An object having its total mass concentrated at centre and having six identical rods (light) of length $l$ each attached to it symmetrically is placed on an inclined rough surface of inclination $α .$ Object looks like a wheel with six equally spaced spokes without rim. Assume that there is no slipping and collision between the spokes and plane is perfectly inelastic and wheel always remains in the vertical plane. Also assume object remains in contact with the plane and initially center of the object is at a distance $l$ from the inclined plane and is released from rest. The acceleration due to gravity is g. Pick the CORRECT statement(s) for the motion of wheel, long time after the motion has started.

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Average speed of the centre of object attains a constant value.

Mechanical energy of object remains conserved.

Instantaneous power delivered by friction to the object is zero.

In steady state, speed of center of object at the instant when a leg is about to hit the plane is $\sqrt{\frac{8 g l \sin α}{3}} .$

answer is A, C, D.

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Detailed Solution

When the pivot point is changed the speed of central point changes suddenly and the kinetic energy is lost. Velocity component perpendicular to new spoke survives. Average velocity becomes constant when the loss in K.E. due to changing contact point = gain in P.E.
$\frac{1}{2} m v_{0}^{2} (1 - \cos^{2} \frac{π}{3}) = m g l .2 \sin \frac{π}{6} \sin α$