January 30, 2014
Rotational Inertia & Kinetic Energy
A rotating object can be viewed as a
combination of many smaller
particles
Rotational Inertia & Kinetic Energy
i
Resistance to Rotation: Demos
1. The kinetic energy of one particle
is:
Ki = ½ mivi2
• Rods
• Rolling
2. Ki = ½ mi(riωi)2
3. Ki= ½ miri2ωi2
4. Total
Ktotal=ΣKi=½ m1r12ω12+½ m2r22ω22+...
5. Common rotational velocity
Ktotal = ½ω2 ( m1r12 + m2r22 + ...)
6. This describes a new type of inertia
based on mass AND position from
pivot point
I = m1r12 + m2r22 + ... = Σmiri2
7. …which defines a new type of
kinetic energy
K = ½Iω2
for a rotating, solid object
Rotational Inertia & Kinetic Energy
• "I" is a moment of inertia or
rotational inertia
Rotational Inertia
• I = m1r12 + m2r22 + ... = Σmiri2
• I is unique to each shape and mass
• If mass is focused at one end, away
from pivot...
m
r
I = mr2
Moment depends on object and
the choice of pivot.
I
I
r
m
I
r
m
I
M
r
m
r
January 30, 2014
Rotational Inertia & Kinetic Energy
Rotational Inertia & Kinetic Energy
Sticks have a moment of …
I=1/12 ML2 if they pivot through
the center of mass.
• Mass resists forces
• Moments resist torques
• Torque ≈ a twisting or turning force
r
• τ=r × F
center of mass
pivot
If the stick doesn’t pivot in the
center of mass:
Iparallel = Icom + Md2
d = distance from center of mass
F
r
ϕ
F
F⊥
pivot
center of mass
d
τ=rFsin(Φ)
Rotational Inertia & Kinetic Energy
• Newton's Second Law for rotation
• Fnet = ma
Rotational Inertia & Kinetic Energy
Trebuchet Calculations
• τnet = I α
F
r
A force of 40 N is applied to the edge
of a disk which has a moment of
inertia of 3 kg m2. The radius of the
disk is 0.50 m. Calculate the…
a. Angular acceleration
b. Tangential acceleration