Torsion Disk Rebound

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90.Torsion Disk Rebound

+10

A uniform disk of mass

M

and radius

R

is mounted on a frictionless axle through its center and is attached to a torsion spring of stiffness

\kappa

(restoring torque

\tau = -\kappa\theta

). The disk oscillates freely about its equilibrium orientation with small angular amplitude

\theta_0

. A small puck of mass

m

slides at speed

v

along a straight frictionless track and strikes the rim of the disk in a head-on elastic collision. The collision occurs at the instant the disk passes through its equilibrium orientation, with the rim point that contacts the puck moving directly toward the puck. For what amplitude

\theta_0

does the puck rebound back along its incoming path at speed

v

$Top-down view. A uniform disk of radius $R$ centered at the origin, labeled $M$, $R$, with a small filled circle at the center marking the vertical axle. A coiled spring symbol drawn around the axle represents the torsion spring of stiffness $\kappa$. A short curved arrow on the disk shows it rotating counterclockwise. At the rightmost rim point $(R, 0)$, draw a tangential velocity arrow pointing in the $+y$ direction labeled $v_\text{rim}$. To the right of the disk, just outside the rim along a horizontal straight track, draw a small filled puck of mass $m$ at position roughly $(R, +y_0)$ with a velocity arrow pointing in the $-y$ direction labeled $v$. Indicate the angular amplitude with a dashed reference radius rotated by $\theta_0$ from the disk's solid reference radius, with a small arc labeled $\theta_0$ between them.$

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