A thin rigid wire of mass M and length L = 1 m rotates about a...

 

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A thin rigid wire of mass M and length L = 1 m rotates about a vertical axis passing through its centre, as illustrated in the diagram below. You may assume its moment of inertia is = ML/12. The wire is threaded through two small beads, each of mass m, which can move along the wire without friction. No external torques act on the system. (a) Initially the beads are at the centre of the wire which rotates with angular velocity wo- Derive an expression for the angular velocity when the beads have moved a distance x from the centre. (b) Let M = 1 kg and m = 0.1 kg. Determine (i) the angular velocity of the wire at the point where the beads fly off, as a function of wo, and (ii) the tangential velocity of the bead at this point, also as a function of wo (c) The wire rotates 1 m above the ground. The system is modified so that the beads are held at the end of wire before releasing them. Calculate the horizontal distance between the point where the bead leaves the wire, and the point where the bead hits the ground, if the initial angular velocity @o= 10 rad/s. You may assume that the bead's radial velocity is sufficiently small that it can be neglected. A thin rigid wire of mass M and length L = 1 m rotates about a vertical axis passing through its centre, as illustrated in the diagram below. You may assume its moment of inertia is = ML/12. The wire is threaded through two small beads, each of mass m, which can move along the wire without friction. No external torques act on the system. (a) Initially the beads are at the centre of the wire which rotates with angular velocity wo- Derive an expression for the angular velocity when the beads have moved a distance x from the centre. (b) Let M = 1 kg and m = 0.1 kg. Determine (i) the angular velocity of the wire at the point where the beads fly off, as a function of wo, and (ii) the tangential velocity of the bead at this point, also as a function of wo (c) The wire rotates 1 m above the ground. The system is modified so that the beads are held at the end of wire before releasing them. Calculate the horizontal distance between the point where the bead leaves the wire, and the point where the bead hits the ground, if the initial angular velocity @o= 10 rad/s. You may assume that the bead's radial velocity is sufficiently small that it can be neglected.

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a To derive the expression for the angular velocity when the beads have moved a distance x from the ... View the full answer