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Consider the bead threaded on a circular hoop of Example

Chapter 9, Problem 9.17

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QUESTION:

Consider the bead threaded on a circular hoop of Example 7.6 (page 260), working in a frame that rotates with the hoop. Find the equation of motion of the bead, and check that your result agrees with Equation (7.69). Using a free-body diagram, explain the result (7.71) for the equilibrium positions.

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QUESTION:

Consider the bead threaded on a circular hoop of Example 7.6 (page 260), working in a frame that rotates with the hoop. Find the equation of motion of the bead, and check that your result agrees with Equation (7.69). Using a free-body diagram, explain the result (7.71) for the equilibrium positions.

ANSWER:

Step 1 of 4

As seen in a frame rotating with the hoop, there are five forces on the bead. The first three, all of which act in the plane of the hoop, are the bead's weight \(m\mathbf{g}\), the centrifugal force \(F_{\mathrm{ef}}=m\omega ^2R\mathrm{sin}\theta \rho\), and the normal force \(\mathbf{N}\) (actually the component of the normal force in the plane of the hoop). The other two are the Coriolis force \(\mathbf{F}_{cor}\) and the component of the normal force normal to the hoop (neither of which is shown in the picture). Since these last two both act normal to the hoop, they cancel one another and need not concern us further. The bead can move only in the tangential direction, and its equation of motion is \(ma_{tang}=F_{tang}\) or

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