Circular Motion Deriving the formula for centripetal acceleration.

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Presentation transcript:

Circular Motion Deriving the formula for centripetal acceleration

Deriving Formula for Centripetal Acceleration r is the radius of the circle that the object is traveling v is the tangential velocity r and v form a right triangle r v

Deriving Formula for Centripetal Acceleration Angle between r of 1 st triangle and 2 nd triangle = θ Angle between hypoteneuse of 1 st triangle and hypotenuse of 2 nd triangle = θ Thus, angle between v of 1 st triangle and 2 nd triangle = θ r v1v1 r v2v2 Θ Θ

Deriving Formula for Centripetal Acceleration r v1v1 r v2v2 l One triangle is formed using the two points where we determined the velocity The angle across from l is θ

Deriving Formula for Centripetal Acceleration r v1v1 r v2v2 l Another triangle is formed with the velocity vectors Note – the vectors are being subtracted because they are tail to tail Δv = v 2 – v 1 The angle across from Δv is θ v1v1 v2v2 Δv

Deriving Formula for Centripetal Acceleration r v1v1 r v2v2 l If the object is being swung with a constant speed then the magnitude of v1 and the magnitude of v2 are equal. Thus – the two triangles are similar v 1 = Δv r l v1v1 v2v2 Δv

Deriving Formula for Centripetal Acceleration Divide both sides by Δt v 1 = Δv r l Δv = v 1 l r Δv = v 1 l Δt Δt r

Deriving Formula for Centripetal Acceleration So Δv = v 1 l Δt Δt r Δv = a Δt v = l Δt

Deriving Formula for Centripetal Acceleration a = v 2 r