Uniform circular motion: Magnitude of centripetal acceleration

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Intuition What kinematic factors can affect the magnitude of the inward acceleration of an object undergoing uniform circular motion?

𝑎𝑎IN > 0

↑ 𝑣𝑣 ⇒ ↑ Δ𝜃𝜃 ⇒ ↑ 𝑎𝑎IN ↑ 𝑣𝑣 ⇒ ↑ |𝐯𝐯�⃑ | ⇒ ↑ 𝑎𝑎IN

↑ 𝑟𝑟 ⇒ ↓ 𝑎𝑎IN

𝑎𝑎IN = 𝑣𝑣TAN2

𝑟𝑟

Uniform circular motion: Magnitude of centripetal acceleration

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Symbolic derivation

Consider very small Δ𝐯𝐯�⃑ and very small Δ𝑡𝑡

|Δ𝐯𝐯�⃑ |𝑣𝑣TAN

≈𝑣𝑣TANΔ𝑡𝑡

𝑟𝑟

|Δ𝐯𝐯�⃑ |Δ𝑡𝑡

≈𝑣𝑣TAN2

𝑟𝑟

𝑎𝑎IN = limΔ𝑡𝑡→0

�Δ𝐯𝐯�⃑Δ𝑡𝑡�

𝑎𝑎IN = limΔ𝑡𝑡→0

|Δ𝐯𝐯�⃑ |Δ𝑡𝑡

𝑎𝑎IN =𝑣𝑣TAN2

𝑟𝑟