Moment of inertia

moment of inertia (I)

• an objects resistance to rotation

• some objects harder to rotate than others

  • this depends on not only mass, but also on how that mass is distributed around the axis of rotation

• I = mr² 

  • m = mass of the object

  • r = distance from the axis of rotation

• SI units: kg × m²

• scalar quantity (no vector components)  

I = mr² 

what is the equation for the moment of inertia (I)? 

an objects resistance to rotation

what is the moment of inertia (I)?

• if the same mass of children on a merry-go-round was concentrated around the rim of the merry-go-round (spinning circle)

• if the same mass of children on a merry-go-round was concentrated at the center of the merry-go-round (spinning circle)

if there are multiple objects, the total moment of inertia is the sum

remember: if there are multiple objects, the total moment of inertia is the sum

I = m1r1² + m2r2² + m3r3² 

total moment inertia equation for 3 diff objects: 

moment of inertia (I) in rigid bodies

• step 1) split up the object into all of its particles

• step 2) add up the moments of inertia (I=mr²) of all the particles

• we are NOT doing this

moment of inertia = ½ mr²

Moment of inertia of Cylinder or disk, about center

• some of its particles are very close to the axis of rotation

• others are further away 

• when u add up all the diff moments of inertia of all of these particles it averages out to ___

moment of inertia = mr² 

Moment of inertia of Cylindrical hoop, about center 

• each particle in the hoop is the same distance (r) away from the axis of rotation

• when u add up all the diff moments of inertia of all of these particles:

  • u get a total moment of inertia of ___

moment of inertia = 2/5 mr²

Moment of inertia of a Solid sphere, about diameter (center)

moment of inertia = 2/3 mr² 

Moment of inertia of a Spherical shell, about diameter (center) 

moment of inertia = 1/12 ma²

Moment of inertia of a Plane or slab, about center

moment of inertia = 1/3 ma² 

Moment of inertia of Plane or slab, about edge (like a door that youre trying to rotate)