Question

Which formulas have been correctly rearranged to solve for radius? Check all that apply.

r = StartFraction G times m Subscript central Baseline over v Superscript 2 Baseline EndFraction.
r = StartFraction F Subscript c Baseline times m over v Superscript 2 Baseline EndFraction.
r = StartFraction a Subscript c Baseline over v Superscript 2 Baseline EndFraction
r = StartFraction v T Over 2 pi EndFraction
r = StartFraction a Subscript c Baseline times T over pi EndFraction

Answer 1

Correct ones: 1 and 4.

Brief derivations/errors:

1) r = G m_central / v^2 — correct. From equating gravity to centripetal force: G m_central m / r^2 = m v^2 / r → G m_central / r = v^2 → r = G m_central / v^2.

2) r = (F_c m) / v^2 — incorrect. From F_c = m v^2 / r → r = m v^2 / F_c (the given expression has F_c and v^2 swapped).

3) r = a_c / v^2 — incorrect. Since a_c = v^2 / r → r = v^2 / a_c (reciprocal of the given).

4) r = v T / (2π) — correct. From v = 2πr / T → r = v T / (2π).

5) r = a_c T / π — incorrect. Eliminating v using v = 2πr / T in a_c = v^2 / r gives r = a_c T^2 / (4π^2), not a_c T / π.