A cylinder whose radius is 0.50 m and whose rotational inertia is 10.0 kgm^2
is rotating around a fixed horizontal axis through its center. A constant force
of 10.0 N is applied at the rim and is always tangent to it. The work done by
on the disk as it accelerates and rotates turns through
g four complete
p
revolutions is ____ J.
Physics 106 Week 4
Work, Energy, Rolling,
SJ 7th Ed.: Chap 10.8 to 9, Chap 11.1
•
•
•
•
Work and rotational kinetic energy
Rolling
Kinetic energy of rolling
Examples of Second Law applied to rolling
Today
2
1
Goal for today
Understanding rolling motion
For motion with translation and rotation about center of mass
ω
vcm
Example: Rolling
K total = K rot + K cm
K rot =
1
I cmω 2
2
Emech = K tot + U
K cm =
1
2
Mvcm
2
U gravity = Mghcm
2
A wheel rolling without slipping on a table
•
The green line above is the path of the mass
center of a wheel.
•
The red curve shows the path (called a cycloid)
swept out by a point on the rim of the wheel.
wheel
•
When there is no slipping, there are simple
relationships between the translational
(mass center) and rotational motion.
s = Rθ
v cm = ωR
ω
vcm
a cm = αR
First point of view about rolling motion
Rolling = pure rotation around CM + pure translation of CM
a) Pure rotation
b) Pure translation
c) Rolling motion
3
Second point of view about rolling motion
Rolling = pure rotation about contact point P
• Complementary views – a snapshot in time
• Contact point “P” is constantly changing
v A = ωP 2R cos(ϕ)
A
v tan g = 2ωPR
v cm = ωPR
φ
R
ωP
φ
P
vcm = ωP R = ωcm R
v tan g = 0
∴ ωP = ωcm
∴α p = α cm
Angular velocity and acceleration are the same about contact
point “P” or about CM.
A bowling ball (a solid sphere with I = (2/5) MR2 ) is rolling without slipping on
flat, level ground with a mass center speed vcm.
Find the ratio of its translational (mass center) kinetic energy to its rotational
kinetic energy around an axis through the ball’s mass center
4
iClicker Q:
A solid sphere and a spherical shell of the same radius r and
same mass M roll to the bottom of a ramp without slipping from
the same height h.
True or false? : “The two have the same speed at the bottom.”
A) True
B) False.
Hint:
ƒ
Rotation accelerates if there is friction
between the sphere and the ramp
‰ Friction force produces the net torque
and angular acceleration.
acceleration
‰ There is no mechanical energy change
because the contact point is always at
rest relative to the surface, so no work is
done against friction
I_(cm, spherical shell) = (2/3) MR^2
I_(cm, solid sphere)=(2/5) MR^2
Example: Use energy conservation to find the speed of the
bowling ball as it rolls w/o slipping to the bottom of the ramp
Given: h=2m
ƒ
Formula: For a solid sphere
2
Icm = 5 MR 2
Hint:
ƒ
Rotation accelerates if there is friction
between the sphere and the ramp
‰ Friction force produces the net torque
and angular acceleration.
‰ There is no mechanical energy change
because the contact point is always at
rest relative to the surface, so no work is
done against friction
5
Example: A uniform circular disk of radius r and mass M is pulled by
constant horizontal force F applied to the center of mass, and is
rolling without slipping. M=2 kg, r=0.5 m, I_(cm,disk)=(1/2)MR^2,
F=5N.
a) Find the angular acceleration.
b) Find minimum coefficient of static friction that makes such rolling
without slipping possible.
r
cm
CCW = +
F
P
fs
6