In Fig. 12-68, an 817 kg construction bucket is suspended by...

In Fig. $12-68,$ an 817 $\mathrm{kg}$ construction bucket is suspended by a cable $A$ that is attached at $O$ to two other cables $B$ and $C,$ making angles $\theta_{1}=51.0^{\circ}$ and $\theta_{2}=66.0^{\circ}$ with the horizontal. Find the tensions in (a) cable $A,$ (b) cable $B$ , and (c) cable $C .$ (Hint: To avoid solving two equations in two unknowns, position the axes as shown in the figure.)

In Fig. $12-68,$ an 817 $\mathrm{kg}$
construction bucket is suspended
by a cable $A$ that is attached at $O$
to two other cables $B$ and $C,$ making angles $\theta_{1}=51.0^{\circ}$ and $\theta_{2}=66.0^{\circ}$
with the horizontal. Find the tensions in
(a) cable $A,$ (b) cable $B$ ,
and (c) cable $C .$ (Hint: To avoid
solving two equations in two unknowns,
position the axes as
shown in the figure.)

Key Concepts

Tension in Cables

This concept involves the force transmitted through a cable when it is pulled tight by forces acting from opposite ends. Understanding how the tension is distributed through cables at different angles is crucial for calculating the individual forces that keep a system in equilibrium.

Optimal Choice of Coordinate Axes

Choosing a coordinate system that aligns with one or more forces simplifies the resolution of forces into components. This strategic selection can reduce the complexity of the calculations by minimizing the number of equations needed, particularly when dealing with angled forces in static equilibrium problems.

Free Body Diagram

Drawing a free body diagram helps visualize all the forces acting on an object. It is a key tool in statics to identify and represent forces such as tensions, weights, and applied loads, and to set up the appropriate equations based on equilibrium conditions.

Force Resolution into Components

When forces act at angles, breaking them up into horizontal and vertical components using trigonometric functions (sine and cosine) simplifies the problem. This allows one to write separate equilibrium equations for each direction, an approach that is fundamental in solving for unknown forces.

Static Equilibrium

This concept states that if an object is not accelerating, then the sum of all forces and moments acting on it is zero. It is essential for analyzing problems involving multiple forces, like the tensions in cables supporting a suspended load, ensuring that horizontal and vertical components of the forces balance.

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