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Opus in profectus

Rotational Inertia

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Problems

practice

  1. Cartoon representation of the problem
    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
  2. Write something.
  3. Write something.
  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)

    Graphical representation of all 12 parts of this problem

conceptual

  1. Spacecraft from the US landed on Mars in 1976 and 1997. By communicating with spacecraft on the surface, NASA scientists were able to determine the orientation of Mars in space. Changes in the orientation between 1976 and 1997 were then used to determine the moment of inertia of Mars. Why would anyone care about the moment of inertia of a planet? What good is it to know this quantity?

algebraic

  1. For many applications, it's better to have the moment of inertia written in terms of the density of the material it's made out of than the mass of the finished object. Do this for each of the following shapes…
    objectmass-momentdensity-moment
    disk, solid cylinder
    I = 1 MR2
    2
     
    annulus, hollow cylinder, thick pipe
    I = 1 M (R22 + R12)
    2
     
    solid sphere
    I = 2 MR2
    5
     
    hollow sphere
    I = 2 M R25 − R15
    5R23 − R13
     
    rectangular plate, solid box (axis perpendicular to face)
    I = 1 M (L2 + W2)
    12
     
    cube (axis perpendicular to face)
    I = 1 MS2
    6
     
    cone (rotated about its central axis)
    I = 3 MR2
    10
     
    cone (rotated about its vertex)
    I = 3 M 
    1 R2 + H2 
    54
  2. Derive the moment of inertia for each of the following flat geometric shapes when they are rotated about a diameter instead of the axis of symmetry using the perpendicular axis theorem.
     
    Iz = Ix + Iy
     
    shape axis of symmetry diameter
    ring or hoop
    I = 1 MR2
    2
     
    annulus
    I = 1 M (R22 + R12)
    4
     
    disk
    I = 1 MR2
    4
     

statistical

  1. earth.txt
    The data in this text file gives the density and gravitational field strength of the Earth at various depths below the surface. Using data analysis software (preferably something that can do numerical integration) determine the moment of inertia of the Earth.
  2. Show that the moment of inertia of a thin-walled, holow cubical box with edges of length s and total mass M is given by…
     
    I =  5  Ms2
    18
     
    …when it is rotated about an axis through its geometric center and perpendicular to one face.

calculus

  1. Derive the moment of inertia for each of the following flat geometric shapes when they are rotated about a diameter instead of the axis of symmetry using the integral equation.
     
    I = 
     r2 dm
     
    shape axis of symmetry diameter
    ring or hoop
    I = 1 MR2
    2
     
    annulus
    I = 1 M (R22 + R12)
    4
     
    disk
    I = 1 MR2
    4
     
  2. Geometric figure
    Show that the moment of inertia of a cone rotated about its vertex is given by…
     
    I = 3 M 
    1 R2 + H2 
    54
     
    Where R is the radius of the base and H is the height.
  3. Geometric figure
    Show that the moment of inertia of a cylindrical shell or thin pipe of radius R and length L is given by…
     
    I = 1 MR2 + 1 ML2
    22
     
    …when it is rotated about a diameter through its center of mass.
  4. Show that the moment of inertia of a ring torus (a doughnut) with major radius R (the mean radius of the ring) and minor radius r (the radius of a cross section) is given by…
     
    I =  MR2 +  3  Mr2
    4
     
    …when it is rotated about its axis like a wheel and by…
     
    I =  1  MR2 +  5  Mr2
    2 8
     
    …when it is rotated about a diameter. (Assume that R > r.)