Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 24, 25
Hw: Chapter 15 problems and exercises
Suppose there were an axle at the origin with a rigid,
but massless rod attached to it with bearings so that
the rod could freely rotate. At the end of the rod, of
length b, there is a block of mass M as shown below:
v0, m
rod
axle
b
x0
A bullet is fired at the block. If the bullet strikes the
block and sticks, what will be the angular velocity of
the block about the axle? Neglect gravity.
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
Ex. 4
A platform can rotate, without friction, about a
vertical axle through its center with a moment of
inertia, Ip. A small bug of mass m is placed on the
platform at a distance b from the center, and the
system is set spinning with angular velocity 0
(Clockwise as viewed from above).
a) What is the total angular momentum of the
system with the bug at rest on the platform?
b) What is the total angular momentum if he
runs in the opposite direction to the platform’s
rotation?
c) Is it possible for a little bug to stop the big
platform from rotating?
Two disks are spinning on a frictionless axle. The one on the
left has mass M, radius RL, and moment of inertia about the
axle IL. It is spinning with L in the direction shown. It is
moving to the right with velocity of magnitude v0. The one
on the right has mass M, radius RR, and moment of inertia
about the axle IR. It is spinning with R in the direction
shown. It is moving to the left with velocity of magnitude v0.
When the two disks collide they stick together. What is the
velocity and the angular velocity of the combined system
after the collision?
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.)
What is the moment of inertia of a disk of
thickness h, radius R and total mass M about an
axis through its center?
For symmetrical objects rotating about their axis of
symmetry:

2
L  I (rhr ); I   mi ri
i
Second Law:

 ext  I (rhr )
1
2
2
KE  m(Vr  V )
2
The rope is assumed not to slip as the pulley
turns. Given m1, m2, R, and I find the
acceleration of mass m1.
I
R
m2
m1
Rotational Kinetic Energy
1 2
KE  I
2
1 2
1
2
KE  mv cm  I cm
2
2
A rigid body in motion about a
moving axis
Motion of a rigid body:
combination of translation
motion of the center of mass
and rotation about an axis
through the center of mass.
1
1
2
2
K  Mvcm  I cm
2
2
A primitive yo-yo is made by wrapping a string
several times around a solid cylinder with mass
M and radius R. You hold the end of the string
stationary while releasing the cylinder with no
initial motion. The string unwinds but does not
slip or stretch as the cylinder drops and rotates.
Use energy considerations to find the speed vcm of
the center of mass of the solid cylinder after it
has dropped a distance h.
Consider the speed of a yo-yo toy
The yo-yo (again)
Gyroscopic precession
• The precession of
a gyroscope
shows up in many
“common”
situations.
A rotating flywheel
Ex. 7
Find the minimum value of the coefficient of
friction in order for a cylinder to roll without
slipping down an inclined plane of angle θ.
Conservation of Momentum
 
dp
 Fext
dt
If

Fext  0,
p x  Const


dp
 0, p  Const
dt
p y  Const
p x (before)  p x (after)
p y (before)  p y (after)
If the collision is perfectly elastic, the kinetic
energy is conserved!
Circular Motion
Fr  mar ; F  ma
2
d r
dr
2
ar  2  r ; a  2   r
dt
dt


r  r ir
dr
d
Vr  ; V  r
 r
dt
dt

i
y

ir

r
x
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
a  R
For symmetrical objects rotating about their axis of
symmetry:

2
L  I (rhr ); I   mi ri
i
Second Law:

 ext  I (rhr )
1
2
2
KE  m(Vr  V )
2
Have a great day!
Hw: Chapter 15 problems and
exercises