# Derive an equation for the moment of inertia using translational and ...

Mechanics

Nov 14, 2013 (4 years and 8 months ago)

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Phys223

Rolling Motion

Derive an equation for the moment of inertia using
translational and rotational
dynamics

and
kinematics
.

Draw

an
extended FBD

for
the hoop.

Include coord system

and label

all quantities

equations. Don’t solve here:

x
F
 

y
F
 

 

In the last equation, indicate with subscripts whether you are considering the torques (and
moment of inertia) about the center of mass or the
point of contact between the hoop and the
track.

List the acceleration constraint (how are

and a
cm

related?):

r
1

r
2

Phys223

Rolling Motion

Use kinematics to express the acceleration a
cm

in terms of quantities you measured (i.e. in terms
of time t, track length d, and angle of incl
ine

. If you measured height h instead of

, express h
in terms of

.

Solve your equations for the moment of inertia about the center of mass. Call this quantity I
M2
.
Note: you should only have measured qu
antities on the right hand side.

M2
=

Phys223

Rolling Motion

d

h

Derive an equation for the moment of inertia using
conservation of energy and
kinematics
.

Use kinematics to find the final speed v of the hoop:

Set up the equation for conservation of energy:

From the
above equation, eliminate the angular velocity (express it in terms of v). Then solve
the equation for the moment of inertia. Make sure the right hand side has only measured
quantities in it: