Suppose a person exerts a force of 50 N on one end of a rope as shown in Fig. 6-82. (a) What ar

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Suppose a person exerts a force of 50 N on one end of a...

Suppose a person exerts a force of $50 \mathrm{~N}$ on one end of a rope as shown in Fig. $6-82$. (a) What are the magnitude and direction of the force at point $A$ exerted on the rope by the ceiling? (b) What are the magnitude and direction of the force exerted on the ceiling by the rope? How does the force get transmitted from one end of the rope to the other? What does the stretching of the rope have to do with this? (c) What are the magnitude and direction of the force the rope exerts on the person's hand at point $B$ ? (d) Draw a diagram with vector arrows indicating the relative magnitudes and directions of the forces the rope exerts on the ceiling at point $A$ and the force the rope exerts on the person's hand at point $B$.

Suppose a person exerts a force of $50 \mathrm{~N}$ on one end of a rope as shown in Fig. $6-82$.
(a) What are the magnitude and direction of the force at point $A$ exerted on the rope by the ceiling?
(b) What are the magnitude and direction of the force exerted on the ceiling by the rope? How does the force get transmitted from one end of the rope to the other? What does the stretching of the rope have to do with this?
(c) What are the magnitude and direction of the force the rope exerts on the person's hand at point $B$ ?
(d) Draw a diagram with vector arrows indicating the relative magnitudes and directions of the forces the rope exerts on the ceiling at point $A$ and the force the rope exerts on the person's hand at point $B$.

Key Concepts

Newton's Third Law

Newton's third law, stating that every action has an equal and opposite reaction, is central to understanding how forces interact in this scenario. When the rope exerts a force on the ceiling, for example, the ceiling exerts an equal and opposite force on the rope, ensuring that the action-reaction pairs are balanced in accordance with this law.

Force Equilibrium

For a system in static equilibrium, the sum of all forces acting on each component must be zero. This means that all forces, including those due to tension, contact forces, and any applied forces, must balance out, permitting the analysis of magnitudes and directions of individual forces acting along the rope.

Elasticity and Stretching of the Rope

While an ideal rope is considered inextensible, real ropes exhibit some elasticity and can stretch slightly under load. This stretching (or elastic deformation) is crucial for the transmission of force because it allows the force applied at one end to be communicated to the other by gradually building up tension through the rope’s material, thereby influencing how the rope reacts under load.

Free-Body Diagrams

Free-body diagrams are essential tools in mechanics for isolating and visualizing all forces acting on a segment or component of a system. They help in identifying the magnitude and direction of each force, such as the forces exerted by the ceiling and the person’s hand on the rope, and are integral to setting up the equilibrium equations needed to determine these forces.

Tension in a Rope

Tension is the force transmitted through a rope when it is pulled tight by forces acting from opposite ends. In an idealized, massless rope under static conditions, the tension is equal in magnitude along its entire length; this means that an applied force at one end is directly transmitted to the other end, assuming no losses or slack.

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