional motion with constant acceleration

The component form of the equations for vfand rfshow us that two-dimensional

motion at constant acceleration is equivalent to two independent motionsone in

the x direction and one in the y directionhaving constant accelerations ax and

ay.


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Special cases of Two dimensional motion

Projectile motion

Uniform circular motion


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Projectile motion ( in our day to day lives)

You see projectile motion in our every day lives too often


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Projectile motion

In projectile motion , we consider these two assumptions

1. Free fall acceleration g is constant during motion

2. Effect of air resistance is negligible.

This is equation of parabola , Hence trajectory of projectile motion is parabola.


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Analyzing projectile motion

when analyzing projectile motion,

consider it to be the superposition of

two motions:

(1) constant-velocity motion in the

horizontal direction

(2) free-fall motion in the vertical

direction.

Note : The horizontal and vertical components of a projectiles motion are completely

independent of each other and can be handled separately, with time t as the

common variable for both components.


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Parameters associated with ProjectileTime of ascent

Maximum Height

Range


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Various angles of projectiles


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Uniform circular motion

A car moving in a circular path withconstant speed v. Such motion is

called uniform circular motion, andoccurs in many situations.

It is often surprising to find that even

though an object moves at a constantspeed in a circular path, it still has anacceleration

Note that the acceleration depends onthe change in the velocity vector. herein this case of circular motion velocityvector changes in direction.


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Magnitude of centripetal acceleration

In uniform circular motion the acceleration is directed inward toward the centerof the circle. There will be no acceleration component parellel to velocity vector. If

it is there , velocity will change and there is no chance of uniform circular motion.

The acceleration acting towards center of circle is called centripetal acceleration

and is given by

In many situations it is convenient to describe the motion of a particle moving with

constant speed in a circle of radius r in terms of the period T


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Tangential and Radial acceleration

Here in this situation velocity changes both in magnitude and direction.

Here Velocity vector is always tangential to path of particle.


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Tangential and Radial acceleration

The total acceleration vector a can be written as the vector sum of thecomponent vectors:

Total acceleration = Radial acceleration + Tangential acceleration

The tangential acceleration component causes the change in the speed of theparticle. at = dV/dt

The radial acceleration component arises from the change in direction of the velocity

vector

At a given speed, aris large when the radius of curvature is small and small when r is large.

at is in the direction or opposition of velocity vector.


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Relative velocity and Relative acceleration

We find that observers in different frames of reference may measure different

positions, velocities, and accelerations for a given particle.

That is, two observers moving relative to each other generally do not

agree on the outcome of a measurement.


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Galilean transformations

We define the time t " 0 as that instant at which the origins of the two reference

frames coincide in space. Thus, at time t, the origins of the reference frames willbe separated by a distance v0t.

If velocity is cosntant Vo = 0 , there fore a' = a

the acceleration of the particle measured by an observer in one frame of reference

is the same as that measured by any other observer moving with constant velocity

relative to the first frame.


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Next lecture continues fromNewton Laws of motion

Classical Mechanics (part 1)

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