A uniform square metal plate of side length and total mass lies in the -plane

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

Problem

A uniform square metal plate of side length

a

and total mass

M

lies in the

xy

-plane. Three different fixed axes, each perpendicular to the plate, are considered: • Axis 0 passes through the plate’s center. • Axis 1 passes through the midpoint of one side. • Axis 2 passes through one corner of the plate. For a uniform square plate, the rotational inertia about Axis 0 is given as

I_0 = \tfrac16 M a^{2}

. (a) Derive an algebraic expression for the rotational inertia

I_1

of the plate about Axis 1. Express your result in terms of

M

and

a

. (b) Using any results you need from part (a), derive an algebraic expression for the rotational inertia

I_2

of the plate about Axis 2 (the corner axis). Express your result in its simplest form in terms of

M

and

a

. (c) Verify that the ratio

I_2/I_0

is independent of the mass

M

and compute its numerical value. (d) Discuss quantitatively how the change in axis location from the center to the corner affects the rotational kinetic energy required to spin the plate at a fixed angular speed

\omega

. Support your discussion with the results from parts (a)–(c).

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