pp. 137–138. Torque. Torque is the rotational analogue of force: it is an influence that tends to change something’s rotation.
The actual formula for torque involves multiplying the vectors of displacement (from the center of rotation) r and force F in a way to obtain another vector, which is the torque τ. This vector multiplication is called the cross product, which is written with the symbol “×,” as r r × F = τ. The textbook does not describe the vector cross product.
Note: In the first paragraph on page 138, the book states that if a boy and girl sitting on opposite sides of a seesaw produce equal torques, the net torque on the seesaw is zero, and the seesaw balances. This isn’t quite true: their torques must have equal magnitudes but opposite directions, so they really need to produce opposite torques.
Recommmended workbook exercise: pp. 41–42.
pp. 150–151. Angular Momentum. Everything in this chapter so far, about forces and accelerations, is stuff we’ve already studied, only considered from a different perspective. This is different: this is new physics. Angular momentum is a physics shorthand for describing in a concise and simple manner the motion of bodies that rotate, or that otherwise change their angular position with respect to something.
What is the formula for angular momentum? This section gives two of them!
It’s not obvious that these two formulas are equivalent. To really understand what is behind the first formula, you must understand rotational inertia. We won’t cover that in this class, because I don’t believe that the added insight gained would justify the time and effort invested.
If you think closely about the first formula, you may realize that rotational velocity is a vector quantity, so that angular momentum must also be a vector quantity. The use of the symbol “×” for multiplication in the book here is unfortunate. This is not a cross product; it is just scalar multiplication of a vector.
Thinking closely about the second formula may also lead you to realize that the tangential velocity v is also a vector, so this formula should give a vector quantity as well. But it is also a bit misleading. The formula given is for the magnitude of the angular momentum; Hewitt is careful to specify that it only uses the magnitudes of the vectors. This is true as long as the velocity v really is tangential and is not carrying the object toward or away from the axis. The real formula for angular momentum involves vector cross-products: l = r × p.
pp. 151–153. Conservation of Angular Momentum. This is cool stuff. It is fairly easy to follow if all masses are point masses: as you decrease the distance of the object from its center of rotation, its velocity increases to keep the angular momentum constant. When you have objects, such as cats and people, that can re-distribute their mass, they can change their rotational inertia and thus change rotational speed while keeping angular momentum constant. It is easy to demonstrate this for yourself: stand up and spin around with your arms held at your sides, and again with your arms held away from your body. Which way lets you spin faster?
Torque and angular momentum are vectors, which are defined in a way that is new to this course. Think about how these vectors are defined, and what they mean in a physical system.
Copyright © 2008, Richard Barrans
Revised: 11 January 2010. Maintained by Richard Barrans.
URL: http://www.barransclass.com/astr1070/rguides/P1050F10rg_02-18.html