I Moment of Inertia I p
I Moment of Inertia I p. 1
1. Point Mass ( I = mr2 )
[pic]
From 'F = ma' we obtain: [pic]
Finally we have a relationship between torque and angular acceleration:
[pic] and if we call mr2, the 'moment of inertia, I' then we have our
[pic] in its rotational form: [pic]
2. Lotsa point masses ( [pic]) That's a Greek sigma for 'S' (summation)!
3. Continuous Mass Distribution ( [pic] ) Just think of the integral symbol as an elongated 'S' for summation!
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−-
4. Hoop or Shell of mass, M, and radius, R ( I = MR2 )
[pic] = ? Well, since all the little masses ( dm's ) are all at a distance of r = R, we can 'slide' out R2 as a constant to get: [pic] (since the dm's add to M)
HOOP VS SOLID CYLINDER ( I = ½ MR2 )
End View Side View
5. Solid Disk or Solid Cylinder of mass, M, and radius, R, (any length!) ( I = ½ MR2 )
The plan here is to sum a lot of hoops with radius, x. (See inside ring above.)
So… [pic] = [pic] (where we will integrate from x = 0 to x = R)
[pic]
So replace 'dm' with [pic] and replace [pic] (see box info below) to get:
I =[pic]
I = [pic]=[pic]
Finally we get: [pic] whew! (I mean Q.E.D)
I Moment of Inertia I p. 2
6. Rod of mass, M, and length, L ( ICM = [pic] ) rotation axis through it's c.m.
Here the plan is to integrate from x = 0 to x = L/2 (from the center out) and double our answer to get the I for the other half of the rod.
I1/2 =[pic]
Doubling and substituting for 'dm' we get:
2 I1/2 = I = [pic]
Finally, we get: [pic] = [pic]
7. Parallel Axis Theorem – When rotating any mass about a line parallel to a line through its center of mass: I = Icm + MD2 (where D is the distance between the two lines)
x = L
c.m.
D = L/2 (distance from c.m.) x = 0
8. Rod of mass, M, and length, L, rotated about a line at one of its ends (parallel lines...)
([pic])
Method 1/ Using the Parallel Axis Theorem, I = [pic]+ MD2 = [pic]+M(L/2)2
So we get: I = [pic]
Method 2/ Integrating (we need the practice!) I =[pic]
This time we'll integrate from x = 0 to x = L (no doubling here!)
So we obtain: I = [pic]
= [pic]= [pic]
I Moment of Inertia I p. 3
9. Hollow Cylinder, mass M and with an outer (R2) and an inner (R1) radius. Same as the 'Solid Cylinder'.
Just integrate from x = R1 to x = R2
I = [pic]
=[pic]=[pic] = [pic]
= [pic] [pic] (This is just an exercise!)
[pic]
10. Rectangular Plate with dimensions 'a' and 'b' , I = [pic]
Here, we'll consider long rods with length, L = b, and integrate from x = 0 to x = a/2
and then double our result to get the other half of the plank or plate!
For each 'dx' there will be a rod with area,[pic].
Using #7 above, the parallel axis theorem:
I = [pic]+ Mx2 (where each rod is D=x away...)
each rod will have an I(or dI) of [pic] + [pic].
I1/2 = [pic] = [pic] = [pic]
I1/2 = [pic]= [pic]
I = 2I1/2 = [pic] = [pic]
I Moment of Inertia I p. 4
Time Out for Geometry! Area of a Frustum: [pic] (L is the slant height)
Step 1/ What's the lateral area of a cone?
L.A. = [pic]
Proof: Cut along a slant height and unroll.
Notice the sector of a circle with radius, [pic], and arc length, s =[pic].
The area of a sector when [pic] is measured in degrees is: Asector = [pic]
When [pic] is measured in radians is: Asector = [pic]
When [pic] is measured in radians the arc length is: [pic]
Using the figure above with s = [pic]= R[pic] =[pic][pic] (since here R =[pic])
we get: [pic][pic]
Step 2/ What's the area of a frustum of a cone? Area = [pic] (say R1 < R2)
The area is simply the big cone area minus the little cone area.
That formula is: Area = [pic] (Different?!)
To show the two formulae are equal we'll use:
(i) L = [pic] and by [pic] (ii) [pic]
Now we'll start with the first area formula and substitute
L with [pic] to get: Area =[pic], now we'll 'foil' and...
get: Area = [pic], but by (ii) above [pic] = 0 (whoa!)
so our first formula becomes: Area = [pic] =[pic] (whoa!)
and that's our second area formula!
I Moment of Inertia I p. 5
Now what was that frustum area all about? We'll actually use that [pic]
to calculate the moment of inertia of a frustum-like piece of a thin spherical shell.
11. Thin Spherical Shell with mass, M, and radius, R ( I = [pic] )
Integrate from x = 0 to x = R and double our result.
Also use: x2 + y2 = R2 [pic]
Obtain slant height 'ds' with the Pythagorean Thm.
[pic]or
[pic] (used for arc length) and
[pic] where 'r' is replaced by 'y', giving us:
I1/2 =[pic] = [pic] and [pic] [pic][pic]
[pic]= [pic]
12. Solid Sphere with mass, M, and radius, R ([pic])
At least the hard work has all been done above! Now we'll sum up (integrate) a lot of thin spherical shells from x = 0 to x = R . (Note: We don't have to double our result this time.)
[pic]
I [pic]
I [pic]
That's All Folks!
-----------------------
HOOP
I = MR2
Area Density: [pic]
Also [pic]
Linear Density: [pic]
Also [pic]
Linear Density: [pic]
Also [pic]
x = L/2
x = 0
Area Density: [pic]
Also [pic]
R1
R2
a
b
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
L
Area Density: [pic]
Also [pic]=[pic] = [pic]
dA =[pic]
where ds = L (slant height) and we replace (R1+R2) with 2r.
[pic]
Volume Density: [pic]
Also: [pic] and with x = r
dV = [pic]
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