Question

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. Determine the angular velocity of the system as a function of time.

Accepted Answer

Understand the problem: The system consists of a woman and a rotating cylindrical platform. The woman moves toward the center of the platform, and we need to determine how the angular velocity of the system changes as a function of time. This is a conservation of angular momentum problem because there is negligible friction, meaning no external torques act on the system. Write the expression for the total angular momentum of the system. Initially, the angular momentum is given by: $$L_0 = I$$_{platform} 	imes ω_0, where I_{platform} = (1/2) M $$R^2 is $$the moment of inertia of the cylindrical platform, and ω_0 is the initial angular velocity. The woman also contributes to the angular momentum, which is L_{woman} = m $$r^2$$ ω_0, where r is her distance from the center at any given time. Apply the conservation of angular momentum: Since there are no external torques, the total angular momentum remains constant. Thus, L_{initial} = L_{final}. Initially, $$L_0 = (1/2) M$$ $$R^2$$ ω_0 + m $$R^2$$ ω_0. At a later time, the woman has moved closer to the center, so her contribution to the moment of inertia changes. The final angular momentum is L_{final} = (1/2) M $$R^2$$ ω + m $$r(t)^2$$ ω, where ω is the new angular velocity and r(t) is her distance from the center as a function of time. Solve for the angular velocity ω: Equating L_{initial} and L_{final}, we get: (1/2) M $$R^2$$ ω_0 + m $$R^2$$ ω_0 = (1/2) M $$R^2$$ ω + m $$r(t)^2$$ ω. Rearrange this equation to isolate ω: ω = \frac{(1/2) M $$R^2$$ ω_0 + m $$R^2$$ ω_0}{(1/2) M $$R^2 + m$$ $$r(t)^2$$}. Express r(t): The woman's distance from the center, r(t), decreases as she walks toward the center with speed υ. If she starts at the edge of the platform (r(0) = R), then r(t) = R - υt. Substitute this expression for r(t) into the equation for ω to get the angular velocity as a function of time: ω(t) = \frac{(1/2) M $$R^2$$ ω_0 + m $$R^2$$ ω_0}{(1/2) M $$R^2 + m (R$$ - υ$$t)^2$$}.

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Key Concepts

Conservation of Angular Momentum

Relative Velocity

Moment of Inertia