A solid cube with sides of a is in equilibrium on top of a cylinder of radius R.

Question

A solid cube with sides of a is in equilibrium on top of a cylinder of radius R. The density of right half of the cube is twice larger than that of left half. The planes of four sides of the cube are parallel to the axis of the cylinder. The contact between the cube and the cylinder is perfectly rough.

a) Find the contact point.

b) Under what condition is the equilibrium stable or not stable?

6. %. A solid cube with sides of a is in equilibrium on top of a cylinder of radius R. The density of right half of the cube is twice larger than that of left half. The planes of four sides of the cube are parallel to the axis of the cylinder. The contact between the cube and the cylinder is perfectly rough. a) Find the contact point. (5 pts.) b) Under what condition is the equilibrium stable or not stable? (15 pts.)

Explanation / Answer

b) The cylinder is assumed to be fixed so it cant rotate, so that the block moves away from the equilibrium point on top of the sphere by rolling off to the side. One approach to analyzing conditions of equilibrium is to calculate the potential energy of the block as a function of the angle by which the cube rotates as it rolls, and to look at the derivatives of the potential energy with respect to . Another approach is to look for restoring torques to maintain a stable equilibrium.

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