In Fig. 12-21, a nonuniform bar is suspended at rest in a...

In Fig. 12-21, a nonuniform bar is suspended at rest in a horizontal position by two massless cords. One cord makes the angle $\theta=30.0^{\circ}$ with the vertical; the other makes the angle $\phi=60.0^{\circ}$ with the vertical. If the length $L$ of the bar is $9.50 \mathrm{~m}$, compute the distance $x$ from the left end of the bar to its center of mass.

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Key Concepts

Static Equilibrium

Static equilibrium occurs when an object is at rest and the sum of all forces and torques acting on it equals zero. This ensures that the object does not experience any linear or rotational acceleration, which is critical when analyzing systems supported by multiple forces at different angles.

Torque (Moment) Equilibrium

Torque equilibrium refers to the condition where the total torque (or moment) about any pivot or reference point is zero. In problems involving objects suspended by cords or forces acting at various angles, calculating torques and setting their sum to zero is essential to determine unknown distances or forces.

Center of Mass

The center of mass of an object is the point where the total mass of the body can be considered to be concentrated. For nonuniform objects, the center of mass does not coincide with the geometric center, and its location is crucial in the analysis of rotational equilibrium, influencing the distribution of weight and the resulting torques.

Lever Arm

The lever arm is the perpendicular distance from the axis of rotation (or pivot point) to the line of action of a force. In torque calculations, the magnitude of the torque is the product of the force and its lever arm, making it a fundamental concept in solving for unknown positions or forces in equilibrium problems.

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