Circular Motion (Chapter 1)

Definitions

• Angular Displacement: It is the angle through which an object moves on a circular path. Its unit is radian.

• Angular Velocity: It is the rate of change of Angular Displacement of An object. Its Unit is ω.

 ω=Δθ/T

ω=2π/T=2πf (Since f=1/T)

• Radian: It is the Angle at which the arc length of a circle becomes equal to its radius.

s=rθ

θ=Arc-length/Radius=s/r

Proof That 1 Radian=2π

We know that s=rθ and we also know that s=2πr. Now equating the two equations we get

rθ=2πr

θ=2π

• Linear Velocity: It is the rate of Change of displacement of an object that is moving in a straight path.

v=rω

• Centripetal Force: Resultant force acting on an object moving in a circle always directed towards the center of the circle.

F=mv²/r

putting in v=rω we get F=mrω²

• Centripetal Acceleration: It is the acceleration of an object moving in uniform circular motion.

a=v²/r

Using v=rω in the equation a = r²ω²/r = rω²

Situations for Vertical circular motion

Case 1

Centripetal force = Tension - Weight 

mrω² =Tension - mg

Case 2

 Centripetal force = Tension + Weight 

 mrω² =Tension + mg

Case 3

When object is at the top

mg-R= mrω² 

When object is at the bottom

R-mg= mrω² 

Situations for Horizontal circular motion

Case 1

friction provides centripetal force

F=mrω²

Case 2

Tcosθ = mg

Tsinθ = mrω²

Case 3

Rsinθ = mg

Rcosθ = mrω²