IB Physics Centrifugal Force?

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A tutorial sheet on rotating reference frames is given below.

  1. "For matter in a circular orbit at radius r around a mass M the centrifugal force and the gravitational attraction balance". Is this statement correct?
  2. A tube containing liquid is spun about a vertical axis through one end. Where in the tube is the pressure greatest? Show that the pressure difference between the end of a straight tube of length L and its axis is ⍴⍵2L2/2, where ⍴ is the density of the fluid in the tube and ⍵ is the angular speed of the tube. The tube rotates about an axis through one end perpendicular to the tube.
  3. A bucket of water spins at a constant rate about an axis through its centre. Why does the water build up at the edge of the bucket? (the fluid at the edge has greatest speed, creating a pressure gradient in the fluid between the centre and the edge pushing the fluid highest at the edges, there is no centrifugal force in an inertial reference frame)
  4. Why does the acceleration due to gravity on the surface a planet decrease if the planet spins faster?
  5. A space station has a large toroidal (doughnut) shape. When it spins are objects inside the space station weightless?
  6. A bead is placed half-way along a smooth horizontal wire. The wire now spins about a vertical axis through one end. Why does the bead move outwards along the wire?
  7. In question 3 there are now two fluids in the bucket, each of different density ⍴1 and ⍴2. What is the shape of the common surface when the bucket is spun at a constant angular speed ⍵?