A uniform solid disk of radius and mass is to be mounted on a shaft so that it can…

PHYSICS · Multi-part · A Level · Physics · Engineering physics · Engineering physics

Problem

A uniform solid disk of radius

0.15\,\text{m}

and mass

3.0\,\text{kg}

is to be mounted on a shaft so that it can spin freely. The shaft axis is required to be parallel to the disk’s central axis but offset by a distance

d

from the disk’s center. Engineers need the rotational inertia of the mounted disk about the shaft to be

0.12\,\text{kg}\cdot\text{m}^2

. (For a solid disk about its central axis,

I_{\text{cm}}=\tfrac{1}{2}MR^{2}

.) (a) Derive an algebraic expression for

d

in terms of

M

R

, and the desired rotational inertia

I

. Do not substitute numerical values yet. (b) Using the given numerical values, compute the required offset distance

d

. (c) The mechanical housing restricts the shaft to offsets no greater than

0.18\,\text{m}

. Determine whether the target rotational inertia can be met with the disk alone. If not, calculate the smallest mass

m

that could be attached as a point mass at the disk’s rim, directly opposite the shaft, to achieve the required rotational inertia while keeping

d=0.18\,\text{m}

. Show all reasoning.