Circular Motion Revision Quiz

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