Definition Of Angular Momentum-Classical Physics-Handouts-Classical Physics-Handouts, Lecture notes of Classical Physics

Download Definition Of Angular Momentum-Classical Physics-Handouts-Classical Physics-Handouts and more Lecture notes Classical Physics in PDF only on Docsity! PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 35 Summary of Lecture 13 – ANGULAR MOMENTUM ( ) ( ) 1. Recall the definition of angular momentum: . The magnitude can be written in several different but equivalent ways, (a) sin (b) sin (c) sin L r p L r p L r p r p L r p r p θ θ θ ⊥ ⊥ = × = = = = = 2. Let us use this definition to calculate the angular momentum of a projectile thrown from the ground at an angle . Obviously, initial angular momentum is zero (why?).θ ( ) ( ) 20 0 0 We know what the projectile's coordinates will be at time after launch, 1 v cos , v sin 2 as well as the velocity components, v v cx t x t y t gtθ θ= = − = ( ) ( ) ( ) 0 2 2 2 0 0 0 os , v v sin . ˆˆ ˆ ˆ ˆ Hence, v v v v v 1 ˆ ˆ v cos v cos v cos . 2 2 ˆ ˆ ˆ In the above, is a unit vector pe y x y x y x gt L r p xi y j i j m m x y k mm gt gt k gt k k i j θ θ θ θ θ = − = × = + × + = − ⎛ ⎞= − = −⎜ ⎟ ⎝ ⎠ = × 2 rpendicular to the paper. You can see here that the angular momentum increases as . 2. Momentum changes because a force makes it change. What makes angular momentum change? Answer: torque. H t ere is the definition again: . Now let us establish a very important relation between torque and rate of change of L. Begin: . At a slightly later time, r F L r p L L r τ = × = × + Δ = + Δ( ) ( ) By subtracting, . r p p r p r p r p r p L r p r p p rL r p r p r p t t t t × + Δ = × + ×Δ + Δ × + Δ ×Δ Δ ×Δ + Δ × Δ Δ Δ = ×Δ + Δ × = = × + × Δ Δ Δ Δ x y 0 v v xv yv O θ docsity.com PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 36 O z L O φΔ LΔ z sinL θ θ ( ) 0 Now divide by the time difference and then take limit as 0 : lim But is v and v ! Also, v v v v t t L dL dL d p d rr p t dt dt dt dt d r drp m p m m dt dt Δ → Δ → Δ = ∴ = × + × Δ = × = × = × ∵ 0. So we arrive at . Now use Newton's Law, . Hence we get the fundamental equation , or . So, just as a particle's momentum changes with time bec dL d p d pr F dt dt dt dL dLr F dt dt τ = = × = = × = ause of a force, a particle's momentum changes with time because of a torque. 3. As you saw in the lecture, the spinning top is an excelllent application of . angular dL dt τ= Start from where sin . But is perpendicular to and so it cannot change the magnitude of . Only the direction changes. Since , you can see from the r F F mg Mgr L L L t τ τ θ τ τ = × = ∴ = Δ = Δ diagram that . So the precession speed sin sin sin is: . As the top slows down due to friction and sin sin decreases, the top precesses faster and faster. 4 P P L t L L Mgr Mgr L t L L L τφ ω θ θ φ τ θω θ θ Δ Δ Δ = = Δ = = = = Δ . Now consider the case of many particles. Choose any origin with particles moving with respect to it. We want to write down the total angular momentum, L L= 1 2 1 1 2 1 1 Since , it follows that . Thus the time rate of change of the total N N n n N N n n N n n n n L L L dL dL dL dL dL dt dt dt dt dt d L d L dt dt τ τ = = = + + ⋅ ⋅ ⋅ + = = + + ⋅ ⋅ ⋅ + = = = ∑ ∑ ∑ angular momentum of a system of particles equals the net torque acting on the system. I showed earlier that internal forces cancel. So also do internal torques, as we shall see. docsity.com