Rotational Inertia

Practice

practice problem 1

Four point objects of mass m are located at the corners of a square of side s as shown in the figure to the right. Determine the moment of inertia of this system if it is rotated about …
  1. the perpendicular bisector of a side
  2. a side
  3. a diagonal
  4. one corner on an axis perpendicular to the plane containing the masses

solution

  1. In the first case, each of the four masses is a distance s/2 from the axis. Thus …
     
    I = ∑ r2m = 4 ⎛
    ⎝    		 s ⎞
    ⎠    		 2 m = ms2
    2  
     
  2. In the second case, two of the masses are on the axis and contribute nothing to the moment of inertia. The other masses are each s away from the axis. Thus …
     
    I = ∑ r2m = 2ms2
     
  3. In the third case, two masses lie on the axis and two are half a diagonal away from the axis. Thus …
     
    I = ∑ r2m = 2 ⎛
    ⎝    		 s√2 ⎞
    ⎠    		 2 m = ms2
    2  
     
  4. In the fourth case, one mass lies on the axis, two masses are a distance s away, and one is a diagonal away from the axis. Thus …
     
    I = ∑ r2m = 2ms2 + m(s√2)2 = 4ms2
     

practice problem 2

Write something.

solution

Answer it.

practice problem 3

Write something.

solution

Answer it.

practice problem 4

Determine the moment of inertia for each of the following shapes. The rotational axis is the same as the axis of symmetry in all but two cases. Use M for the mass of each object.
  1. ring, hoop, cylindrical shell, thin pipe
  2. annulus, hollow cylinder, thick pipe
  3. disk, solid cylinder
  4. spherical shell
  5. hollow sphere
  6. solid sphere
  7. rod, rectangular plate (perpendicular bisector)
  8. rod, rectangular plate (axis along edge)
  9. rectangular plate, solid box (axis perpendicular to face)
  10. cube (axis perpendicular to face)
  11. cone (rotated about its central axis)
  12. cone (rotated about its vertex)

[magnify]

solution

  1. ring, hoop, cylindrical shell, thin pipe

    There isn't much of a proof here. Since all the mass is located the same distance R away from the axis of rotation, the moment of inertia is the same as that for a point mass located a distance R from the axis, namely …

     I =  ⌠
    ⌡    		 r2 dm = R2 ⌠
    ⌡    		 dm 

    which has a trivial solution …

    I = MR2

    Note how the height of the hoop is not a factor. This formula would work equally well for a long thin tube or a flat thin ring.

  2. annulus, hollow cylinder, thick pipe

    A hollow cylinder is basically a series of infinitesimally thin nested cylindrical shells all added together. The way to write this in calculus is …

     I =  ⌠
    ⌡    		 r2 dm

    The mass of each infinitesimal slice (dm) is the overall density (ρ) times the infinitesimal volume (dV) of the slice.

     I =  ⌠
    ⌡    		 r2 ρ dV

    The infinitesimal volume is the surface area of a cylindrical shell (2πrh) times its infinitesimal thickness (dr).

     I =  ⌠
    ⌡    		 r2 ρ 2πrh dr

    The last piece of the puzzle is density, which is mass divided by volume.

    I =  ⌠
    ⌡    		  r2  M  2πrh dr
    V

    The volume of a hollow cylinder is the volume of the outer cylinder minus the volume of the inner cylinder.

    V = πR22h − πR12h = π (R22 − R12) h

    Putting it altogether and integrating from the inner radius (R1) to the outer radius (R2) yields …

    R2
     I =  ⌠
    ⌡    		  r2  M  2πrh dr =  2M ⌠
    ⌡    		  r3 dr =  2M   R24 − R14
    π (R22 − R12) h R22 − R12 R22 − R12 4
    R1

    which simplifies to  …

    I =  M (R22 + R12) (R22 − R12)
    2 (R22 − R12)

    and eventually simplifies to  …

     I = 1 M (R22 + R12)
    2

    Note how height cancelled out of this equation a few steps back. This formula would work for a long, thick-walled pipe or a flat, hollowed out disk (also known as an annulus).

  3. disk, solid cylinder

    A solid cylinder is a hollow cylinder with an inner radius of zero, so this proof is similar to the previous one. Start with the definition of the moment of inertia and substitute density times volume (ρ dV) for mass (dm).

     I =  ⌠
    ⌡    		 r2 dm =  ⌠
    ⌡    		 r2 ρ dV

    The infinitesimal volume is the surface area of a cylindrical shell (2πrh) times its infinitesimal thickness (dr). The density of a uniform cylinder is its total mass (M) divided by its total volume (πR2h).

     I =  ⌠
    ⌡    		 r2 ρ dV =  ⌠
    ⌡    		  r2  M  2πrh dr
    πR2h

    Now, integrate all the infinitesimal shells from r = 0 to r = R …

    R
     I =  2M ⌠
    ⌡    		  r3 dr =  2M   R4
    R2 R2 4
    0

    and simplify …

     I = 1 MR2
    2

    Once again, height is not a factor affecting the moment of inertia of this shape. This formula would work for a long solid cylinder or a flat solid disk.

  4. spherical shell

    This is a tough proof. As always, start with the basic formula.

     I =  ⌠
    ⌡    		 r2 dm =  ⌠
    ⌡    		 r2 ρ dV

    Now the hard part. How do we slice this thing up? I recommend rings. Imagine the standard unit circle from trig class. Start on the x axis as is the usual way and walk counterclockwise across the circumference of the circle measuring and angle θ that starts at 0 radians and ends at π radians taking teeny, tiny dθ steps. (I'll use the x axis as the axis of rotation. I hope that's OK.) The radius of each ring is R sin θ, which means its circumference is 2πR sin θ. The width of one of these rings would be R dθ and its thickness would be something small. Something that will hopefully go away in the math we're about to start. Let's call it t. This gives us a volume element dV = (2πR sin θ)(R dθ)(t) and an integral …

     I =  ⌠
    ⌡    		 (R sin θ)2 ρ (2πR sin θ Rdθ t)

    We're getting closer. Replace density with mass per volume. The volume of a spherical shell would equal the surface area of the shell (4πR2) times its thickness (t).

     I =  ⌠
    ⌡    		 (R sin θ)2  m  (2πR sin θ R dθ t) =  ⌠
    ⌡    		 (R sin θ)2  M  (2πR sin θ R dθ) 
    V 4πR2 t

    Clean this beast. I beg you.

     I =  MR2 ⌠
    ⌡    		 sin3 θ dθ
    2

    Wow! What happened to all the symbols? I'm telling you this algebra stuff is magic. Oops, I forgot the limits of integration. Let's put them in.

    π
     I =  MR2 ⌠
    ⌡    		 sin3 θ dθ
    2
    0

    Hmmm, I don't quite know how to solve this one. May I suggest looking up the result in an integral table? Or maybe, perhaps, letting a machine do the work for you? If you tell this one to find the integral of (Sin[x])^3 it will return something like this expression without the constants in the front or the limits at the end …

    π
    I =  MR2   1 ⎡
    ⎣    		 cos 3θ − 9 cos θ ⎤
    ⎦
    2 12
    0

    The limits of this integral are … well … something. I feel so lazy today after finding all these moments of inertia. Let me use another online source to calculate the upper limit …

    cos(3 * pi) - (9 * cos(pi)) = 8

    and the lower limit …

    cos(3 * 0) - (9 * cos(0)) = -8

    of the quantity in the square bracket.

     I =  MR2   1 ⎡
    ⎣    		  (+8) − (−8)  ⎤
    ⎦    		  =  16   MR2
    2 12 24

    I see the final answer approaching.

     I = 2 MR2
    3

    I am now officially happy.

  5. hollow sphere

    What is a hollow sphere but a series of spherical shells piled on top of one another. Do not use the basic formula.

    do not use   I =  ⌠
    ⌡    		  r2 dm  do not use

    Start with something we just dervied a second ago — the moment of inertia of a spherical shell.

     Ispherical shell =  2  MR2 
    3

    Break the hollow sphere up into a series of infinitesimal spherical shells and integrate these infinitesimal moments.

    R2
     I = ⌠
    ⌡    		   2  r2 dm
    3
    R1

    Replace dm with ρ dV. Replace density with total mass (M) over total volume (4/3 π (R23 − R13)). Replace dV with the surface are of a sphere (4πr2) times its infinitesimal thickness (dr).

    R2 R2
     I = ⌠
    ⌡    		   2  r2 ρ dV = ⌠
    ⌡    		   2  r2  M  4πr2 dr
    3 3 4/3 π (R23 − R13)
    R1 R1

    This can be simplified to …

    R2
     I =  2M ⌠
    ⌡    		  r4 dr
    R23 − R13
    R1

    which certainly is simple to integrate.

    R2
    I =  2M   ⎡
    ⎣    		   r5   ⎤
    ⎦
    R23 − R13 5
    R1

    Put the limits in …

    I =  2M   R25 − R15
    R23 − R13 5

    and clean it up a bit.

    I = 2 M R25 − R15
    5R23 − R13

    This is as simple as I can make it.

  6. solid sphere

    You want an easy proof? What is a solid sphere but a hollow sphere with no inner radius. Start with the hollow sphere formula 

    Ihollow sphere =  2  m  R25 − R15
    5 R23 − R13

    Let R2 = R and take the limit as R1 → 0

    I =  2  m  R5
    5 R3

    Simplify and we're done.

     I = 2 MR2
    5

    You want a harder proof? A solid sphere is built like an onion from layer upon layer of thin spherical shells. Each shell has moment of inertia equal to 

     Ispherical shell =  ⌠
    ⌡    		   2  r2 dm =  ⌠
    ⌡    		   2  r2 ρ dV
    3 3

    Again, density is total mass (M) divided by total volume (4/3 πR3) and infinitesimal volume (dV) is the surface area of a spherical shell (4πr2) times its infinitesimal thickness (dr). Substitute these values and simplify …

     I =  ⌠
    ⌡    		   2  r2  M  4πr2 dr = 2M ⌠
    ⌡    		 r4 dr
    3 4/3 πR3

    Yet another simple integral …

    R R
     I = 2M  ⌠
    ⌡    		 r4 dr = 2M  ⎡
    ⎣    		 r5 ⎤
    ⎦
    5
    0 0

    and it gives us the right answer …

     I = 2 MR2
    5

    Dare I try another proof? What is a solid sphere but a stack of disks.

     Idisk =  ⌠
    ⌡    		   1  r2 dm =  ⌠
    ⌡    		   1  r2 ρ dV
    2 2

    Review your analytical geometry. The formula for a circle is …

    R2 = x2 + y2

    The disks of our sphere have radii (represented by the symbol y) that vary according to this formula.

    y2 = R2 − x2

    Again, density is total mass (M) divided by total volume (4/3 πR3), but now the infinitesimal volume (dV) is the surface area of a circular disk (πy2) times its infinitesimal thickness (dx). Substitute, simplify, …

     I =  ⌠
    ⌡    		   1  (R2 − x2)  M  π(R2 − x2) dx =  3M ⌠
    ⌡    		 (R2 − x2)2 dx
    2 4/3 πR3 8R

    and integrate. It's an ugly one. Viewer discretion is advised.

    +R +R
     I =  3M ⌠
    ⌡    		 (R2 − x2)2 dx =  3M   ⎡
    ⎣    		 x5  −  2R2x3  + R4x ⎤
    ⎦    		  =  3M   16R5
    8R3 8R3 5 3 8R3 15
    −R −R

    All of the stuff in square brackets reduces to 16/15 R5. Trust me. I've checked it several times. One last bit of simplification and we're done.

     I = 2 MR2
    5
  7. rod, rectangular plate (perpendicular bisector)

    Let M and L be the mass and length of the plate respectively. Then …

    λ =  M
    L

    is its linear density. Divide the rectangle up into thin strips that run parallel to the axis of rotation. The width of these strips, dx, times the linear density is the infinitessimal mass of each. Plop this into the moment of inertia formula and integrate from the left edge of the plate (−½L) to the right edge (+½L).

    +½L +½L
     I =  ⌠
    ⌡    		 r2 dm =  ⌠
    ⌡    		 x2 λ dx =  ⌠
    ⌡    		 x2  M  dx =  ⎡
    ⎣    		 Mx3 ⎤
    ⎦
    L 3L
    −½L −½L

    Stuff cancels, and with a minimal amount of work you end with …

     I = 1 ML2
    12
  8. rod, rectangular plate (axis along edge)

    Use the same set up as in the previous proof. Integrate from the left edge of the plate to the right edge; that is, from 0 to L.

    L L
     I =  ⌠
    ⌡    		 r2 dm =  ⌠
    ⌡    		 x2 λ dx =  ⌠
    ⌡    		 x2  M  dx =  ⎡
    ⎣    		 Mx3 ⎤
    ⎦
    L 3L
    0 0

    Easy peasy, here's the answer …

     I =  1  ML2
    3

    You could also try using the parallel axis theorem.

    I = Icm + mL2

    The moment of inertia about the center of mass was determined in the previous proof. Just add on a little correction and we're done.

     I =  1  ML2 + m(½L)2 =  ⎛
    ⎝    		 1  +  1 ⎞
    ⎠    		 ML2
    12 12 4

    This simplifies to the answer …

     I = 1 ML2
    3
  9. rectangular plate, solid box (axis perpendicular to face)

    Start with the basic formula, but make one sup change. We'll replace the volume density (ρ = M/V) with surface density (σ = M/A) since the thickness of the plate doesn't contribute anything to the moment of inertia about this axis.

     I =  ⌠
    ⌡    		 r2 dm =  ⌠
    ⌡    		 r2 σ dA =  ⌠
    ⌡    		 r2  M  dA
    A

    Now let's dice the plate up into rectangular strips dx long by dy wide and any old height whatsoever.

     I =  ⌠⌠
    ⌡⌡    		 r2  M  dx dy 
    LW

    Since I like food preparation analogies, imagine we're slicing the plate up into infinitessimal french fries. Each french fry has coordinates (x, y) relative to the axis, which means their distances from the axis can be found using pythagoras' theorem.

    r2 = x2 + y2

    Now, put everything altogether and set the limits of integration. For a plate of length L and width W, the appropriate limits would be ±½L and ±½W.

    +½W +½L
     I =  ⌠
    ⌡    		 ⌠
    ⌡    		 (x2 + y2)  M  dx dy 
    LW
    −½W −½L

    Integrate first over x while y stays constant …

    +½W +½L +½W
     I =  M ⌠
    ⌡    		 ⎡
    ⎢
    ⎣    		 x3  + xy2 ⎤
    ⎥
    ⎦    		  dy =  M ⌠
    ⌡    		 ⎛
    ⎝    		 L3  + Ly2 ⎞
    ⎠    		  dy
    LW 3 LW 12
    −½W −½L −½W

    then integrate over y …

    +½W
    I =  M ⎡
    ⎢
    ⎣    		 L3y  +  Ly3 ⎤
    ⎥
    ⎦    		  =  M ⎛
    ⎝    		 L3W  +  LW3 ⎞
    ⎠
    LW 12 3 LW 12 12
    −½W

    and then simplify.

     I = 1 M (L2 + W2)
    12
  10. cube (axis perpendicular to face)

    A cube is a plate with length and width equal. Start with the results of the previous proof …

     I =  1  M (L2 + W2)
    12

    and set L = W = S.

     I =  1  M (S2 + S2)
    12

    Voila, c'est finis!

     I = 1 MS2
    6
  11. cone (rotated about its central axis)

    A cone is a series of infinitesimally thin disks of varying radius. If we add up the moments of inertia of all these very, very thin slices we'll get the moment of inertia of the whole cone. Adding up a lot of very small pieces to create a whole is called integration.

      I =  ⌠
    ⌡    		  Islice dx =  ⌠
    ⌡    		 1  mslicer2 dx
    2

    Replace mass with density times volume and proceed.

     I =  ⌠
    ⌡    		 1  ρA r2 dx =  ⌠
    ⌡    		 1  ρ (πr2) r2 dx =  ⌠
    ⌡    		 1  πρr4 dx
    2 2 2

    The "trick" to solving this part of the problem is determining how the radius of the slices vary from the vertex (x = 0) to the base (x = H). We need a function that begins at 0, ends at R, and increases linearly. May I suggest …

    r =  R  x
    H

    Make the switch and integrate.

    H H
    I = ⌠
    ⌡    		 1  πρ  ⎛
    ⎝    		 R  x  ⎞
    ⎠    		 4  dx =  πρR4 ⎡
    ⎣    		 x5 ⎤
    ⎦    		  =  πρR4H
    2 H 2H4 5 10
    0 0

    Recall that the volume of a cone is …

     V =  1  πR2H
    3

    Do you see the volume hidden inside the moment of inertia? It's in there.

     I =  πρR4H  = ρ  ⎛
    ⎝    		 1  πR2 H ⎞
    ⎠    		 ⎛
    ⎝    		 3  R2 ⎞
    ⎠    		  = ρV  3  R2
    10 3 10 10

    Density times volume is mass. Therefore …

     I = 3 MR2
    10
  12. cone (rotated about its vertex)

    Here's the answer …

     I = 3 M ⎛
    ⎝    		 1 R2 + H2 ⎞
    ⎠
    54

    I'll leave it to the bold reader to work out the solution. Gotta leave something for homework.

The Physics Hypertextbook

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