Circular Motion in Kinematics Short Notes PDF
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Uploaded by EffusiveSard7556
RNS Pre University College
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Summary
These handwritten notes provide a summary of circular motion in kinematics. Topics included are angular displacement, velocity, and acceleration. Equations, diagrams, and examples of uniform and non-uniform motion are also presented.
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# Circular Motion in Kinematics - When an object moves on the circumference of a circle. - Reference point - centre of the circle - [Image Description: Diagram showing a circle with radius R, point A at (0,0) and point B on the circumference. The arc length between A and B is labeled "Arc (dist.)"...
# Circular Motion in Kinematics - When an object moves on the circumference of a circle. - Reference point - centre of the circle - [Image Description: Diagram showing a circle with radius R, point A at (0,0) and point B on the circumference. The arc length between A and B is labeled "Arc (dist.)", and the displacement vector is labeled "Disp"]. ## Angular displacement (θ) - θ is an axial vector, dimensionless. - Anticlockwise → outward - Clockwise → inward - Distance = Rθ (Arc) - Disp = 2R sin(θ/2) - Position vector along radius away from the centre. - Velocity is perpendicular to the radius along the tangent. - Axial vectors: all angular parameters are perpendicular to the plane, axis of rotation (θ, ω, α) ## Angular Velocity (ω) 1. **Average angular velocity (in time 't')**: - ωavg = (θf-θi)/Δt - ωavg = ∫θdt/∫dt 2. **Inst. angular velocity (at time 't')**: - ω = dθ/dt → slope of θ-t graph is angular speed - unit → radians - dimension → T<sup>-1</sup> - Scalar ## Relation with Angular velocity (ω) & frequency (f) - f = 1/T - ω = 2πf = 2π/T [No of rev = 2π, 4π, 6π, etc] ## Uniform Angular Velocity - Rate of change of angle is constant. - Same angle travelled in the same time. - Arc length is also constant. - Speed constant. ## Non-uniform angular velocity - Different angle travelled in the same time. - Arc length is different. - Speed changing. ## Angular Acceleration (α) - a<sub>avg</sub> (in time 't') - Δω/Δt - Δω/Δt = ∫ω dt/∫dt - a<sub>avg</sub> = ∫ωdt/∫dt - a<sub>inst</sub> (at time 't') - a<sub>inst</sub> = dω/dt - Rate of change of angular velocity - a<sub>inst</sub> = dω/dt = dθ/dt = ωdω/dθ - Unit → rad/sec<sup>2</sup> - Dimension → T<sup>-2</sup> - ω = constant → α = 0 → ω α 0 - ω ≠ constant → α ≠ 0 - Parallel: ω, α - Antiparallel: ω, α - ω↗, α↗ Ang speed ↑ - ω↗, α↓ Ang speed ↑ ## Relation between linear & Ang. Speed - V = Rω - a<sub>T</sub> = Rα → Tangential acceleration | Acceleration | Tangential Acceleration | Centripetal Acceleration | Angular Acceleration | |---|---|---|---| | a = dv/dt | a<sub>t</sub> = dv/dt | a<sub>c</sub> = v<sup>2</sup>/R | α = dω/dt | | a = a<sub>t</sub> + a<sub>c</sub> | a<sub>T</sub> = Rα | | | ## Uniform Acceleration (UCM) - Avg. speed → constant - Speed → constant - Velocity → changing (direction changing) - KE → constant - Momentum (P=mv) → changing - Ang. Acc (α = dω/dt) = 0 - Tangential acc (a<sub>T</sub>) = 0 - Acc (a = a<sub>T</sub> + a<sub>c</sub>) ≠ 0 - a<sub>c</sub> ≠ 0 - Work = ΔKE = 0 - Force ≠ 0 - Power = Work/Time ## Non-uniform circular Motion (NUCM) - ω (ang. velocity) → Variable - Ang. Speed → Variable ( |a<sub>T</sub>| ≠ 0) - Direction → Variable (a<sub>c</sub> ≠ 0) - KE → Variable - Momentum → Variable - Work ≠ 0 - Power ≠ 0 ## Speed ↑  - |a| = √(a<sub>T</sub><sup>2</sup> + a<sub>c</sub><sup>2</sup>) - |a|<sub>inst</sub> = v/R - tanθ = a<sub>c</sub>/a<sub>T</sub> - θ < 90<sup>o</sup> - a<sub>T</sub> = a cosθ - ΔT = a sinθ - dV/dt = R X dθ/dt ## Speed ↓  - |a| = √(a<sub>T</sub><sup>2</sup> - a<sub>c</sub><sup>2</sup>) - tanθ = a<sub>c</sub>/a<sub>T</sub> - θ =180 - θ - a<sub>c</sub> = a cosθ - a<sub>T</sub> = a sinθ - dV/dt = R X dθ/dt ## Equation of Motion in circular Motion | linear (SI) | angular (rad/s) | |---|---| | V (m/s) | ω (rad/s) | | a (m/s<sup>2</sup>) | α (rad/s<sup>2</sup>) | | V = dx/dt | ω = dθ/dt | | a = dv/dt | α = dω/dt | | a = v dv/dx | α = ω dω/dθ | ## v = u + at ## S = ut + 1/2 at<sup>2</sup> ## v<sup>2</sup> = u<sup>2</sup> + 2 aS ## S<sub>nth</sub> = u + (2n -1) a ## ω<sub>f</sub> = ω<sub>i</sub> + αt ## θ = ω<sub>i</sub>t + 1/2 αt<sup>2</sup> ## ω<sub>f</sub><sup>2</sup> = ω<sub>i</sub><sup>2</sup> + 2αθ ## θ<sub>nth</sub> = ω<sub>i</sub> + (2n-1) α # Note: - The document is handwritten. - Image quality was very poor, so I have had to make several assumptions based on guesses. - Some equations have had to be re-written due to unclear handwritten content. - This transcription has been attempted to the best of my ability.
