778_Dynamics 11ed Manual

778_Dynamics 11ed Manual - Engineering Mechanics - Dynamics...

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Unformatted text preview: Engineering Mechanics - Dynamics Chapter 21 Solution: Also, ⎛ a + r cos ( θ ) ⎞ φ' ⎟ r ⎝ ⎠ rψ' = a + r cos ( θ ) φ' Since no sipping occurs, ( 2) ω = φ' + ψ' F = m aφ' 2 mr Ix = Iy = 2 ψ' = ⎜ N − mg = 0 2 Iz = m r ω = φ' sin ( θ ) j + ( −ψ' + φ' cos ( θ ) k ωx = 0 Thus, ωy = φ' sin ( θ ) ωz = −ψ' + φ' cos ( θ ) ω' = φ' × ψ' = −φ' ψ' sin ( θ ) ω'x = −φ' ψ' sin ( θ ) ω'y = ω'z = 0 Applying Σ Mx = Ixω'x + ( Iz − Iy) ωzωy 2⎞ ⎛ mr (−φ' ψ' sin (θ ) ) + ⎜mr2 − m r F r sin ( θ ) − N r cos ( θ ) = 2 2 ⎝ 2 Solving we find m aφ' r sin ( θ ) − m g r cos ( θ ) = 2 2 ⎛ −m r2 ⎞ 2 a + r cos ( θ ) ⎞ ⎛ mr ⎞ ⎛ a ⎞ 2 ⎜ ⎟ φ' sin ( θ ) ⎛ ⎟ ⎜ ⎟ φ' sin ( θ ) −⎜ ⎜ ⎟ r ⎝2⎠ ⎝ ⎠ ⎝ 2 ⎠⎝ r ⎠ 2g cos ( θ ) = aφ' sin ( θ ) + rφ' sin ( θ ) cos ( θ ) 2 ⎟ ( −ψ' + φ' cos ( θ ) ) ( φ' sin ( θ ) ) ⎠ 2 φ' = 2g cot ( θ ) a + r cos ( θ ) Problem 21-70 The top consists of a thin disk that has weight W and radius r. The rod has a negligible mass and length L. If the top is spinning with an angular velocity ωs, determine the steady-state precessional angular velocity ωp. 776 ...
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