What's the deal with force and velocity components?

[Identity]

I have two questions about components which are closely related.

The first is about how 'real' the component of a force is. Take a force $$F$$ in diagram 1. You can componentize it along two perpendicular axes. Now, $$F\sin{(a)}$$ is a force directed along the vertical axis. If we now componentize THAT force along perpendicular axes, we get $$F\sin{(a)}\cos{(a)}$$ perpendicular to $$F$$.
Why is it that you cannot componentize a force component?

The second is about componentizing velocity. In special relativity, the length of an object moving at a relative speed to you is contracted. Say we have a spaceship moving diagonally, as in diagram 2. If we componentize its velocities, there are components of speed vertically and horizontally. Does this mean that those directions are also contracted?
In fact, since we can choose orthogonal components at any angle, does that mean that the only direction unaffected by length contraction will be the cross-section perpendicular to $$v$$?
Presumably if this is right then it also affects time dilation? That would be tricky.

 

[tiny-tim]

Hi Identity!

The first is about how 'real' the component of a force is. Take a force $$F$$ in diagram 1. You can componentize it along two perpendicular axes. Now, $$F\sin{(a)}$$ is a force directed along the vertical axis. If we now componentize THAT force along perpendicular axes, we get $$F\sin{(a)}\cos{(a)}$$ perpendicular to $$F$$.
Why is it that you cannot componentize a force component?

You can't "componentize" a component on its own, it makes no sense.

You can "componentize" the whole original force in that direction, but you would have to "componentize" both the original components, and add the results.

In special relativity, the length of an object moving at a relative speed to you is contracted. Say we have a spaceship moving diagonally, as in diagram 2. If we componentize its velocities, there are components of speed vertically and horizontally. Does this mean that those directions are also contracted?
In fact, since we can choose orthogonal components at any angle, does that mean that the only direction unaffected by length contraction will be the cross-section perpendicular to $$v$$?

Yes, the only direction completely unaffected by length contraction is perpendicular to v.

It is as if you squashed the whole graph along the v direction.

 

[Identity]

Oh I see, if you add the other component you have $$F\sin{a}\cos{a}$$ canceling itself out!

The relativity thing is pretty cool too :]

Thanks tiny-tim