Parallel-Axis Theorem Pre-Lab
Axis through the center of mass
Rotational Inertia of a slab about a perpendicular axis through its center of mass
Parallel-Axis Theorem Pre-Lab
Where
M = mass of the slab
a = width of the slab
b = length of the slab
Rotational Inertia of a slab about a perpendicular axis
through its center
a b
rx y
dmrI 2
But the density of the disk is constant. Hence,
dAdm
AM
dxdyabM
dm
dxdydm
abM
Also,222 yxr
Parallel-Axis Theorem Pre-Lab
Rotational Inertia of a slab about a perpendicular axis through its center
a b
rx y
2
2
2
2
22
a
a
b
b
dxdyabM
yxI
2
2
2
2
22
a
a
b
b
dxdyyxabM
I
2
2
2
2
23
3
b
b
a
adyxy
xabM
I
2
2
2323
224224
b
b
dyayaaya
abM
I
Take the integral with respect to x with constant y
Parallel-Axis Theorem Pre-Lab
Rotational Inertia of a slab about a perpendicular axis through its center
a b
rx y
2
2
23
12
b
b
dyaya
abM
I
2
2
33
312
b
b
ayyaabM
I
24242424
3333 abbaabbaabM
I
1212
33 abbaabM
I
22
12ba
MIslab
Where
M = mass of the slab
a = width of the slab
b = length of the slab
Take the integral with respect to y with constant x
Parallel-Axis Theorem Pre-Lab
Purpose: To calculate graphical and mathematical representations of the relationship between the rotational inertia of a body of mass M around a parallel axis of rotation not through the center of mass and the distance h from the center of mass to that parallel axis.
Axis through the center of mass
Parallel Axis
h
sinrF
pulley the of radius rstring the in TFF
90
TrF
r
The clamp-on Super Pulley must be adjusted at an angle, so that the thread
runs in a line tangent to the point where it leaves the 3-step Pulley and straight down the middle of the groove on the clamp-on
Super Pulley (Figure 1.2).
Parallel-Axis Theorem Pre-Lab
sinrF
pulley the of radius rstring the in TFF
90
TrF
gF
TF
Tg FF Since the masses are accelerating
downward.
yy amF
maFF gT
maFF gT
mamgFT
agmFT
Since the string doesn’t slip, the linear acceleration of the masses is equal to the tangential acceleration
of the outside of the pulley.
pulley the of masses the of taa
rat
a
find To
Parallel-Axis Theorem Pre-Lab
TrF agmFT rat
rgrm
Newton’s Second Law I
I
rgrmI
Experimental Rotational Inertia about a parallel axis a perpendicular distance h from the center of mass
Parallel-Axis Theorem Pre-Lab