**A rigid tube is placed in the vertical plane. It has the shape of an arbitrary curve, but has start and end points of the same height. A chain with uniform mass distribution lies in the tube. Show that the chain lies stationary in the tube, no matter its shape. **

**A stick of mass density per unit length **** rests on a circle of radius R. The stick makes an angle **** with the horizontal and is tangent to the circle at its upper end. Friction exists at all points of contact, and assume that it is large enough to keep the system at rest. Find the friction force between the ground and the circle.**

**A large number of sticks (with mass density per unit length p and circles (with radius R) lean on each other. Each stick makes an angle with the horizontal and is tangent to the next circle at its upper end. The sticks are hinged to the ground, and every other surface is frictionless. In the limit of a very large number of sticks and circles, what is the normal force between a stick and the circle it rests on, very far to the right? Assume that the last circle leans against a wall, to keep it from moving.**