Understanding Mechanics: The Relationship Between Radians and Vectors

[Jason x]

Hi! Here's a simple part of the lecture on mechanics (Fig.). And the trivial question is why the dα (which is in radian) equals dτ (which is vector), the idea of the question is why vector value could be equal to radian? Isn't it strange? Vector must contain three numbers, or maybe length and angle value, but dα is just a angle!

Thanks for attention)

 

[David Lewis]

What does ρ stand for?

 

[beamie564]

why the dα (which is in radian) equals dτ (which is vector), the idea of the question is why vector value could be equal to radian

Although I don't get the context of your question( it's not in English), I'll try.
While normal angles don't follow the laws of vector addition, infinitesimal angles do. So $d\alpha$ can be equated to a vector.
Also, in your equations $d\alpha$ isn't equal to $d\tau$. You can't do this because the angle(radians) is a dimensionless quantity and torque isn't. In the bottom line $d\alpha$ is equal to $\frac{d\tau}{|\tau|}$; this is okay, since the RHS is a unit vector (which is also dimensionless).

 

[Jason x]

Thank you for your reply! It was not necessary to go through the context.

But I didn't get the last thing:

a unit vector (which is also dimensionless).

why is it also dimensionless? It has direction and that's how we can find the sense of dτ. Sorry for stupid question (in case of it).

 

[Jason x]

What does ρ stand for?

it is the radius of curvature, and an is a normal acceleration and aτ is a tangential acceleration.

 

[beamie564]

why is it also dimensionless? It has direction and that's how we can find the sense of dτ.

True. It does have a direction, but it has no units. For example, suppose there's a position vector $\vec a = 3(m)\hat i + 4(m)\hat j$. Then its magnitude $|\vec a|= \sqrt {3^2+4^2} m = 5m$. The unit vector, being $\frac{\vec a}{|a|}$ has no units(and is dimensionless).