In different contexts, my book(Principles of Physics by Resnick, Halliday ,Walker) , they wrote
For torque, the path need no longer be a circle and we must write the torque as a vector $\vec{\tau}$ that may have any direction. …. Note carefully that to have angular momentum about $O$ , the particle does not itself have to rotate around $O$ .
That's what they wrote. But I am really confused why they wrote so. In fact I can't imagine torque & angular momentum without circular motion. Why did they tell so? What is the cause?? Please explain.
Best Answer
Actually, your book is correct. Even if the most usual uses of angular momentum involve circular or rotating motion, this is not the general case. An object moving in a straight line has angular momentum in a reference frame in which the origin does not fall on the the line.
To see this simply remember the definition of angular momentum $\overrightarrow{L}=\overrightarrow{r}\times m\overrightarrow{v}$, and torque $\overrightarrow{\tau}=\overrightarrow{r}\times \overrightarrow{F}$. Both can defined for any object regardless of their motion. One example where you use it for linear motion is when solving problems involving conservation of angular momentum and a mass gets ejected from a rotating one. The ejected mass will move in a straight line, but total angular momentum is conserved when you include that of straight moving mass too. If you compute the angular momentum of a free moving mass (no applied forces), you will obtain that it is constant across the entire path, as you would expect.
In the picture below it's shown both: that the angular momentum of a stright moving objects exists, and that it is a constant (if the velocity is constant) This is because $r \sin{\alpha}$ is constant along the trajectory (is the distance between the two parallel lines in the picture.