Table 1 The selection of positive and negative signs of \(N_r\), \(N_\varphi\), \(f_r\) and \(f_\varphi\) in Eqs. (7)–(10) with different ranges of \(\omega\), n and \(\theta _0\).
From: Orbit constraint of a small bead on a rotating large circular hoop in a horizontal plane
\(\omega\) | \(>0\) | \(<0\) | ||||||
|---|---|---|---|---|---|---|---|---|
n | 2k | \(2k+1\) | 2k | \(2k+1\) | ||||
\(\theta _0\) | \((0,\pi )\) \((-\pi ,0)\) | \((\pi ,2\pi )\) \((-2\pi ,-\pi )\) | \((0,\pi )\) \((-\pi ,0)\) | \((\pi ,2\pi )\) \((-2\pi ,-\pi )\) | \((0,\pi )\) \((-\pi ,0)\) | \((\pi ,2\pi )\) \((-2\pi ,-\pi )\) | \((0,\pi )\) \((-\pi ,0)\) | \((\pi ,2\pi )\) \((-2\pi ,-\pi )\) |
\(N_r\) | − | \(+\) | \(+\) | − | − | \(+\) | \(+\) | − |
\(N_\varphi\) | − | \(+\) | \(+\) | − | − | \(+\) | \(+\) | − |
\(f_r\) | \(+\) | − | − | \(+\) | − | \(+\) | \(+\) | − |
\(f_\varphi\) | − | \(+\) | \(+\) | − | \(+\) | − | − | \(+\) |
