018 moment of inertia

7/27/2019 018 Moment of Inertia

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Moment of Inertia ( I )

The property of an object that serves

as a resistance to angular motion.Chapter 7 in text


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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.


3/17

Mathematically Defining I

An object can be thought to be composed of manyparticles 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 changein angular motion.

The units are mass * squared length: kg*m2

2

1iia rmI

N

i


4/17

Calculate I for the baseball bat

using

0.4 m

0.8 m

2

1iia rmI

N

i

0.4 m 0.4 m

1 kg2 kga

1

1 kgb 2 kg

2


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Solutions

2

1 iia

rmIN

i

1 Ia = (1 kg)(0.4 m)2 + (2 kg)(0.8 m)2

Ia = 1.44 kg*m2

2 Ib = (1 kg)(0.4 m)2 + (2 kg)(0.4 m)2

Ib = 0.48 kg*m2


6/17

Radius of Gyration (k)

The distance from the axis of rotation to a pointwhere all of the mass can be concentrated toyield the same resistance to angular motion.

An averaging out of the radii (ri) of all the mass

particles. This allows all the mass to berepresented by a single radius (k).

The distribution of an objects mass has a muchgreater affect on the moment of inertia thanmass.

2

aa mkI


7/17

Moments of Inertia about 3 Axes of a Block

Axis 1

r1

r2

Axis 2

r3 r1

Axis 3

r3 r2

2

13

Applying the axis with the

greatest radius of gyration (k) will have thegreatest moment of inertia because the

mass of the block doesnt change.

2

aa mkI

Rotating about Axis 1, the distribution of the

blocks mass has the greatest average radius (k).


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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 byits 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


9/17

Transverse Axis

Longitudinal

Axis

3 Principal Axes for a Human in

Anatomical Position

Frontal = Imax

(Cartwheel)

Transverse = Iint

(Back flip)

Longitudinal = Imin

(Discus throw)


10/17

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?

Cluhead 2 has a greater distribution of mass, therefore agreater moment of inertia and a greater resistance to angular

motion.

Perimeter weighted clubs are more forgiving on off center hits.


11/17

Cavity Back Putters


12/17

Cavity Back Irons


13/17

Muscle Back Irons


14/17

The Modern Tennis Racquet

Early Version Transition

Twisting on

off centerHits

Larger moment

of inertia reducestwisting from off

center hits

Modern Version


15/17

Force and Torque

F = ma and = I

Newtons 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 torqueproduces an angular acceleration resisted by

the moment of inertia.


16/17

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.


17/17

Comparison of Linear and

Angular Quantities

LINEAR ANGULAR

Force (F) Torque ()Mass (m) Moment of Inertia (I)

Linear Momentum (mv) Angular Momentum (I)

Linear Impulse (Ft) Angular Impulse (t)

Linear Velocity (v) Angular Velocity ()

Linear Acceleration (a) Angular Acceleration ()

Linear Displacement (D) Angular Displacement ()

Time (t)

018 moment of inertia

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