Class 11Physics Chapter 4 Circular Motion in a Plane

Examples of circular motion are the rotation of the earth around its own axis and the motion of the moon around the earth. there are two types of displacement in a circular motion, one is linear displacement and angular displacement .linear displacement is the length of the arc covered by an object is s and the radius of the circle is r then angular displacement is s/r radian, linear displacement divided by time results linear velocity and angular displacement divided by time results angular velocity.

 

Class 11Physics Chapter 4 Circular Motion in a Plane

Let the angular displacement of a rotating object is θ and linear displacement is s

Angular displacement = Arc/Radius

θ = s/r

s =rθ, is the relationship between linear and angular displacement

Linear velocity =Linear displacement/Time

Let there is Δs change in linear displacement, change in angular displacement is Δθ in time Δt, then average velocity and average angular velocity are given by

Average linear velocity =Change in linear displacement/Change in time and Average angular displacement =Change in angular displacement/Change in time

v_av = Δs/Δt and w_av = Δθ/Δt

If Δt is infinitesimally small then corresponding linear displacement and angular displacement are ds and dθ then instantaneous linear velocity and instantaneous angular velocity is evaluated as following

v_ins = ds/dt and w_ins = dθ/dt

Angular velocity is an axial vector, so the direction of angular velocity is in outward of the plane. If the object is moving clockwise then the direction of angular velocity is otward of the plane and if the object is moving anticlockwise then the direction of angular velocity is inward of the plane.

The relationship between angular velocity and linear velocity:

The equations are known to us

v = Δs/Δt …..(i) w = Δθ/Δt…….(ii)

Dividing equation (i) by equation (ii)

v/w = Δs/ Δθ

Putting the value of

s =rθ

v/ω= Δ(rθ)/ Δθ

v/ω = r(Δθ)/ Δθ[ r is constant)

v = rω

Angular velocity of an object which is moving in a straight line:

Let there is a point B wherefrom we are going to observe the angular velocity of an object at A which is moving at the velocity of v in a straight line.

ω = v/r,here v and r should be perpendicular to each other

Now, let us consider the angular velocity at C, at point C the linear velocity remains the same as it is supposed that the object is moving with a uniform velocity v but its angular velocity changes with the change in the value of θ(i.e angular displacement)

For evaluating the value of angular velocity we have to consider the component of the velocity which is directed perpendicular to its distance from B(i.e r’) which is vcosθ

Therefore in this case the angular velocity will be

ω = vcosθ/r

Angular velocity of a moving object in a straight line with respect to another moving object:

Let the velocity of an object at A is v_A

and the velocity of another object which is at a distance of AB and moving with a velocity of  v_B by an angle θ with the distance AB

Angular velocity of the object B with respect to A = ω_BA

First of all let’s calculate the component of the linear velocity v_B which is perpendicular to the distance AB

The vertical component of v_B is =v_B sinθ

The relative velocity of the object B with respect to A = v_BA = v_B -v_A 

Forces and Newton’s First Laws of motion:Class 11 Physics Chapter 5 CBSE

Addition of Vectors: CBSE Class 11 Physics Chapter 4 -Motion in a Plane

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