Remember Newton’s 2nd Law? For linear motion : F=ma a m F For rotational motion : =I I  

Torque and Angular acceleration: Relation between t and a is analogous to relation between F and a linear Inertia Rotational Inertia r m F a Rotational Inertia: I (kgm2) Gives a measure of how ‘reluctant’ an object is to changing its angular speed. High I means harder to accelerate, ie more τ required.

The rotational inertia of an object depends on: Its mass Distribution of its mass relative to the center (If an object’s mass is distributed further from the axis of rotation, its inertia will be larger…)

Inertia Rods Two batons have equal mass and length. Which will be “easier” to spin? A) Mass on ends B) Same C) Mass in center I = S m r2 Further mass is from axis of rotation, greater moment of inertia (harder to spin)

See rotation_masses

Biscuit inertia

Moment of Inertia of a Hoop All of the mass of a hoop is at the same distance R from the center of rotation, so its moment of inertia is the same as that of a point mass rotated at the same distance.

A Dumbbell Use the definition of moment of inertia to calculate that of a dumbbell-shaped object with two point masses m separated by a distance of 2r and rotating about a perpendicular axis through their center of symmetry.

Rotational Inertia (AKA Moment of Inertia) Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Ring = m Mass of disk = m  r   r 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Rod = m Mass of Rod = m  L   L 

Moments of Inertia (no need to memorize)

Rotational Inertia Which has a greater rotational inertia? (assume the balls to be point masses and the rods have negligible mass) A. B. C. They have the same D. Can’t be determined 2m m m  L   2L 

Rotational Inertia (AKA Moment of Inertia) Which has a greater Rotational Inertia? A. B. C. They have the same D. Can’t be determined Mass of Disk = m Mass of Cylinder = m  r   r 

Moments of Inertia (no need to memorize)

Rotational Inertia Which has a greater rotational inertia? (assume the balls to be point masses and the rods have negligible mass) A. B. C. They have the same D. Can’t be determined Mass of Ring = 2m m m  2L   L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Ring = m Mass of Rod = m  2L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Rod = m Mass of Rod = m  2L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of piece of pie = m Mass of Rod = m  L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of piece of pie = m Mass of disk = m  2L   L 

(results from calculus) Recall: m = mass = resistance to translation Now: I = rotational inertia = resistance to rotation Also called “moment of inertia” How to calculate rotational inertia (I): For discrete particles: I = mr2 For solid continuous objects: see table (results from calculus)

See rotation_spin_multi

Moments of Inertia (no need to memorize)

More Moments

1 2 Notes on Rotational Inertia In general: the more mass far from the axis the harder it is to rotate the larger the rotational inertia value. Moments of inertia add. Example: Find the rotational inertia for a Solid Sphere and a Solid Cylinder rotating together m = 2.00 kg r = 15.0 cm m = 3.00 kg r = 18.0 cm 1 2

What is the mass of a basketball whose diameter is 30 cm and whose moment of inertia is 0.0075 kg·m2? R = 0.15 m I = 0.0075 kg·m2 M = ? Use I = (2/3) MR2 [hollow sphere] M = (3/2)I / R2 M = 0.5 kg How much torque is needed to angularly accelerate a 3 kg·m2 fan blade at 12 rad/s2? I = 3kg·m2 α = 12 rad/s2 τ = Iα τ = 36 N·m