Cradled Descent

HardPractice Problems

55.Cradled Descent

+10

A solid uniform sphere of mass

m

and radius

R

rolls without slipping down a long right-angle V-groove (two flat walls meeting at

90^\circ

) whose central axis is tilted at angle

\theta

below the horizontal. What is the magnitude of the sphere's translational acceleration along the groove?

$Cross-section of the V-groove taken perpendicular to the direction of travel. Two flat walls meet at the bottom to form a 90-degree V, each wall drawn at $45^\circ$ from vertical. A circle of radius $R$ representing the sphere sits nestled in the V, tangent to both walls. Label the center of the circle as $O$, the contact point on the left wall as $A$, and the contact point on the right wall as $B$. Draw a dashed horizontal segment from $A$ to $B$ passing below $O$, and mark the perpendicular distance from $O$ down to segment $AB$ as $R/\sqrt{2}$. Indicate the $45^\circ$ angle between each wall and the vertical.$

\frac{2}{7} g \sin\theta

\frac{2}{5} g \sin\theta

\frac{5}{9} g \sin\theta

\frac{5}{7} g \sin\theta

g \sin\theta

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