11
Rolling, Chap torque, and angular momentum
Note this is an outline of the lecture, print this and take notes on it in lecture. It is not the complete lecture. You should attend lecture to get the complete notes
Images taken from Halliday/Resnick/Walker, Fundamentals of Physics, 7th Edition, John Wiley and Sons Inc., by permission.
7. Use conservation of energy to find the speed at which ball leaves the roof. Remember to take account of the translational and rotational KE. Treat ball as a projectile after it has left the roof. 8. The com of the ball undergoes circular motion in the loop. The radius of this motion is R r. The ball must have a certain minimum v to make it around p pp y p the loop. Apply Newton's 2nd law to the ball at the top of the loop to find this v. Now apply conservation of energy. 43. This is a completely inelastic collision between two rotating objects. Since there are no external torques, angular momentum is conserved (Li = Lf ) 47. b) [angular momentum of two skaters before they grab pole] = [angular momentum after they grab pole]. d) When the two skaters pull themselves inward, their angular momentum does not change (Why?)
Hints - HW
Rolling = Translation + rotation
Rolling and speed
Rolling motion = translational + rotational motion s = R ds d = R dt dt vcom = R
+
=
KE of rolling
Static friction and rolling
In order for the wheel to roll (rather than slide) there must be a friction f present. Is it kinetic or static friction? What is the direction of f ? What is the Newton's 3rd law reaction force?
1 1 2 K = I com 2 + Mvcom 2 2 rotational translational
1
SP11-1
Approximate each wheel on the car Thrust SSC as a disk of uniform thickness and mass M = 170 kg, and assume smooth rolling. When the car's speed was 1233 km/h, what was the kinetic energy of each wheel?
Object rolling down slope
K + U = 0 1 1 2 I com 2 + Mvcom 2 2 1 1 2 2 I com + Mvcom - 0 + (0 - mgh ) = 0 2 2 Kf =
2 2 I com vcom / R 2 + Mvcom = 2mgh 2 vcom =
h
2mgh I com / R 2 + M
SP11-2
A uniform ball, of mass M = 6.00 kg and radius R, rolls smoothly from rest down a ramp at angle = 30.0 (Figure 11-8). (a) The ball descends a vertical height h = 1.20 m to reach the bottom of the ramp. What is its speed at the bottom? (b) What are the magnitude and direction of the frictional force on the ball as it rolls down the ramp?
Race Demo and calculation
Race between Same R, M hoop solid disc (so what's different here?)
Race between solid discs aluminum wood Same R, Different M
Rolling down a ramp
Friction force fs point up slope Choose x up the slope
Newton' s 2nd law (translation) f s - Mg sin = Macom , x
Rolling down a ramp-NO
x
Friction force fs point up slope Choose x up the slope
Newton' s 2nd law (translation) f s - Mg sin = Macom , x Two unknowns f s and acom , x
Newton' s 2nd law (rotational) Rf s = I com
Newton' s 2nd law (rotational) Rf s = I com
Magnitude of = acom , x / R Rf s = I com acom , x / R f s = - I com acom , x / R 2 - I com acom , x / R 2 - Mg sin = Macom , x acom , x = - g sin 1 + I com / MR 2
2
Yo-Yo
Newton' s 2nd law (translation) T - Mg = Macom , x (chose + x direction to be down)
Torque
= rF
Magnitude = rF sin
x z
z y
z
Newton' s 2nd law (rotational) R0T = I com
y y
x x
Angular momentum
l = r p = m(r v ) magnitude
l =rp
Newton's 2nd law in angular form
l = rmv
Fnet sin =
net
l
dp dt dl = dt
translational angular
SP11-5
In Figure 11-14, a penguin of mass m falls from rest at point A, a horizontal distance D from the origin O of an xyz coordinate system. (The positive direction of the z axis is directly outward from the plane of the figure.) (a) What is the angular momentum of the falling penguin about O? (b) About the origin O, what is the torque on the penguin due to the gravitational force ?
Angular momentum of a system of particles
L = l1 + l2 + l3 + l4 + ....
1 = net
dl dl1 dl , 2 = 2 , 3 = 3 dt dt dt dL = System of particles dt
3
Angular momentum of a rigid body L = I
To prove this : angular momentum of small peice of mass mi is li = mvr sin 900 = mi vri Now sum these li ' s
SP11-6
George Washington Gale Ferris, Jr., a civil engineering graduate from Rensselaer Polytechnic Institute, built the original Ferris wheel (Figure 11-16) for the 1893 World's Columbian Exposition in Chicago. The wheel, an astounding engineering construction at the time, carried 36 wooden cars, each holding as many as 60 passengers, around a circle of radius R = 38 m. The mass of each car was about g 1.1 104 kg. The mass of the wheel's structure was about 6.0 105 kg, which was mostly in the circular grid from which the cars were suspended. The wheel made a complete rotation at an angular speed F in about 2 min. (a) Estimate the magnitude L of the angular momentum of the wheel and its passengers while the wheel rotated at F. (b) Assume that the fully loaded wheel is rotated from rest to F in a time period t1 = 5.0 s. What is the magnitude tavg of the average net external torque acting on it during t1?
Conservation of angular momentum
net =
dL dt If net = 0, L = constant
In this example there are no external torques on person
Diver
or Li = L f
As diver pulls arms and legs in they spin faster why? As they extend arms and legs they spin slower why?
Spacecraft
If flywheel starts to rotate one way, the spacecraft rotates in the opposite direction why?
White dwarfs and Neutron stars
As star collapses it spins more rapidly (e.g. Prob. 46)
Spinning slowly
Spinning more rapidly Why?
4
sitting on a stool that can rotate freely SP11-7 A student The student, initially at rest, is holding a about a vertical axis. bicycle wheel whose rim is loaded with lead and whose rotational inertia Iwh about its central axis is 1.2 kg m2. The wheel is rotating at an angular speed wwh of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum of the wheel points vertically upward. The student now inverts the wheel (Figure 11-20b) so that, as seen from overhead, it is rotating clockwise. Its angular momentum is now . The inversion results in the student, the stool, and the wheel's center rotating together as a composite rigid body about the stool's rotation axis, with rotational inertia Ib = 6.8 kg m2. (The fact that the wheel is also rotating about its center does not affect the mass distribution of this composite body; thus, Ib has the same value whether or not the wheel rotates.) With what angular speed wb and in what direction does the composite body rotate after the inversion of the wheel?
SP11-7
Initially the person is not rotating. When they turn the spinning wheel upside down they start rotating. Why?
5