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
Page 2 of 2 DAVID LIAO.COM
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
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