Rotational Inertia

1
Rotational InertiaKinetic Energy
2
Linear Angular
Linear Angular
Displacement x ?
Velocity v ?
Acceleration a ?
Inertia m I
KE ½ mv2 ½ I?2
N2 F ma ? I?
Momentum P mv L I?
3
Rolling Motion
If a round object rolls without slipping,
there is a fixed relationship between the
translational and rotational speeds
4
Rolling Motion
We may also consider rolling motion to be a
combination of pure rotational and pure
translational motion
5
Rolling Motion
We may also consider rolling motion at any
given instant to be a pure rotation at rate w
about the point of contact of the rolling object.
6
A Rolling Tire
A car with tires of radius 32 cm drives on a
highway at a speed of 55 mph. (a) What is the
angular speed w of the tires? (b) What is the
linear speed vtop of the top to the tires?
7
Rotational Kinetic Energy

• Consider a mass M on the end of a string being
  spun around in a circle with radius r and angular
  frequency w
• Mass has speed v w r
• Mass has kinetic energy
• K ½ M v2
• K ½ M w2 r2
• Rotational Kinetic Energy is energy due to
  circular motion of object.

M
24
8
Rotational Inertia I

• Tells how much work is required to get object
  spinning. Just like mass tells you how much
  work is required to get object moving.
• Ktran ½ m v2 Linear Motion
• Krot ½ I w2 Rotational Motion
• I S miri2 (units kg m2)
• Note! Rotational Inertia (or Moment of Inertia)
  depends on what you are spinning about
  (basically the ri in the equation).

13
9
Inertia Rods

• Two batons have equal mass and length.
• Which will be easier to spin?
• A) Mass on ends
• B) Same
• C) Mass in center

I S m r2 Further mass is from axis of
rotation, greater moment of inertia (harder to
spin)
10
A Dumbbell
Use the definition of momentof inertia to
calculate that of adumbbell-shaped object
withtwo point masses m separatedby a distance
of 2r and rotatingabout a perpendicular axis
throughtheir center of symmetry.
11
Nose to the Grindstone
A grindstone of radius r 0.610 m is being used
to sharpen an axe. If the linear speed of the
stone is 1.50 m/s and the stones kinetic energy
is 13.0 J, what is its moment of inertia I ?
12
Moment of Inertia of a Hoop
All of the mass of a hoop is at the same
distance R from the center of rotation, so its
moment of inertia is the same as that of a point
mass rotated at the same distance.
13
Moments of Inertia
14
More Moments
15
I is Axis Dependent
16
Rotation Plus Translation
17
Rolling Objects
18
Like a Rolling Disk
A 1.20 kg disk with a radius 0f 10.0 cm rolls
without slipping. The linear speed of the disk
is v 1.41 m/s. (a) Find the translational
kinetic energy. (b) Find the rotational kinetic
energy. (c) Find the total kinetic energy.
19
Preflight Rolling Race (Hoop vs Cylinder)

• A hoop and a cylinder of equal mass roll down a
  ramp with height h. Which has greatest KE at
  bottom?
• A) Hoop B) Same C) Cylinder
• 20 50 30

20
Preflight Rolling Race (Hoop vs Cylinder)

• A hoop and a cylinder of equal mass roll down a
  ramp with height h. Which has greatest speed at
  the bottom of the ramp?
• A) Hoop B) Same C) Cylinder
• 22 30
  48

21
Rolling Down an Incline
0
0
22
Compare Heights
A ball is released from rest on a no-slip
surface, as shown. After reaching the lowest
point, it begins to rise again on a frictionless
surface. When the ball reaches its maximum height
on the frictionless surface, it is higher, lower,
or the same height as its release point?
The ball is not spinning when released, but will
be spinning when it reaches maximum height on the
other side, so less of its energy will be in the
form of gravitational potential energy.
Therefore, it will reach a lower height.
23
Spinning Wheel
A block of mass m is attached to a string that is
wrapped around the circumference of a wheel of
radius R and moment of inertia I, initially
rotating with angular velocity w that causes the
block to rise with speed v . The wheel rotates
freely about its axis and the string does not
slip. To what height h does the block rise?
24
A Bowling Ball
A bowling ball that has an 11 cm radius and a
7.2 kg mass is rolling without slipping at 2.0
m/s on a horizontal ball return. It continues to
roll without slipping up a hill to a height h
before momentarily coming to rest and then
rolling back down the hill. Model the bowling
ball as a uniform sphere and calculate h.
25
Insert Title Here

• medium