Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_32

⚡ Mecademy AIENG정역학 · ch3  Problem Statement A uniform ring of mass and radius carries an eccentric mass at a radius and is in an equilibrium position on the incline, which makes an angle with the horizontal. If the contacting surfaces are rough enough to prevent slipping, write the expression for the angle which defines the equilibrium position. Problem 3/32 (a) Equilibrium of the Ring and Eccentric Mass ● Calculation Process 1. Present Final Formula: To find the equilibrium position, we apply the principle of static equilibrium. For a body resting on a single contact point, the most efficient method is to sum moments about the contact point . 2. Substitute Values: Consider the horizontal distances of the weights from the vertical line passing through the contact point . The ring of mass has its center of gravity at its center . The horizontal distance from the contact point to the center is . The eccentric mass is located at a distance from the center . Its angular position is measured from the downward vertical line passing through . Its horizontal distance from the vertical line through is . The net horizontal distance of from the vertical line through is (assuming the mass is positioned such that it balances the tendency of the ring to roll down). Substituting these into the moment equilibrium equation: 3. Partial Operations: Factor out the common term and rearrange the equation to isolate the terms involving masses and geometry: mrm 0 b α θ C M = ∑ C 0 C mO COd = O rsi