# What will be the angular velocity of a line about a line that is at some angle with the plane of motion?

In this book it is written that angular velocity of a rigid body is time derivate of "angular displacement" of any line in the plane of motion of the body. The angular position of the line is measured from any conveniently fixed reference axis in the plane of motion. For example in the figure shown below, the authors have chosen the horizontal line to be the reference axis. The angular positions of lines $$1$$ and $$2$$ are $$\theta_{1}$$ and $$\theta_{2}$$ respectively.

I don't understand how this can be the definition used to find the angular velocity of the lines (1 and 2), which move in the plane, from a reference line that make some angle to the plane of motion. What will be angular positions for lines 1 and 2, from this angled reference axis? What is angular displacement?

For concrete example suppose in the figure below I take $$OM$$ as reference axis and I want to find the angular velocity of the line $$OP$$ that is rotating anticlockwise maintaining the same height $$PM$$ and $$NP$$ (although they don't look to be the same in my very bad drawing). In starting $$OM$$ and $$OP$$ makes angle $$\alpha$$ at some time and then after infinitesimal seconds later the line moves and now makes angle $$\beta$$ with the line $$OM$$. So should the angular displacement be $$\beta -\alpha$$? This seems very wired; how is this difference related to the rotation?

I have read many books on this topic but they all give the same definition which involves the reference axis to be in the plane of motion. It seems that we can't take any line as a reference axis.

• Yes you can take any line as reference. It is the change in angle of a line in the body. Take a line in the body, any line. Measure how much it’s angle changes. That change tells you how much angular displacement there was. The reference line is meaningless. It’s just a base, in order to have two numbers to subtract for calculating the difference Sep 3, 2021 at 9:46
• Of course the line may have moved. We are measuring its angle. And how much its angle changes. Change in its angle from vertical, in its angle from horizontal, in its angle from a 45-line. Change in its angle. Thats angular displacement Sep 3, 2021 at 9:49
• There. Ok sorry not any line: any line in the plane of the moving line can be the reference line. But you dont need a reference line. Can take the line that has moved in a plane and measure angle between before motion and after motion. Sep 3, 2021 at 11:51
• @Al Brown So we cant define the angular velocity of $OP$ with respect to $OM?$ Sep 3, 2021 at 11:51
• We need an axis to have an angular position of another line, but as you point out we don't need a reference axis for defining angular velocity. Angular velocity is the derivative of angular displacement wrt to time. Angular displacement is the angle made between a line at two instants. Sep 3, 2021 at 12:12

• I don't see how this directly answers the question. I am asking if I can take the line $OM$ as a reference axis for defining the angular velocity of the other line. Thanks. Sep 3, 2021 at 13:47