The blocks B and C in the Fig. 3.90 have mass m each. The...

The blocks $B$ and $C$ in the Fig. $3.90$ have mass $m$ each. The strings $A B$ and $B C$ are light, having tensions $T_{1}$ and $T_{2}$, respectively. The system is in equilibrium with a constant force $m g$ acting on $C$. (A) $\tan \theta_{1}=1 / 2$ (B) $\tan \theta_{2}=1 / 2$ (C) $T_{1}=\sqrt{5} \mathrm{mg}$ (D) $T_{2}=\sqrt{2} \mathrm{mg}$

The blocks $B$ and $C$ in the Fig. $3.90$ have mass $m$ each. The strings $A B$ and $B C$ are light, having tensions $T_{1}$ and $T_{2}$, respectively. The system is in equilibrium with a constant force $m g$ acting on $C$.
(A) $\tan \theta_{1}=1 / 2$
(B) $\tan \theta_{2}=1 / 2$
(C) $T_{1}=\sqrt{5} \mathrm{mg}$
(D) $T_{2}=\sqrt{2} \mathrm{mg}$

Key Concepts

Static Equilibrium

This concept refers to the state in which all forces and moments acting on a system cancel out, resulting in no net motion. In problems involving blocks, strings, and applied forces, the sum of forces in both the horizontal and vertical directions must be zero. This principle allows the formulation of equations that ensure the system remains at rest.

Tension in Strings

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. In systems with multiple masses connected by strings, each string can carry a different tension depending on the forces acting on the masses it supports. Analyzing tension typically involves drawing a free-body diagram for each mass and applying equilibrium equations.

Force Decomposition

When forces act at angles, they are often resolved into perpendicular components (usually horizontal and vertical) using trigonometric functions. This decomposition simplifies the analysis of the forces in equilibrium by handling one direction at a time. It is a key method for solving problems where applied forces are not aligned with the standard coordinate axes.

Trigonometric Relationships in Force Analysis

Trigonometric functions, such as sine, cosine, and tangent, are used to relate the angles of forces to their horizontal and vertical components. Specifically, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In the context of equilibrium problems, these relationships help to link the angles at which strings or forces act to the magnitudes of the forces or tensions.

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