When introducing the cross product, one thing that always catches my students' attention is a demonstration of precession. We have a bike wheel with a heavy solid rubber tire. (You may be able to borrow one from your physics department.) I get it spinning with its axis in the horizontal plane, and then suspend it by a rope from one side. They are gratifyingly surprised when they see it precess in the horizontal plane rather than flopping. This connects to torque expressed as a vector cross product.

Another example I like to do is to have them imagine a spinning cylindrical space station and lead them through a series of thought experiments leading them to figure out the rules by which the centrifugal and Coriolis forces operate. E.g., a person standing on the deck releases a ball. In the frame of the stars, the ball travels straight. What appears to happen in the rotating frame? Once they see that the centrifugal force operates like a radially directed gravitational field, I do examples that show the existence of a velocity-dependent force. E.g., what happens if the person standing on the deck throws the ball opposite to the direction of rotation, such that he exactly cancels its motion in the frame of the stars? What if a ball is released on the axis? What if the ball is released on the axis with some small radial velocity? Based on this sort of thing, it's fairly easy to get them to conclude that the Coriolis force is an odd function of the ball's velocity, and also an odd function of the angular velocity of the space station. There is only one good way to get a vector-valued function with these properties, so it must be $\mathbf{F}=c\mathbf{\omega}\times\mathbf{v}$. They enjoy imagining the fun and strange things that happen on the space station, and it's a cool example where the uniqueness of certain mathematical operations makes it possible to determine the answers to problems without grotty calculations.