PowerPoint Presentation: Circular Motion By Team Chakra
PowerPoint Presentation: A particle is said to be in circular motion when it moves along the circumference of a circle
Angular Displacement (θ): Angular Displacement ( θ )
PowerPoint Presentation: The angle through which the radius vector turns in an interval of time t is called angular displacement Its SI Unit is radians (rad)
Angular Velocity (ω): Angular Velocity ( ω ) Angular displacement per unit time. ω = d θ dt
Circular Motion: Circular Motion Uniform Non Uniform
Uniform Circular motion: Uniform Circular motion Linear speed (v) = R θ /t Angular speed( ω ) = θ /t Instantaneous angular speed ( ω ) = d θ /dt = v/R Time period to transverse a circle 2 π = T = 2 π R/v = 2 π / ω Frequency (no. of cycles per second)( ν ) = 2 π R/v = 2 π / ω
Centripetal Acceleration: Centripetal Acceleration
If we take two points seperated by a small angle Δθ and covered during a small time interval Δt as shown.: If we take two points seperated by a small angle Δθ and covered during a small time interval Δ t as shown . The change in velocity Δ v is v(t)-v(t+ Δ t) as shown. Let ǀ v(t) ǀ = ǀ v(t+ Δ t) ǀ = v If we take Δθ as small as possible, ǀ Δ vǀ = v Δθ and Δ v is perpendicular to v(t) ,ie. Radially along radius. The acceleration vector a = Rate of change of velocity = Δ v/ Δ t . Direction of a is direction of Δ v, ie. Radially inward and,
ǀaǀ = ǀΔv/Δtǀ = ǀvΔθ/Δtǀ = vω= (v.v)/R =V2/R = ω2R: ǀaǀ = ǀ Δ v/ Δ tǀ = ǀv Δθ / Δ tǀ = v ω = (v.v)/R =V 2 /R = ω 2 R Because it is radially inward, it is called Centripetal Acceleration. Therefore a particle in uniform circular motion is under centripetal acceleration. The acceleration vector has a constant magnitude but varying direction because at any instant, it is along the radius vector at that point (and towards centre). Centripetal acceleration is generally given by a symbol a r , a radial or an (a normal )
Non uniform circular motion: Non uniform circular motion In this motion, there are two types of changes in velocity, which will result in two types of accelerations. Centripetal Acceleration Tangential Acceleration a t = dv / dt
PowerPoint Presentation: V = u + a t Δ t V/R = u/R + (a t/ R) Δ t ω = ω 0 + αΔ t where ω 0 is the initial angular velocity and ω is the final angular velocity. α = a t /R
Angular Acceleration(α): α = a t /R It is a vector quantity directed along the axis of rotation. Its SI unit is rad s -1 Angular Acceleration( α )
Similarly,: Similarly, ω = ω 0 + α t θ = ω 0 t + ½ α t 2 θ =(( ω 0 + ω )/2)t ω 2 - ω 0 2 = 2 αθ
Net acceleration = : Net acceleration = a net = (a r 2 + a t 2 ) 1/2 = [(v 2 /r) 2 +(dv/dt) 2 ] 1/2
PowerPoint Presentation: tan φ = a t / a r