12 Questions
What is the condition for uniform circular motion?
The speed of the particle remains constant
What is the scalar form of angular displacement?
?S = r?θ
What is the relation between linear velocity and angular velocity?
v = rω
What is the vector form of angular acceleration?
α = Δω/Δt
What is the equation of motion for circular motion?
a = ω^2/r
What is the definition of frequency?
The number of rotations made by the particle per second
What is the direction of centripetal force?
Along the radius towards the center
What is the magnitude of centrifugal force?
F = mv2/r
What is the relationship between centripetal and centrifugal forces?
They are equal in magnitude but opposite in direction
What is the velocity of the cyclist when the radius of curvature is smaller?
Greater
What is the minimum velocity required to take a body round a vertical circle?
√5gr
What is the condition for oscillation in a conical pendulum?
VA > √2gl
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
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.
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