Parallel-Axis Theorem Pre-Lab

Axis through the center of mass

Rotational Inertia of a slab about a perpendicular axis through its center of mass

Parallel-Axis Theorem Pre-Lab

Where

M = mass of the slab

a = width of the slab

b = length of the slab

Rotational Inertia of a slab about a perpendicular axis

through its center

a b

rx y

dmrI 2

But the density of the disk is constant. Hence,

dAdm

AM

dxdyabM

dm

dxdydm

abM

Also,222 yxr

Parallel-Axis Theorem Pre-Lab

Rotational Inertia of a slab about a perpendicular axis through its center

a b

rx y

2

2

2

2

22

a

a

b

b

dxdyabM

yxI

2

2

2

2

22

a

a

b

b

dxdyyxabM

I

2

2

2

2

23

3

b

b

a

adyxy

xabM

I

2

2

2323

224224

b

b

dyayaaya

abM

I

Take the integral with respect to x with constant y

Parallel-Axis Theorem Pre-Lab

Rotational Inertia of a slab about a perpendicular axis through its center

a b

rx y

2

2

23

12

b

b

dyaya

abM

I

2

2

33

312

b

b

ayyaabM

I

24242424

3333 abbaabbaabM

I

1212

33 abbaabM

I

22

12ba

MIslab

Where

M = mass of the slab

a = width of the slab

b = length of the slab

Take the integral with respect to y with constant x

Parallel-Axis Theorem Pre-Lab

Purpose: To calculate graphical and mathematical representations of the relationship between the rotational inertia of a body of mass M around a parallel axis of rotation not through the center of mass and the distance h from the center of mass to that parallel axis.

Axis through the center of mass

Parallel Axis

h

sinrF

pulley the of radius rstring the in TFF

90

TrF

r

The clamp-on Super Pulley must be adjusted at an angle, so that the thread

runs in a line tangent to the point where it leaves the 3-step Pulley and straight down the middle of the groove on the clamp-on

Super Pulley (Figure 1.2).

Parallel-Axis Theorem Pre-Lab

sinrF

pulley the of radius rstring the in TFF

90

TrF

gF

TF

Tg FF Since the masses are accelerating

downward.

yy amF

maFF gT

maFF gT

mamgFT

agmFT

Since the string doesn’t slip, the linear acceleration of the masses is equal to the tangential acceleration

of the outside of the pulley.

pulley the of masses the of taa

rat

a

find To

Parallel-Axis Theorem Pre-Lab

TrF agmFT rat

rgrm

Newton’s Second Law I

I

rgrmI

Experimental Rotational Inertia about a parallel axis a perpendicular distance h from the center of mass

Parallel-Axis Theorem Pre-Lab