Moments of Inertia

Question 1

Solid cylinder through a circular face

Answer 1

(1/2)(M)(R^2)

Question 1

Solid cylinder through the diameter of the middle of its height

Answer 1

(1/4)(M)(R^2) + (1/12)(M)(L^2)

Question 2

Hoop through a diameter

Answer 2

(1/2)(M)(R^2)

Question 3

Hoop through its centre

Answer 3

(M)(R^2)

Question 4

Sphere through a diameter

Answer 4

(2/5)(M)(R^2)

Question 5

Thin Spherical Shell through a diameter

Answer 5

(2/3)(M)(R^2)

Question 6

Thin rod through the middle

Answer 6

(1/12)(M)(L^2)

Question 7

Thin rod about its end

Answer 7

(1/3)(M)(L^2)

Question 8

Flat Plate of dimensions a by b, through the centre of the square and the side of length a

Answer 8

(1/12)(M)(a)^2

Question 9

Parallel Axis Theorem

Answer 9

I_x = I_o + Mh^2 where I_o is the moment of inertia about the centre of mass, I_x is an axis parallel to that moment of inertia, and h is the distance between I_x and I_o

Question 10

Perpendicular axis theorem

Answer 10

I_z = I_x + I_y where the object is in the x-y plane and 2 dimensional