Download Lectures 34

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 34, 35,
36
Hw: Chapter 14 problems and exercises
Torque and Angular Momentum
Conservation
of
Angular
Momentum
  
  r F
  
Lrp

dLtot 
  ext
dt
   

dr 
2
L  r  p  r  m[ ir  ri ]  mr  (rhr )
dt
“Walk the Plank”
x1
Mg
x2
N = mg
Mgx1 = mgx 2
M
x2 =
x1
m
Map of Texas
t = r^× F = 0
Because r = 0
^
F = mg
t = r^× F ¹ 0
F = mg
t = r^× F = 0
Because r = 0
^
F = mg
Brady, TX
Skyhooks
The skyhook alone won ’ t
balance on your finger, but when
you put a belt on it, it does! This
is all because adding the belt
which curves under as it hangs
actually moves the center of
mass right under your finger!
Motion along the straight line: momentum


P  mV
Vector directed along velocity
The larger the momentum, the
larger force you need to apply
in order to change its
magnitude or direction
Rotational motion: angular momentum
Moment of inertia


L  I
Vector
along Angular
axis of rotation velocity


P  mV
I  mr


L  I
2
Conservation of angular momentum:
when radius decreases, rotation
velocity goes up
L(before) = L(after)
I bb  I aa
I  mr
2
Torque and Angular Momentum
Conservation
of
Angular
Momentum
  
  r F
  
Lrp

dLtot 
  ext
dt
   

dr 
2
L  r  p  r  m[ ir  ri ]  mr  (rhr )
dt
Pr. 1
A bullet of mass m is fired in the negative x
direction with velocity of magnitude V0,
starting at x = x0, y=b. (y remains constant)
What is its angular momentum, with respect to
the origin, as a function of x? Neglect gravity.
Pr. 2
A ball of mass m is dropped from rest from the
point x = B, y=H. Find the torque produced by
gravity about the origin as a function of time.
Problem 6 p.267
Consider a massless teeter-totter of length R,
pivoted about its center. One kid of mass m2 sits
on the right end and another
of mass m1 sits on

the left end. What is dL as a function of θ, the
angle the board makesdtwith horizontal?
Two men of equal mass are skating in a circle on a
perfectly frictionless pond. They are each holding
onto a rope of length R. What happens to the
magnitude of momentum p of each man if they
both pull on the rope, “hand over hand”, and
shorten the distance between them to R/2. (Assume
the men again move in a circle and the magnitude
of their momenta are equal).
An ant of mass m is standing at the center of a massless
rod of length l. The rod is pivoted at one end so that it
can rotate in a horizontal plane. The ant and the rod
are given an initial angular velocity 0. If the ant
crawls out towards the end of the rod so that his
distance from the pivot is given by l  bt 2 , find the
2
angular velocity of the rod as a function of time,
angular momentum, force exerted on the bug by the
rod, torque about the origin.
Moment of Inertia
For symmetrical objects rotating about their axis of
symmetry:

2
L  I (rhr ); I   mi ri
i
A man stands on a platform which is free to
rotate on frictionless bearings. He has his arms
extended with a huge mass m in each hand. If he
is set into rotation with angular velocity 0 and
then drops his hands to his sides, what happens
to his angular velocity? (Assume that the man’s
mass is negligible and that his arms have length
R when extended and are R/4 from the center of
his body when at his sides.)
Newton’s law of gravitation
Orbital motion
Conservation of Angular Momentum
A block of mass M is cemented to a circular platform
at a distance b from its center. The platform can
rotate, without friction, about a vertical axle through
its center with a moment of inertia, Ip. If a bullet of
mass m, moving horizontally with velocity of
magnitude vB as shown, strikes and imbeds itself in
the block, find the angular velocity of the platform
after the collision.
b
vB
axle
top view
Have a great day!
Hw: Chapter 14 problems and
exercises