Subject: physics

Topic: circular motion

Class: 11

## #UNIFORM CIRCULAR MOTION

When a
particle moves in a horizontal surface with a constant speed the motion of the
particle is called uniform circular motion.

OR

OR

## Uniform Circular Motion

Uniform circular motion can be described as the motion of an object in a circle at a constant speed. As an object moves in a circle, it is constantly changing its direction. At all instances, the object is moving tangent to the circle. Since the direction of the velocity vector is the same as the direction of the object's motion, the velocity vector is directed tangent to the circle as well. The animation at the right depicts this by means of a vector arrow.

An object moving in a circle is accelerating. Accelerating objects are objects which are changing their velocity - either the speed (i.e., magnitude of the velocity vector) or the direction. An object undergoing uniform circular motion is moving with a constant speed. Nonetheless, it is accelerating due to its change in direction. The direction of the acceleration is inwards. The animation at the right depicts this by means of a vector arrow.

The final motion characteristic for an object undergoing uniform circular motion is the net force. The net force acting upon such an object is directed towards the center of the circle. The net force is said to be an inward or

*centripetal*force. Without such an inward force, an object would continue in a straight line, never deviating from its direction. Yet, with the inward net force directed perpendicular to the velocity vector, the object is always changing its direction and undergoing an inward acceleration.## #HOW IT IS CONSTANT SPEED

It is in
Constant speed because the distance between the ground and moving body remain
same. So it is called constant speed.

The relation between the angular displacement and linear displacement of circular motion is as the angular displacement increase the linear displacement also increase.

we know that θ=arc /radian

θ=x/r

Where

θ= angular displacement

X = linear displacement

Angular displacement unit = radian

OR

## # ANGULAR MOTION

Wen cicular motion take place the angle changes so this is given a name called angular motion. When a partical move in straight line it is called linear motion.## # ANGULAR DISPLACEMENT (Δθ)

The angle made between the two radius of a circular is called angular displacement and angular displacement is denoted by (θ) and the distance between the two point of radius that make a straight line is called linear displacementThe relation between the angular displacement and linear displacement of circular motion is as the angular displacement increase the linear displacement also increase.

we know that θ=arc /radian

θ=x/r

Where

θ= angular displacement

X = linear displacement

Angular displacement unit = radian

OR

## # ANGULAR DISPLACEMENT (Δθ)

Angular
displacement of a particle in a circular motion is defined as the angle trussed
out by the radius vector at the center of circle in the given time it is
denoted by (Δθ)

Therefore ∠QθP= Δθ = Angular displacement

##
# ANGULAR VELOCITY (ω)
Angular velocity of a particle making circular motion

As we know that velocity = displacement upon time

ω= angular displacement angular /time taken

ω = θ/T

angular velocity unit is = radian per second

OR

##
#
ANGULAR VELOCITY (**ω**)

Angular
velocity of particle in a circular motion is defined as the time rate of change
of its angular displacement.

Therefore
Angular velocity (

**ω**) = Δθ/ΔT
If ΔT= 0 then angular velocity becomes equal to
instantaneous velocity.

##
# INSTANTANEOUS VELOCITY

##
# INSTANTANEOUS VELOCITY
If there is less time and the angle is very small its called instantaneous velocity .

ω=dθ/dT

OR

ω=dθ/dT

OR

## # INSTANTANEOUS VELOCITY

LT =Δθ/ΔT =dθ/dT -----------(1)

IF ΔT = 0

##
#Frequency of circular motion
Frequency of circular motion = number of cycle in 1 second. For an example a particle moves 10 cycle in one second it frequency will be 10.

##
# 1 Radian Displacement
When a point is making a circular motion hence the length of arc is equal to radius between two position
that angle subtended at centre is 1 radian.

##
# 1 Radian Displacement
When a point is making a circular motion hence the length of arc is equal to radius between two position
that angle subtended at centre is 1 radian.

## # RELATION BETWEEN LINEAR VELOCITY AND ANGULAR VELOCITY

Consider
a particle moving in a horizontal circle with linear velocity (V) and angular
velocity (

**ω**) At anytime (T) then particle is at point P where the radius vector OP = R
After
time T+ΔT the particle reached at point Q the angle made at the center is θ

OP = R

**ω =**

**Δθ/ΔT**

Δθ =

**ω**AT -------- (1)
If the
linear displacement of particle is PQ =ΔL in
time (ΔT) then

Linear
displacement

V = ΔL/ΔT

V = ΔL/ΔT

ΔL= VΔT

The
angle Δθ= ΔL/R

Now
putting the value

**ω.**

**Δ**T = V

**Δ**T/R

**ω.**R ---------(3)

## #ANGULAR ACCLEERATION (α)

Angular acceleration of a particle in a
circular motion is defined

as the time rate of change of its angular velocity.

##
Δ
ω The change in angular velocity of the particle in
the time ΔT then

Angular acceleration = Δω / ΔT

If ΔT tends to
be 0 then angular acceleration becomes equal to Instantaneous angular acceleration.

∴ Instantaneous acceleration (α)
= LT Δω
/ ΔT

= dω / dT

= dω / dT

## #RELATION BETWEEN LINEAR ACCLERATION AND ANGULAR ACCELERATION

(a)
Scalar Form

If ω

_{1 }and ω_{2}be the angular velocity of particle with time t_{1}and t_{2}then
Angular acceleration = change in velocity /
change in time

Angular acceleration = ω

_{1 }- ω_{2 }/ t_{2}- t_{1}
[V= rω , V

_{1}=rω_{1},V_{2}=rω_{2}]
α = (V

_{2}/r - V_{1}/r)/(T_{2 }– T_{1})
α =1/r (V

_{2}- V_{1})/(T_{2 }– T_{1})
α = a/r

r. α = a ------------------------------------(1)

(b) Vector Form

We know that V is linear velocity

V= ω *
r ----------------------------------(1)

Different the above equation (1) with respect to
time T

d/dt * v = d/dt (ω *
r)

a= dω/dt * r + ω *dr/dt

a= α * r + ω*v
-----------------------(2)

The linear acceleration is composed of two components.

(1)
Tangential acceleration (at) =
α * r according to the rule of cross
product of this component acts along the tangent there for this component is
called Tangential acceleration.

(2)
Radial acceleration or centripetal acceleration
(ac)= ω*v According to the law of cross product this
component acts along the radius towards the center of circle therefore this acceleration
is called Radial acceleration or Centripetal acceleration.

Net acceleration (a) can be calculated from the
prallogram law

(a)=
√ac

^{2}+at^{2}-------------------(3)
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