A mass M is suspended by a rope from a rigid support at A as...

A mass $M$ is suspended by a rope from a rigid support at $A$ as shown in figure. Another rope is tied at the end $B$ and it is pulled horizontally with a force $F$. If the rope $A B$ makes an angle $\theta$ with the vertical, then the tension in the string $A B$ is (a) $F \sin \theta$ (b) $F / \sin \theta$ (c) $F \cos \theta$ (d) $F / \cos \theta$

A mass $M$ is suspended by a rope from a rigid support at $A$ as shown in figure. Another rope is tied at the end $B$ and it is pulled horizontally with a force $F$. If the rope $A B$ makes an angle $\theta$ with the vertical, then the tension in the string $A B$ is
(a) $F \sin \theta$
(b) $F / \sin \theta$
(c) $F \cos \theta$
(d) $F / \cos \theta$

Key Concepts

Static Equilibrium

This concept states that an object at rest experiences no net force in any direction. For the mass suspended by the rope, the sum of forces must cancel out both horizontally and vertically. Understanding static equilibrium is crucial for setting up the force balance equations that determine unknown forces such as tension.

Force Resolution

Force resolution involves breaking down a force into components along specified axes, typically horizontal and vertical. In this problem, the tension in the rope can be decomposed such that its horizontal component is responsible for balancing the applied horizontal force. Recognizing which trigonometric function applies to the respective component is key in solving for the unknown tension.

Tension in a Rope

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In static scenarios, the rope transmits the same force along its length. Understanding the nature of tension allows one to relate the horizontal force to the components of tension, thereby determining its magnitude.

Trigonometric Relationships

Trigonometric relationships connect the angles in a geometric figure with the ratios of the sides of the triangle. In this context, applying the sine function relates the unknown tension in the rope to the horizontal force component. This connection is crucial in solving for the tension as it sets the relation T*sin(?) equal to the applied horizontal force.

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