Motion is defined as continuous change of position of a body. If the body moves so that every particle of the body follows a straight-line path, then the motion of the body is said to be rectilinear.
When a body moves from one position to another, the effect may be described in terms of motion of the center of mass of the body from a point A to a point B (see illustration). If the center of mass of the body moves along a straight line connecting points A and B, then the motion of the center of mass of the body is rectilinear. If the body as a whole does not rotate while it is moving, then the path of every particle of which the body is composed is a straight line parallel to or coinciding with the path of the center of mass, and the body as a whole executes rectilinear motion.
Kinematics (from Greek, , to move) is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion.
The simplest application of kinematics is for particle motion, translational or rotational. The next level of complexity is introduced by the introduction of rigid bodies, which are collections of particles having time invariant distances amongst themselves. Rigid bodies might undergo translation and rotation or a combination of both. A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints.
Particle Kinematics
Particle kinematics is the study of the kinematics of a single particle. The results obtained in particle kinematics are used to study the kinematics of collection of particles, dynamics and in many other branches of mechanics.
Position & Reference Frames
The position of a point in space is the most fundamental idea in particle kinematics.
To specify the position of a point, one must specify three things:
1. the reference point (often called the origin),
2.distance from the reference point and
3.the direction in space in which the straight line from the reference point to the particle makes.
From example:
Consider for example a tower 50 m south from your home. The reference point is home, the distance 50 m and the direction south
Rest & Motion
Once the notion of position is firmly established, the ideas of rest and motion naturally follow.
Motion: If the position vector of the particle (relative to a given reference frame) changes with time, then the particle is said to be in motion with respect to the chosen reference frame. Rest: the position vector of the particle (relative to a given reference frame) remains same with time, then the particle is said to be at rest with respect to the chosen frame.
Note that rest and motion are relative to the reference frame chosen. It is quite possible that a particle at rest relative to a particular reference frame is in motion relative to the other.
Path
The path of the particle can be defined as the locus of endpoints of the particle's position vector over time
Displacement
Displacement is a vector describing the difference in position between two points, i.e. it is the change in position the particle undergoes during the time interval. If point A has position rA = (xA,yA,zA) and point B has position rB = (xB,yB,zB), the displacement rAB of B from A is given by
Geometrically, displacement is the shortest distance between the points A and B. Displacement, distinct from position vector, is independent of the reference frame.
Distance
Distance is a distinct quantity from either position or displacement. It is a scalar quantity, describing the length of the path between two points along which the particle has traveled.
If the position of the particle is known as a function of time (r = r(t)), the distance s it travels from time t1 to time t2 can be found by
Velocity Average
Average velocity is defined as
where Δr is the change in displacement and Δt is the interval of time over which displacement changes.The direction of v is same as the direction of the displacement Δr as Δt>0.
Velocity
Velocity is the measure of the rate of change in displacement with respect to time; that is, how the displacement of a point changes with each instant of time. Velocity also is a vector. Instantaneous velocity (the velocity at an instant of time) can be defined as the limiting value of average velocity Δt becomes smaller and smaller. Both and Δt approach zero but the ratio v approaches a non-zero limitv. This can be expressed as
where dr is an infinitesimally small displacement and dt is an infinitesimally small length of time.
Speed
The speed of an object is the magnitude |v| of its velocity. It is a scalar quantity.
The distance traveled by a particle over time is a non-decreasing quantity. Hence, is non-negative, which implies that speed can't be negative.
Acceleration
Acceleration is the vector quantity describing the rate of change with time of velocity. Instantaneous acceleration (the acceleration at an instant of time) is defined as the limiting value of average acceleration as Δt becomes smaller and smaller. Under such a limit, a → a.
where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time.
Types Of Motion Based On The Velocity & Acceleration Of The Particle
The acceleration of a particle may be constant or variable. If the acceleration is constant the motion is said to be motion with constant acceleration. On the other hand, if the acceleration is variable, the motion is called motion with variable acceleration
A special case arises for zero acceleration, If the acceleration of a particle is zero, then the velocity of the particle is constant over time, and the motion is called uniform. If the velocity of the particle isn't constant over time (which would include motion with non-zero constant acceleration and motion with variable acceleration), the motion is called non-uniform.
Motion with uniform acceleration
consider the motion of the body with uniform acceleration
let u- initial velocity
v-final velocity
t- time taken for change of velocity from u to v
acceleration is defined as the rate of change of velocity.
A= (1)
V= u+ at (2)
Displacement (S) is given by:
S=average velocity Ñ… time;
S= (3)
Substituting the value of v from (1) into (3)
Then , S=
S= ut+1/2at2 (4)
From equation (1)
t= (5)
substitutes it into (3)
then,
S= Ñ…
S= u2 - v2 / 2a
2aS= u2 -v2 (6)
Thus,equation of motion of a body moving with constant acceleration are:
V=u+ at
S= ut+1/2 at2
\2aS= u2 -v2
DERIVATIONS OF EQUATION OF RECTILINEAR MOTION:
From definition of acceleration
(a)
Since 'a' is acceleration
V=at+C1 (b)
Where C1 is the contant of integeration
When t=0, velocity =initial velocity , u
Then
U=0 +at
U=at,
Then from (b)
V= u+ at (A)
From definition of velocity:
= V=u+at..
=> ds =(u+at)dt (b)
Then
S=ut+1/2 at2 +C2
Where C2 is the constant of integeration
When t=0, s=0;
Then, C2=0;
Hence,
S=ut+1/2 at2 (B)
From definition of acceleration of acceleration,
A= Ñ…
S= v (c)
A ds =v dv
By integeration
A ʃ ds = ʃvu v dv
As = [(v2/2)]
As = V2//2 - u2/2
V2- u2=2as (c)
Relative velocity
To describe the motion of object A with respect to object B, when we know how each is moving with respect to a reference object O, we can use vector algebra. Choose an origin for reference, and let the positions of objects A, B, and O be denoted by rA, rB, and rO. Then the position of A relative to the reference object O is
Consequently, the position of A relative to B is
The above relative equation states that the motion of A relative to B is equal to the motion of A relative to O minus the motion of B relative to O. It may be easier to visualize this result if the terms are re-arranged:
or, in words, the motion of A relative to the reference is that of B plus the relative motion of A with respect to B. These relations between displacements become relations between velocities by simple time-differentiation, and a second differentiation makes them apply to accelerations.
For example, let Ann move with velocity relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity , each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity ), the equation above gives:
To find we simply rearrange this equation to obtain:
Examples based on rectilinear motion
1) The car negotiating a curve
2) a plane figure bounded by straight line are called is rectilinear motion
Problem : If the position of a particle along x - axis varies in time as :
x=2t2−3t+1
Then :
What is the velocity at t = 0 ?
When does velocity become zero?
What is the velocity at the origin ?
Solution : We first need to find out an expression for velocity by differentiating the given function of position with respect to time as :
v= d/dt(2t2−3t+1)=4t−3
(i) The velocity at t = 0,
v=4x0−3=−3m/s
(ii) When velocity becomes zero :
For v = 0,
4t−3=0
⇒t=3/4=0.75 sec
(iii) The velocity at the origin :
At origin, x = 0,
x=2t2−3t+1=0
⇒2t2−2t−t+1=0
⇒2t(t−1)−(t−1)=0
⇒t=0.5 s, 1 s.
This means that particle is twice at the origin at t = 0.5 s and t = 1 s. Now, v(t=0.5 s)=4t−3=4x0.5−3=-1 m/s.
Negative sign indicates that velocity is directed in the negative x - direction.
v(t=1 s)=4t−3=4x1−3=1 m/s.
Application of rectilinear motion
The motion of body under gravity is a uniformly accelerated motion hence all the equation of motion for uniformly accelerated motion along the straight line is applicable to the motion of bodies under gravity.
In the case of a body moving under gravity, acceleration is acceleration due to gravity and distance coved is equal to the height through which the object fails.
The equations of motion for a freely falling body are:
S= ut+1/2at2
2aS= v2-u2
V=u+at..
In the case of a body moving aganist gravity ,the force of gravity reduces the speed of the object and hence we take a= - g.
The equations of motion for a body moving against gravity are:
V=u-at
S=ut-1/2at2
V2-u2= - 2aS
When a object thrown from a height h with initial velocity is zero. And object attain the maximum height then final velocity is of body is zero.
EXAMPLE:
1. ROLLER BEARING FOR A RECTILINEAR MOTION:
The rollers bearing for an infinite rectilinear motion are composed of a long truck rail; It is a casing mounted astride the truck rail, and having truck faces in opposition to the truck faces of the truck rail, and having return passages for rollers are arranged in a parallel-roller type between their faces of the truck rail and the casing. The Side plates
are fixed on the both longitudinal ends of the casing, and the direction change passages connect the truck face and the return passage. A shifting passage is formed at the connection of the truck face with the direction change passage to cause the rollers to go away off the truck rail without changing the angle between the rollers axis and the horizon. In Further, the twisting passage follows the shifting passage to change the angle between the roller axis and the horizon. Those passages enable rollers to smoothly go in and out of the truck face with relatively less resistance. As a four row parallel rollers type, the present bearing can be made to have the smallest sectional height, and at a reasonable price. Incidentally the twisting passage can be made in the manner that it is twisted while gradually going away off the truck rail.
ROLLER BEARING FOR AN RECTILINEAR MOTION EFFECT:
The roller can be smoothly go inside, or go out of the truck face, and the resistance due to Sliding, offer by bearing, can be reduced.
(2) To sectional height can be attained due to four - row parallel type.
(3) The manufacturer cost of the ball bearing can be reduced and it is in simple form as well as tracks.
(4) The loading capacity is heavy with respect to other.
