等速円運動 (3回目)

2025/12/7(日)
 
等速円運動 (3回目)
 
(uniform circular motion)
 
■ 等速円運動
▼ ベクトル3重積
A×(B×C)
= A×(ByCz-BzCy, BzCx-BxCz, BxCy-ByCx)
= (Ay(BxCy-ByCx) - Az(BzCx-BxCz),
  (Az(ByCz-BzCy) - Ax(BxCy-ByCx),
  (Ax(BzCx-BxCz) - Ay(ByCz-BzCy))
= (AyBxCy-AyByCx-AzBzCx+AzBxCz + (AxBxCx - AxBxCx), … (=0を足す)
   AzByCz-AzBzCy-AxBxCy+AxByCx + (AyByCy - AyByCy), … (=0を足す)
   AxBzCx-AxBxCz-AyByCz+AyBzCy + (AzBzCz - AzBzCz)) … (=0を足す)
= (Bx(AxCx+AyCy+AzCz) - Cx(AxBx+AyBy+AzBz),
   By(AzCz+AxCx+AyCy) - Cy(AzBz+AxBx+AyBy),
   Bz(AxCx+AyCy+AzCz) - Cz(AxBx+AyBy+AzCz))
= (A･C)B-(A･B)C
A×(B×C) = (A･C)B-(A･B)C
 
▼ 角速度ω 
ω = (0, 0, ω)
 
R = (x, y, z) = (rcosωt, rsinωt, 0)
V = ω×R
= (ω_yz-ω_zy, ω_zx-ω_xz, ω_xy-ω_yx)
= (-ω_zy, ω_zx, 0)
= (-ωy, ωx, 0)
 
A = ω×V
= (ω_yv_z-ω_zv_y, ω_zv_x-ω_xv_z, ω_xv_y-ω_yv_x)
= (-ω_zv_y, ω_zv_x, 0)
= (-ω(ωx), ω(-ωy), 0)
= (-ω²x, -ω²y, 0)
 
A = ω×(ω×R) = (ω･R)ω - (ω･ω)R  … (ω⊥R)より
= -(ω･ω)R
= -|ω|²R
 
▼ 向心力F
r = √(x² + y²)
v = |V| = |ω × R| = √{(-ωy)² + (ωx)²}
= ω√(y² + x²) = rω
ω = v/r
 
F = mA = m(a_x, a_y, a_z) = m(-ω²x, -ω²y, 0) = (-mω²x, -mω²y, 0)
|F| = F･F = √{(-mω²x)² + (-mω²y)²} = mω²√(x² + y²)
= mrω² = mr(v/r)² = mv²/r
 
 
■ 結果
▼ 角速度ω 
ω = (0, 0, ω)
R = (x, y, z) = (rcosωt, rsinωt, 0)
V = ω × R = (-ωy, ωx, 0)
A = ω × V = (-ω²x, -ω²y, 0)
A = ω×(ω×R) = -|ω|²R
 
▼ 向心力F
F = mA = (-mω²x, -mω²y, 0)
|F| = mv²/r