Well, I'm not going to answer that question, but hopefully, we can reason through this. What do you think? What ideas have you had?is this the correct answer?
So there is a stone ball that has a diameter of about 1.3m and it fits in a hemispherical shaped stone cup almost exactly. The ball weighs at least a ton. There is also a water jet at the bottom of the cup so that the stone ball is suspended on a thin film of water, making the friction of the ball with the cup almost 0. The ball is easy to set in motion and once set into motion, it continues to rotate in the direction you start it for a long time. Suppose you can only access the ball from near the top and you can make the ball rotate along any horizontal axis but you do not have enough grip to make it rotate around the vertical axis.
Can you make the ball rotate around the vertical axis anyway?
i think the answer is yes but i came up with the answer without really writing anything down. is this the correct answer?
Sorry if this might be in the wrong section. this was a challenge problem in my linear algebra book.
well i was thinking that since you can only spin the ball along a horizontal axis, you first spin the ball in one direction then slightly change the direction by spinning along a new horizontal axis that is the original axis except shifted by a small angle. you continue to change the horizontal axis little by little by constantly spinning it with your hand and theoretically (in my mind) you can make it spin all the way around eventually. i am not sure how to express this mathematically though.
You're on the right track, I think. My idea would be this: think of a coordinate system x, y, z. Suppose z is in the vertical, and x and y are constructed so that you have a right-handed coordinate system. That is, i x j = k, where i is the unit vector in the x direction, j the same in the y, and k the same in the z. Now, start the ball rotating about the x axis. This you can do by pushing the top of the ball either in the positive or negative y direction. Once you have that rotation going, simply push in the x direction, either positive or negative. Then, when the rotating axis gets to the vertical (probably a little sooner to counteract inertia), stop the second rotation.
All of this is predicated on the non-trivial assumption that you can do the second rotation without affecting the first. That might require some extra thinking.
Make sense?