Last Time: Finish Ch 9 Start Ch 10 Today: Chapter 10 Monday Ch 9 examples Rota:on of a rigid body Torque and angular accelera:on Today Solving problems with torque Work and power with torque Angular momentum Conserva:on of angular momentum (We will save precession for Monday)
Example Rota:on about a moving axis You make a primi:ve yo- yo by wrapping a massless string around a solid cylinder with mass M and radius R. You hold the end of the string and the cylinder unwinds the string as it falls. Find the speed v cm aoer it has descended a distance h. Using torque find the downward accelera:on and the tension in the string. Remember: = F r " = I" a tan = "R I cyl = 1 2 MR2 K trans = 1 2 mv2 K rot = 1 2 I 2 K i +U i +W other = K f +U f
Example Rota:on about a moving axis cont. Find the speed v cm aoer it has descended a distance h. Using torque find the downward accelera:on and the tension in the string. Remember: = F r " = I" a tan = "R I cyl = 1 2 MR2 K trans = 1 2 mv2 K rot = 1 2 I 2 K i +U i +W other = K f +U f
Clicker Ques7on Strings are wrapped around the circumference of two solid disks and pulled with iden:cal forces. Disk 1 has a bigger radius, but both have the same moment of iner:a. Which disk has the biggest angular accelera:on? ω 1 ω 2 A) Disk 1 B) Disk 2 C) same F F Remember = F r " = I" a tan = "R K trans = 1 2 mv2 K rot = 1 2 I 2 I cyl = 1 2 MR2 K i +U i +W other = K f +U f
Work and Power in rota7onal mo7on We just use the same defini:on as before: Start with W by a torque = = " F tan Rd 2 " = " z d 1 " F d l W = " 2 1 d Units are (N*m)*rad = J (like you d expect)
Work and Power in rota7onal mo7on Power is s:ll the rate of change in the work P = W t P = " # = d" dt Work in terms of change in kine:c energy " 2 W = d" " 1 d" = # 2 W = I# d# = 1 I# 2 2 2 " 1 I# 2 1 # T. S:egler 1 11/5/2014 Texas A&M University 2
Clicker Ques7on You apply equal torques to two different cylinders, one of which has a moment of iner:a twice as large as the other. AOer one complete rota:on, which cylinder has the greatest kine:c energy? a) The cylinder with the larger moment of iner:a b) The cylinder with the smaller moment of iner:a c)they have the same Remember: = F r " = I" a tan = "R I rod = 1 12 MR2 = 0 +" 0 t + 1 2 #t 2 " = " 0 +#t W = " $ d W = $# P = $" 2 = 0 2 + 2"# K trans = 1 2 mv2 K rot = 1 2 I 2
Example Simple rota:onal work An airplane propeller is 2.08m in length and has a mass of 117kg. When the engine is first started it applies a constant torque τ = 1950 N*m to the propeller, which starts from rest. a) What is the angular accelera:on of the propeller if you model it as a slender rod? b) What is its angular speed aoer 5 revolu:ons? c) How much work is done by the engine during the first 5 revolu:ons? Remember: = F r " = I" a tan = "R I rod = 1 12 ML2 = 0 +" 0 t + 1 2 #t 2 " = " 0 +#t W = " $ d W = $# 2 = 0 2 + 2"# K trans = 1 2 mv2 P = $" K rot = 1 2 I 2
Angular momentum The rota:onal analog of linear momentum is angular momentum, it is a vector quan:ty that is the cross product of the posi:on vector and the linear momentum. For a rigid body rota:ng about a symmetry axis with angular speed ω L = = i L i r i " m i vi = = ( m i r 2 i ) i = I L = r p m i r i2 i L = I Angular momentum points in the same direc:on as ω, + or defined by direc:on of rota:on.
Conserva7on of angular momentum The rota:onal analog to Newton s 2 nd can be wrigen in terms of angular momentum = I = I d net dt = d An external torque is needed to change the angular momentum of a system of par:cles (just like an external force is needed to change the linear momentum of a system) If the net external torque is zero then the angular momentum is conserved = d L dt = 0 L = constant L dt
Conserva7on of angular momentum
Prelecture Angular momentum; ques:on 1 A solid disk of mass M, radius R, and ini:al angular velocity ω o falls onto a sta:onary second disk having the same radius but three :mes the mass as shown. If there are no external torques ac:ng on the system, what is the final angular velocity of the two disks? a) ω o / 2 b) ω o / 3 c) ω o / 4 = d L dt = 0 L o = L f I o o = I f f I o o = 1 $ # " 2 MR2 % & = 1 $ # o " 2 4MR2 & f %
Prelecture Angular momentum; ques:on 2 A planet orbits a star as shown below. The magnitude of the angular momentum of the center of mass of the planet as it revolves around the star is L Rev, and the magnitude of the angular momentum due to the planets rota:on around its center of mass is L Rot. What is the magnitude of the total angular momentum of the planet about an axis through its sun? a) L Rev + L rot b) L Rev - L Rot
Prelecture Angular momentum; Checkpoint 1 Consider the two collisions shown above. In both cases a solid disk of mass M, radius R, and ini:al angular velocity ω 0 is dropped onto an ini:ally sta:onary second disk having the same radius. In Case 2 the mass of the bogom disk is twice as big as in Case 1. If there are no external torques ac:ng on either system, in which case is the final kine:c energy of the system biggest? a) Case 1 1 $ case 1: # b) Case 2 " 2 MR2 % & = 1 $ # o " 2 2MR2 & f case 1: K % f = 1 1 $ # 2 2 2MR2 & 2 f = 1 " % 8 MR2 2 o c) Same f = o 2 1 $ case 2: # " 2 MR2 % & = # 1 $ case 2: K f = 1 1 $ # o " 2 3MR2 & f 2 " 2 3MR2 & 2 f = 1 % 12 MR2 2 o % f = o 3
Prelecture Angular momentum; Checkpoint 2 The angular momentum of a freely rota:ng disk around its center is L disk. You toss a heavy block horizontally onto the disk along the direc:on shown. Fric:on acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. We will choose the ini:al angular momentum of the disk to be posi:ve. What is the L total, the magnitude of the angular momentum of the disk- block system? a) L total > L disk b) L total = L disk L = r p = r m v c) L total < L disk
Prelecture Angular momentum; Checkpoint 3 The angular momentum of a freely rota:ng disk around its center is L disk. You toss a heavy block horizontally onto the disk along the direc:on shown. Fric:on acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. We will choose the ini:al angular momentum of the disk to be posi:ve. What is the magnitude of the final angular momentum of the disk- block system? a) L total > L disk b) L total = L disk L = r p = r m v c) L total < L disk
Clicker Ques7on A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kine:c energy K? A. L and K both increase. B. L stays the same; K increases. C. L increases; K stays the same. D. L and K both stay the same.
Angular momentum of a point- like par7cle Calcula:ng the angular momentum is similar to calcula:ng the torque. Both involve a cross- product which means you need a perpendicular component and the R.H.R. to get direc:on. A par:cle of mass M with a velocity p = Mvcos θ R θ p = M v v is traveling offset from an axis of rota:on Using or L = I and L = I v tan R = v tan R = (MR 2 ) vcos R = MRvcos L = r p = rp tan = RMvcos T. S:egler 11/3/2014 Texas A&M University
Angular momentum of a point- like par7cle Calcula:ng the angular momentum is similar to calcula:ng the torque. Both involve a cross- product which means you need a perpendicular component and the R.H.R. to get direc:on. A par:cle of mass M with a velocity p = M v v is traveling offset from an axis of rota:on Using L = I and = v tan R p = Mvcos θ θ R or L = r p
Example Angular momentum A door can rotate freely about a ver:cal axis through its hinges. A bullet with mass m and a speed of v strikes the center of the door in a perpendicular direc:on and embeds itself. Find the door s angular speed aoer the collision. v m d Remember: L = r p L = I tot M D I door = 1 3 Md 2
Example Conserva:on of L and E A pumpkin of mass M a is dropped from a height H above one end of a uniform bar that pivots at its center. The bar has a mass m bar and length d. At the other end of the bar is another pumpkin of mass M b. The dropped pumpkin s:cks to the bar aoer it collides with it, how high will the other one go aoer the collision? M a H M b d Remember: K trans = 1 2 mv2 K rot = 1 2 I 2 v tan = R a tan = "R I bar = 1 12 ML2 L = r p = rmv " L = I tot K trans, i + K rot, i +U i +W other = K trans, f + K rot, f +U f
Example Conserva:on of L and E cont. M a H M b d Remember: K trans = 1 2 mv2 K rot = 1 2 I 2 v tan = R a tan = "R I bar = 1 12 ML2 L = r p = rmv " L = I tot K trans, i + K rot, i +U i +W other = K trans, f + K rot, f +U f
Clicker Ques7on A student holding a ball sits on the outer edge a merry go round which is ini:ally rota:ng counterclockwise. Which way should she throw the ball so that she stops the rota:on? A) To her leo B) To her right C C) Radially outward ω A B top view: ini:al final