Physics 101: Lecture 15 Rolling Objects

1
Physics 101 Lecture 15Rolling Objects

• Todays lecture will cover Textbook Chapter
  8.5-8.7

• James Scholars
• Outline due April 14
• Final paper due April 28
• Details http//online.physics.uiuc.edu/courses/ph
  ys101/spring08/honors_101.html
• Need another volunteer!

2
Overview

• Review
• Krotation ½ I w2
• Torque Force that causes rotation
• t F r sin q
• Equilibrium
• S F 0
• S t 0
• Today N2L for rotation
• S t I a
• Energy conservation revisited

3
Linear and Angular

• Linear Angular
• Displacement x q
• Velocity v w
• Acceleration a a
• Inertia m I
• KE ½ m v2 ½ I w2
• N2L F ma t Ia
• Momentum p mv

Today
4
Rotational Form Newtons 2nd Law

• S t I a
• Torque is amount of twist provide by a force
• Signs positive CCW
• Moment of Inertia like mass. Large I means hard
  to start or stop from spinning.
• Problems Solved Like N2L
• Draw FBD
• Write N2L

5
Work Done by Torque

• Recall W F d cos q
• For a wheel
• W Ftangential d
• Ftangential r q (q in radians)
• t q
• P W/t t q/t
• t w

6
The Hammer!

• You want to balance a hammer on the tip of your
  finger, which way is easier?
• A) Head up
• B) Head down
• C) Same

23 71 6
hammer demo
Because the center of gravity would be closer to
your hand making it easier to handle and balance.
Because angular acceleration decreases with R,
the larger the R the easier it is to balance.
15
7
The Hammer!

• You want to balance a hammer on the tip of your
  finger, which way is easier?
• A) Head up
• B) Head down
• C) Same

23 71 6
t I a m g R sin(q) mR2 a
Angular acceleration decreases with R!, so large
R is easier to balance.
Torque increases with R
Inertia increases as R2
g sin(q) / R a
8
Example Falling weight pulley

• A mass m is hung by a string that is wrapped
  around a pulley of radius R attached to a heavy
  flywheel. The moment of inertia of the pulley
  flywheel is I. The string does not slip on the
  pulley.Starting at rest, how long does it take
  for the mass to fall a distance L.
• (no numbers ?? algebra)

I
?
R
T
m

• What method should we use to solve this problem?
• Conservation of Energy (including rotational)
• St Ia and then use kinematics

mg
a
L
Either would work, but since it asks for time, we
will use B.
9
Falling weight pulley...

• For the hanging mass use SF ma
• mg - T ma
• For the flywheel use St I?
• TR sin(90) I?
• Realize that a ?R
• Now solve for a using the above equations.

I
?
R
T
m
mg
a
L
10
Falling weight pulley...

• Using 1-D kinematics we can solve for the time
  required for the weight to fall a distance L

I
?
R
T
m
mg
a
L
11
Rolling on a surface ACT
bike wheel
y

• A wheel is spinning clockwise such that the speed
  of the outer rim is 2 m/s.
• What is the velocity of the top of the wheel
  relative to the ground?
• What is the velocity of the bottom of the wheel
  relative to the ground?

x
2 m/s
2 m/s
You now roll the wheel to the right at 2 m/s.
What is the velocity of the top of the wheel
relative to the ground? A) -4 m/s B) -2 m/s C)
0 m/s D) 2m/s E) 4 m/s What is the velocity
of the bottom of the wheel relative to the
ground? A) -4 m/s B) -2 m/s C) 0 m/s D) 2m/s
E) 4 m/s
12
Example Rolling down a plane

• An object with mass M, radius R, and moment of
  inertia I rolls without slipping down a plane
  inclined at an angle ? with respect to
  horizontal. What is its acceleration?
• Consider CM motion and rotation about the CM
  separately when solving this problem

I
R
M
?
13
Example Rolling down a plane

• Static friction f causes turning. It is an
  unknown, so we must solve for it.
• First consider the free body diagram of the
  object and use SF Ma
• In the x direction Mg sin ? - f Ma
• Now consider rotation about the CMand use S?
  I? realizing that
• ? Rf and a ?R

M
R
?
14
Example Rolling down a plane...
Mg sin ? - f Ma

• We have two equations
• We can combine these to eliminate f

I
A
R
M
?
15
Energy Conservation!

• Friction causes object to roll, but if it rolls
  w/o slipping friction does NO work!
• W F d cos q d is zero for point in contact
• No dissipated work, energy is conserved
• Need to include both translation and rotation
  kinetic energy.
• K ½ m v2 ½ I w2

16
Translational Rotational KE

• Consider a solid cylinder with radius R and mass
  M, rolling w/o slipping down a ramp. Determine
  the ratio of the translational to rotational KE.

Translational KT ½ M v2 Rotational
KR ½ I w2
Rotational KR ½ (½ M R2) (V/R)2
¼ M
v2
½ KT
KT / KR 2
H
17
Rolling ACT
ramp demo

• Two uniform cylinders are machined out of solid
  aluminum. One has twice the radius of the other.
• If both are placed at the top of the same ramp
  and released, which is moving faster at the
  bottom?

(a) bigger one (b) smaller one
(c) same
Ki Ui Kf Uf
does not depend on R or M!
18
Summary

• t I a
• Energy is Conserved
• Need to include translational and rotational
• P t w

19
Tension ACT
F
m3
Compare the tensions T1 and T2 as the blocks are
accelerated to the right by the force F. A) T1 lt
T2 B) T1 T2 C) T1 gt T2
T1 lt T2 since T2 T1 m2 a. It takes force
to accelerate block 2.
T2 gt T1 since RT1 RT2 I2 a. It takes force
(torque) to accelerate the pulley.