Uniform circular motion: Magnitude of centripetal AP Physics 1/4.2 UCM... Uniform circular motion:...

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Transcript of Uniform circular motion: Magnitude of centripetal AP Physics 1/4.2 UCM... Uniform circular motion:...

  • Uniform circular motion: Magnitude of centripetal acceleration

    Page 1 of 2 DAVID LIAO.COM

    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 = 𝑣𝑣TAN 2

    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