Circular Motion

• Uniform Circular Motion:-Circular motion is said to the uniform if the speed of the particle (along with the circular path) remains constant.

• Angular Displacement:-

Scalar form:- ?S = r?θ

Vector form:- 

• Angular Velocity:-

• Relation between linear velocity (v) and angular velocity (ω):-

Scalar form:- v = rω

Vector form:- 

• Angular Acceleration:- 

• Relation between linear acceleration (a) and angular acceleration (α):-

Scalar form:- a= rα

 Vector form:- 

Here, tangential component, 

Radial component, 

• Relation betweenlinear acceleration (a), angular velocity (ω) and linear velocity (v):-

            a=v²/r = ω²/r

• Equations of rotational kinematics:-

  (a)  Angular velocity after a time t second:- ω=ω₀+αt

  (b) Angular displacement after t second:- θ = ω₀t + ½ αt²

  (c) Angular velocity after a certain rotation:- ω² – ω₀² = 2αθ

  (d) Angle traversed in ‘nth’ second:- θ_nth = ω₀ +α/2 (2n-1)

• Time period:- It is the time taken by the particle to complete one rotation.

             T= 2π/ω

• Frequency:- The number of rotations made by the particle per second is called  the frequency of rotation.

            If f is the frequency, the particle describes 2πf radians per second.

            ω = 2πf

            So, f = 1/T

• Centripetal force:- The force, acting along the radius towards the center, which is essential to keep the body moving in a circle with uniform speed is called centripetal force. It acts always along the radius towards the center. A centripetal force does no work.

            F = mv²/r = mrω²

• Centrifugal force:- Centrifugal force is the fictitious force which acts on a body, rotating with uniform velocity in a circle, along the radius away from the center. Magnitude of centrifugal force is,

            F = mv²/r

• Centripetal and centrifugal forces are equal in magnitude and opposite in direction. They cannot be termed as action and reaction since action and reaction never act on same body.

• Banking of Roads:-

            Road offering no frictional resistance, θ = tan^-1 (v²/rg)

            Road offering frictional resistance, v_max = √rg(µ+tanθ/1-µtanθ)

• Bending of Cyclist:- θ = tan^-1 (v²/rg)

  (a) Velocity of the cyclist:- Greater the velocity, greater is his angle of inclination with the vertical.

  (b) Radius of curvature:- Smaller the radius, greater is the angle with the vertical.

• Time period of conical pendulum:-T = 2π √lcosθ/g

• Motion in a vertical circle/looping the loop:-

The minimum velocity of the body, at the lowest point, required to take the body round a vertical circle is √5gr.

So, v = √5gr

The minimum tension in the string, at the lowest point, required to take the body around the vertical circle is equal to six times the     weight of the body.

So, T₁= 6 mg

(a) For lowest point A and highest point B, T_A – T_B = 6 mg

V_A² = V_B²+4gl

V_A≥√5gl

V_B≥√gl

(b) Condition for oscillation:- V_A?√2gl

(c)  Condition for leaving circular path:- √2gl< V_A?√5gl

• Non-uniform circular motion:-

(a) The velocity changes both in magnitude as well as in direction.

(b)  The velocity vector is always tangential to the path.

(c) The acceleration vector is not perpendicular to the velocity vector.

(d)  The acceleration vector has two components.

(i) Tangential acceleration a_t changes the magnitude of velocity vector and is defined as, a_t = dv/dt

(ii) Normal acceleration or centripetal acceleration a_c changes the direction of the velocity vector and is defined as,a_c =v²/r

(iii) The total acceleration is the vector sum of the tangential and centripetal acceleration.

So, a = √a_t²+a_c²