Understanding Mechanics: The Relationship Between Radians and Vectors

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In summary, the question is why vector and radian values can be equal to each other, when vector values typically contain three numbers or a length and angle value while radians are just an angle. The answer is that while normal angles do not follow the laws of vector addition, infinitesimal angles do, allowing for dα to be equated to a vector. Additionally, while dα is not equal to dτ, it can be expressed as a unit vector of dτ divided by its magnitude. This unit vector is dimensionless, meaning it has no units. ρ stands for the radius of curvature and aτ is the tangential acceleration.
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Jason x
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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!

Снимок экрана 2016-07-31 в 1.29.44.png


Thanks for attention)
 
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  • #2
What does ρ stand for?
 
  • #3
Jason x said:
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).
 
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Thank you for your reply! It was not necessary to go through the context.

But I didn't get the last thing:
Aniruddha@94 said:
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).
 
  • #5
David Lewis said:
What does ρ stand for?
it is the radius of curvature, and an is a normal acceleration and aτ is a tangential acceleration.
 
  • #6
Jason x said:
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).
 
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What is a simple mechanics question?

A simple mechanics question is a question related to the basic principles and laws of motion and forces.

What are some examples of simple mechanics questions?

Examples of simple mechanics questions include "What is Newton's first law?", "How do you calculate force?", and "What is the difference between velocity and acceleration?"

Why is it important to understand simple mechanics?

Understanding simple mechanics is important because it helps us understand how objects move and interact with each other in the physical world. This knowledge is essential in many fields, such as engineering, physics, and even everyday activities like driving a car.

How can I improve my understanding of simple mechanics?

To improve your understanding of simple mechanics, you can study the basic principles and laws, practice solving problems, and conduct experiments to see these principles in action. It can also be helpful to seek guidance from a teacher or mentor.

Where can I find resources for learning more about simple mechanics?

There are many resources available for learning more about simple mechanics, such as textbooks, online tutorials, and educational videos. You can also attend workshops or courses, or join a study group to learn from others and discuss concepts. Your local library or university may also have resources available for you to use.

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