Circular Motion Revision Quiz

Study Notes

Circular Motion

Circular motion is said to be uniform if the speed of the particle remains constant along the circular path.

Angular Displacement

Scalar form: ΔS = rΔθ
Vector form: ΔS = rΔθ (vector product)

Angular Velocity

Relation between linear velocity (v) and angular velocity (ω): scalar form - v = rω, vector form - v = rω (vector product)

Angular Acceleration

Relation between linear acceleration (a) and angular acceleration (α): scalar form - a = rα, vector form - a = rα (vector product)

Tangential and Centripetal Acceleration

Tangential acceleration: changes the magnitude of velocity vector
Centripetal acceleration: changes the direction of the velocity vector, a = v²/r = ω²/r

Equations of Rotational Kinematics

Angular velocity after a time t second: ω = ω₀ + αt
Angular displacement after t second: θ = ω₀t + ½ αt²
Angular velocity after a certain rotation: ω² - ω₀² = 2αθ
Angle traversed in 'nth' second: θn = ω₀ + α/2 (2n-1)

Time Period and Frequency

Time period: time taken by the particle to complete one rotation, T = 2π/ω
Frequency: number of rotations made by the particle per second, ω = 2πf, f = 1/T

Centripetal Force and Centrifugal Force

Centripetal force: force acting along the radius towards the center, F = mv²/r = mrω²
Centrifugal force: fictitious force acting along the radius away from the center, F = mv²/r
Centripetal and centrifugal forces are equal in magnitude and opposite in direction

Banking of Roads and Bending of Cyclist

Banking of roads: θ = tan⁻¹ (v²/rg) for roads offering no frictional resistance
Banking of roads: vmax = √rg(µ+tanθ/1-µtanθ) for roads offering frictional resistance
Bending of cyclist: θ = tan⁻¹ (v²/rg)

Conical Pendulum and Motion in a Vertical Circle

Time period of conical pendulum: T = 2π √(lcosθ/g)
Minimum velocity of the body at the lowest point required to take the body round a vertical circle: v = √5gr
Minimum tension in the string at the lowest point required to take the body around the vertical circle: T₁ = 6 mg

Non-uniform Circular Motion

Velocity changes both in magnitude and direction
Velocity vector is always tangential to the path
Acceleration vector is not perpendicular to the velocity vector
Acceleration vector has two components: tangential and normal (centripetal) acceleration

Description

Test your understanding of circular motion, including uniform circular motion, angular displacement, angular velocity, and angular acceleration. Review the scalar and vector forms of these concepts and their relationships with linear velocity and acceleration.