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Q) A plane spiral made of stiff smooth wire is rotated with a constant angular velocity $\omega$ in a horizontal plane about the fixed vertical axis O. A small sleeve M slides along that spiral without friction. Find its velocity $v'$ relative to the spiral as a function of the distance $r$ from rotation axis O if the initial velocity of sleeve is equal to $v_0$.

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The questions has been solved in reference frame fixed to the spiral. However, in the book it is given that of all the forces, work is performed only by the centrifugal force of inertia. The remaining forces, gravity, coriolis, force of reaction are all perpendicular to the $v'$ and do not perform any work. My question is that wouldn't centrifugal force also be perpendicular to the velocity moving and would not do any work?

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My question is that wouldn't centrifugal force also be perpendicular to the velocity moving and would not do any work?

The centrifugal force points radially outwards. This means that the only paths that are perpendicular to the centrifugal force are circular paths about the center of rotation. Since the sleeve is traveling along a spiral, not a circle, the centrifugal force does have a component along the path of the sleeve, and hence it does work.

Another way to see this is to note that in order to travel outward on a spiral, your motion has to have some radial component, otherwise you would be traveling along a circle. This gives you the radial component of motion that is in the same direction as the centrifugal force.

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