Moment of Inertia ( I ) The property of an object that serves as a resistance to angular motion. Chapter 7 in text 5/14/2019 Dr. Sasho MacKenzie HK 376

Moment of Inertia ( I )  Mass Moment of inertia is the angular equivalent of mass. Moment of inertia is affected by both the mass and how the mass is distributed relative to the axis of rotation. Unlike mass which remains constant regardless of the direction of motion, the moment of inertia of an object changes depending on the axis of rotation. 5/14/2019 Dr. Sasho MacKenzie HK 376

Mathematically Defining I An object can be thought to be composed of many particles of mass. Hence, Ia = moment of inertia about axis a mi = mass of particle i ri = radius from particle i to the axis of rotation Each particle provides some resistance to change in angular motion. The units are mass * squared length: kg*m2 5/14/2019 Dr. Sasho MacKenzie HK 376

Calculate I for the baseball bat using 1 kg 2 kg a 1 0.4 m 0.8 m 1 kg b 2 kg 2 0.4 m 5/14/2019 Dr. Sasho MacKenzie HK 376

Solutions 1 2 Ia = (1 kg)(0.4 m)2 + (2 kg)(0.8 m)2 Ia = 1.44 kg*m2 Ib = (1 kg)(0.4 m)2 + (2 kg)(0.4 m)2 Ib = 0.48 kg*m2 5/14/2019 Dr. Sasho MacKenzie HK 376

Radius of Gyration (k) The distance from the axis of rotation to a point where all of the mass can be concentrated to yield the same resistance to angular motion. An averaging out of the radii (ri) of all the mass particles. This allows all the mass to be represented by a single radius (k). The distribution of an object’s mass has a much greater affect on the moment of inertia than mass. 5/14/2019 Dr. Sasho MacKenzie HK 376

Moments of Inertia about 3 Axes of a Block 2 1 3 Axis 1 ra rb Axis 2 rc ra Axis 3 rc rb Applying, , the axis with the greatest radius of gyration (k) will have the greatest moment of inertia because the mass of the block doesn’t change. Rotating about Axis 1, the distribution of the block’s mass has the greatest average radius (k). 5/14/2019 Dr. Sasho MacKenzie HK 376

3 Principal Axes for any Object Maximum Moment of Inertia Axis (Imax) Axis that has the largest moment of inertia Minimum Moment of Inertia Axis (Imin) Axis that has the smallest moment of inertia Intermediate Moment of Inertia Axis (Iint) Has an intermediate moment of inertia. Determined not by its moment of inertia value, but rather because it is perpendicular to the both Imax and Imin. Note: All three axes are perpendicular to each other 5/14/2019 Dr. Sasho MacKenzie HK 376

3 Principal Axes for a Human in Anatomical Position Longitudinal Axis Frontal = Imax (Cartwheel) Frontal Axis Transverse = Iint (Back flip) Transverse Axis Longitudinal = Imin (Discus throw) 5/14/2019 Dr. Sasho MacKenzie HK 376

Moments of inertia in kgm2 19 12 4 2 1 5/14/2019 Dr. Sasho MacKenzie HK 376

5/14/2019 Dr. Sasho MacKenzie HK 376

Golf Club Heads Clubhead 1 Clubhead 2 Top View If both clubheads have the same mass, which one has the greatest moment of inertia in the plane shown? Clubhead 2 has a greater distribution of mass, therefore a greater moment of inertia and a greater resistance to angular motion. Perimeter weighted clubs are more forgiving on off center hits. 5/14/2019 Dr. Sasho MacKenzie HK 376

Cavity Back Putters 5/14/2019 Dr. Sasho MacKenzie HK 376

Cavity Back Irons 5/14/2019 Dr. Sasho MacKenzie HK 376

Muscle Back Irons 5/14/2019 Dr. Sasho MacKenzie HK 376

The Modern Tennis Racquet Modern Version Early Version Transition Twisting on off center Hits Larger moment of inertia reduces twisting from off center hits 5/14/2019 Dr. Sasho MacKenzie HK 376

5/14/2019 Dr. Sasho MacKenzie HK 376

Force and Torque F = ma and  = I Newton’s Laws also apply to angular motion. For every linear term, there is an equivalent angular term. For example, torque is the angular effect of force. Just like a net force produces an acceleration resisted by the mass, a net torque produces an angular acceleration resisted by the moment of inertia. 5/14/2019 Dr. Sasho MacKenzie HK 376

Linear Impulse and Angular Impulse Ft = mv and t = I A net force acting for a period of time produces a linear impulse that results in a change in linear momentum. Likewise, a net torque acting for a period of time produces an angular impulse that results in a change in angular momentum. Where angular momentum is the product of the moment of inertia and angular velocity. 5/14/2019 Dr. Sasho MacKenzie HK 376

Comparison of Linear and Angular Quantities Time (t) Linear Displacement (D) Angular Displacement () Linear Velocity (v) Angular Velocity () Linear Acceleration (a) Angular Acceleration () Torque () Force (F) Mass (m) Moment of Inertia (I) Linear Momentum (mv) Angular Momentum (I) Angular Impulse (t) Linear Impulse (Ft) 5/14/2019 Dr. Sasho MacKenzie HK 376