A baton used in a marching band can be modeled as a thin uniform rod of mass and length

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

A baton used in a marching band can be modeled as a thin uniform rod of mass

0.80\,\text{kg}

and length

1.20\,\text{m}

. The baton will be twirled about an axis that is parallel to the rod and lies in the plane of the rod at a distance

d

from the baton’s center of mass. For smooth performance, the designer wants the rotational inertia of the baton about this axis to be exactly

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

. (For a thin rod of length

L

about an axis through its center and perpendicular to the rod,

I_{\text{cm}}=\tfrac{1}{12}ML^{2}

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

I

of the baton about the twirling axis in terms of

M

L

, and

d

. Solve this expression for

d

symbolically. (b) Calculate the numerical value of

d

that satisfies the designer’s requirement. Include appropriate units. (c) State whether the twirling axis is located inside the physical length of the baton or outside it, and explain how your calculation leads to that conclusion using physical reasoning about mass distribution.

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