Transcript Topic 2_3_Ext A__Rotational work and kinetic energy

Topic 2.3 Extended
A – Rotational work and kinetic energy
ROTATIONAL WORK AND POWER
Recall that work W is given by a force F times a
distance s: W = Fs
If the force is tangent to a rotating object, s = r so
that
W = Fs
W = Fr
W = 
Rotational Work
Recall that power P is the amount of work done in time
t: P = W/t.
For force, this
translates to
For torque, this
translates to
W
t
W
t
P = Fs
t
P =
P = Fv
P = 
P =
P = 
t
Rotational Power
Topic 2.3 Extended
A – Rotational work and kinetic energy
ROTATIONAL KINETIC ENERGY AND THE WORK-ENERGY THEOREM
The formula for rotational work, W = , can be
manipulated as follows:
W = 
W = I
Given
substitution of  = I
2W = I(2)
Why?
2W = I(2 - 02)
2 = 02 + 2
W =
1
I2
2
W = K - K0
K =
1
I2
2
-
1
I02
2
Work-Kinetic
Why?
Energy Theorem
The Work-Energy theorem
Rotational
Kinetic Energy
Topic 2.3 Extended
A – Rotational work and kinetic energy
CONSERVATION OF ENERGY
Suppose a hoop, a disk, and a solid sphere, all of the
same radius R and mass M are released to roll from the
same place at the same time on a ramp. Which one will
reach the bottom first?
M,R,I
v0 = 0
Let’s solve this
problem using energy:
h
K – K0 + U – U0 = 0
v
U0 = K
U0 = Ktrans + Krot
Mgh =
1
Mvcm2
2
Mgh =
vcm =
+
1
Mvcm2
2
1
Iω2
2
+
2Mgh
I
M +
R2
1
2
vcm = Rω
2
v
cm
ω2 =
R2
I
vcm2
2
R
the ramp
roll race
Topic
Don’t forget: The work
done by 2.3
friction Extended
is ZERO in rolling motion,
because
slippage occurs. work and kinetic energy
A – no
Rotational
CONSERVATION OF ENERGY
Suppose a hoop, a disk, and a solid sphere, all of the
same radius R and mass M are released to roll from the
same place at the same time on a ramp. Which one will
reach the bottom first?
Now we can finally answer the question:
Ihoop = MR2
vcm =
vcm =
2Mgh
MR2
M +
R2
gh
Idisk =
1
2
MR2
vcm =
2Mgh
MR2
M +
2R2
vcm =
4
3
gh
Isphere =
2
2
MR
5
vcm =
2Mgh
2MR2
M +
5R2
vcm =
10
gh
7
Now we can finally answer the question:
The sphere has the greatest velocity – therefore it
wins the race… followed by the disk, then the hoop.
Topic 2.3 Extended
A – Rotational work and kinetic energy
SO MANY G&% D@#&!D FORMULAS...
But Brain, how are we
going to remember all
these formulas?
Easy, Pinky. Just remember
the correspondences between
linear and rotational variables!
F
m
a
v
s
↔
↔
↔
↔
↔
τ
I
α
ω
θ
Translation
between
linear and
rotational
formulas