Simple Nature - Light and Matter

k / Example 27.torque to the left would therefore tend to make the angular momentumvector precess in the clockwise direction as seen by thethrower. This would cause the disc to roll to the right, and thereforefollow a curved trajectory. Some specialized discs, used inthe sport of disc golf, are actually designed intentionally to showthis behavior; they’re known as “understable” discs. However, thetypical frisbee that most people play with is designed to be stable:as the disc rolls to one side, the airflow around it is altered in waythat tends to bring the disc back into level flight. Such a disc willtherefore tend to fly in a straight line, provided that it is thrownwith enough angular momentum.Finding a cross product by components example 27⊲ What is the torque produced by a force given by ˆx + 2ŷ + 3ẑ (inunits of Newtons) acting on a point whose radius vector is 4ˆx + 2ŷ(in meters)?⊲ It’s helpful to make a table of the components as shown in thefigure. The results areτ x = r y F z − F y r z =15 N·mτ y = r z F x − F z r x =−12 N·mτ z = r x F y − F x r y =3 N·mTorque and angular momentum example 28In this example, we prove explicitly the consistency of the equationsinvolving torque and angular momentum that we provedabove based purely on symmetry. Starting from the definition oftorque, we haveτ = dLdt= d ∑r i × p idt= ∑ iiddt (r i × p i ) .The derivative of a cross product can be evaluated in the sameway as the derivative of an ordinary scalar product:τ = ∑ [( ) (dri× p i + r i × dp )]idtdtiThe first term is zero for each particle, since the velocity vector isparallel to the momentum vector. The derivative appearing in thesecond term is the force acting on the particle, soτ = ∑ ir i × F i ,which is the relationship we set out to prove.284 Chapter 4 Conservation of Angular Momentum