A small smooth particle P of mass m is free to move under gravity in a thin smooth circular tube of radius r and centre O, fixed in a vertical plane. The particle is projected horizontally from the lowest point of the tube with speed `sqrt (3gr)` .

Explain why the law of conservation of energy can be applied for the motion of the particle.

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The law of conservation of energy is true in all cases. For an isolated system the total energy in the system does not change.

In the problem, the system consisting of the tube and the particle P of mass m placed in it can be considered to be a closed system. The total energy of the particle remains the same. As the particle moves within the pipe its height changes. This results in the kinetic energy and potential energy of the particle changing. The sum of the two however is constant. Any movement only results in some of the kinetic energy getting converted to potential energy or vice versa.

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