In Fig. 12-69 , a package of mass m hangs from a short cord...

In Fig. $12-69$ , a package of mass $m$ hangs from a short cord that is tied to the wall via cord 1 and to the ceiling via cord $2 .$ Cord 1 is at angle $\phi=$ $40^{\circ}$ with the horizontal; cord 2 is at angle $\theta .($ a) For what value of $\theta$ is the tension in cord 2 minimized? (b) In terms of $m g,$ what is the minimum tension in cord 2$?$

In Fig. $12-69$ , a package of mass
$m$ hangs from a short cord that is tied
to the wall via cord 1 and to the ceiling
via cord $2 .$ Cord 1 is at angle $\phi=$
$40^{\circ}$ with the horizontal; cord 2 is at
angle $\theta .($ a) For what value of $\theta$ is the
tension in cord 2 minimized? (b) In
terms of $m g,$ what is the minimum tension in cord 2$?$

Key Concepts

Trigonometric Relationships

Trigonometric relationships are used to relate the angles and sides of triangles, which is particularly useful when working with forces acting at angles. These relationships allow for the accurate resolution of forces into their components and are crucial in solving problems involving inclined forces.

Force Decomposition

Force decomposition involves breaking a force into components along specified coordinate axes, typically horizontal and vertical. This technique simplifies the analysis of forces, especially when they act at various angles, and facilitates the application of equilibrium conditions.

Calculus-Based Optimization

Calculus-based optimization is a method used to find the extreme (minimum or maximum) values of a function by taking its derivative and setting it equal to zero. In physics, this technique is used to optimize parameters such as force or tension, subject to the constraints imposed by the system.

Static Equilibrium

Static equilibrium refers to the condition in which all the forces and moments acting on a body balance each other, resulting in no net force or torque. This concept is fundamental in analyzing systems where objects remain at rest under the action of multiple forces.

Free-Body Diagrams

Free-body diagrams are graphical representations that isolate a body and depict all external forces acting on it. They assist in visualizing force interactions and are essential in writing down the equations of equilibrium for the system.

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