Consider a bead of mass m slides without friction in a uniform gravitational field on a vert circular hoop of radius R. The hoop is constrained to rotate at a fixed angular velocit- about is vertical diameter. Let 0 be the position of the bead on the hoop measured from lowest point (see figure). (a) Write down the Lagrangian L(0,0). (b) Deduce the Euler-Lagrange equation of motion for the bead.
Consider a bead of mass m slides without friction in a uniform gravitational field on a vert circular hoop of radius R. The hoop is constrained to rotate at a fixed angular velocit- about is vertical diameter. Let 0 be the position of the bead on the hoop measured from lowest point (see figure). (a) Write down the Lagrangian L(0,0). (b) Deduce the Euler-Lagrange equation of motion for the bead.
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Transcribed Image Text:5. Consider a bead of mass m slides without friction in a uniform gravitational field on a vertical
circular hoop of radius R. The hoop is constrained to rotate at a fixed angular velocity 2
about is vertical diameter. Let 0 be the position of the bead on the hoop measured froan its
lowest point (see figure).
(a) Write down the Lagrangian L(0,0).
(b) Deduce the Euler-Lagrange equation of motion for the bead.
(e) Find how the equilibriun values of 8 depends on f.
(Hint: Equilibrium occurs where 0
0).
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R.
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