Physics 201: Lecture 15

1
Physics 201 Lecture 15

• Rotational Kinematics
• Analogy with one-dimensional kinematics
• Kinetic energy of a rotating system
• Moment of inertia
• Discrete particles
• Continuous solid objects

2
Lecture 15 Preflight 1

• Consider the following situation You are driving
  a car with constant speed around a horizontal
  circular flat track. On a piece of paper, draw a
  Free Body Diagram (FBD) for the car. Is the car
  being accelerated?
• Yes, radially inward
• Yes, radially outward
• No
• Insufficient information

3
Rotational Variables...

• In Uniform Circular Motion ? was constant
• Now suppose ? can change as a function of time
• We define the angular acceleration

?

• Consider the case when ?is constant.
• Integrate ? and ? wrt time

?
?

• Recall for a point a distance R from the axis of
  rotation
• x ?R
• Differentiating
• v ?R
• a ?R

4
Summary (with comparison to 1-D kinematics)

• Angular Linear

And for a point at a distance R from the rotation
axis
x R????????????v ?R ??????????a ?R
5
Lecture 15 Preflight 2

• A ladybug sits at the outer edge of a
  merry-go-round that is turning and is slowing
  down. The vector expressing her angular velocity
  is
• In the x direction
• In the x direction
• In the y direction
• In the y direction
• In the z direction
• In the z direction
• zero

Right hand rule Curl around right hand fingers
along angular motion - thumb points to angular
velocity direction.
6
Lecture 15 Preflight 3

• A ladybug sits at the outer edge of a
  merry-go-round and a gentleman bug sits halfway
  between her and the axis of rotation. The
  merry-go-round makes a complete revolution once
  each second. The gentleman bugs angular speed is
• Half the ladybugs angular speed
• The same as the ladybugs angular speed
• Twice the ladybugs angular speed
• Impossible to determine

Rate of change of angle is the same! Tangential
speed v is different.
7
Center of Mass

• How do we describe a system made up of many
  parts?
• We defined the Center of Mass (average
  position).
• For a collection of N individual point-like
  particles whose masses and positions we know

Today we will define the Moment of Inertia (mass
distribution about CM)
8
Rotation Kinetic Energy

• Consider the simple rotating system shown below.
  (Assume the masses are attached to the rotation
  axis by massless rigid rods).
• The kinetic energy of this system will be the sum
  of the kinetic energy of each piece

9
Rotation Kinetic Energy...

• So but vi ?ri

which we write as
Define the moment of inertia about the rotation
axis
I has units of kg m2.
10
Rotation Kinetic Energy...

• The kinetic energy of a rotating system looks
  similar to that of a point particle Point
  Particle Rotating System

v is linear velocity m is the mass.
? is angular velocity I is the moment of
inertia about the rotation axis.
11
Moment of Inertia

• So where

• Notice that the moment of inertia I depends on
  the distribution of mass in the system.
• For a given object, the moment of inertia will
  depend on where we choose the rotation axis
  (unlike the center of mass).
• We will see that in rotational dynamics, the
  moment of inertia I appears in the same way that
  mass m does when we study linear dynamics!

12
Calculating Moment of Inertia

• We have shown that for N discrete point masses
  distributed about a fixed axis, the moment of
  inertia is

where r is the distance from the mass to the
axis of rotation.
Example Calculate the moment of inertia of four
point masses (m) on the corners of a square whose
sides have length L, about a perpendicular axis
through the center of the square
m
m
L
m
m
13
Calculating Moment of Inertia...

• Now calculate I for the same object about an axis
  through the center, parallel to the plane (as
  shown)

r
L
14
Calculating Moment of Inertia...

• For a single object, I clearly depends on the
  rotation axis!!

I 2mL2
I mL2
I 2mL2
m
m
L
m
m
15
Lecture 15 Preflight 4

• You are using a wrench and trying to loosen a
  rusty nut. Which of the arrangements shown below
  is least effective in loosing the nut?
• Arrangement 1
• Arrangement 2
• Arrangement 3
• Arrangement 4

Only the component perpendicular to the wrench
contributes.
16
Lecture 15 Preflight 5

• Suppose Earth had no atmosphere and a ball were
  fired from the top of Mt. Everest in a direction
  tangent to the ground. If the initial speed were
  high enough to cause the ball to travel in a
  circular trajectory around Earth, the ball's
  acceleration would
• Be much less than g (because the ball doesnt
  fall to the ground)
• Be approximately g
• Depends on the balls speed

Acceleration due to gravity is approximately
constant anywhere near the surface of the Earth.