A uniform, 458 -g metal bar 75.0 cm long carries a current I in a uniform, horizontal 1.25 - T

step 1 Question we have length of the bar l equals to 75 cm that is 0 .75 meter okay and mass of the ba

A uniform, 458 -g metal bar 75.0 cm long carries a current...

A uniform, 458 -g metal bar 75.0$\mathrm { cm }$ long carries a current $I$ in a uniform, horizontal $1.25 -$ T magnetic field as shown in Fig. P27.81. The directions of $I$ and $\vec { B }$ are shown in the figure. The bar is free to rotate about a frictionless hinge at point $b$ . The other end of the bar rests on a conducting support at point $a$ but is not attached there. The bar rests at an angle of $60.0 ^ { \circ }$ above the horizontal. What is the largest value the current $I$ can have without breaking the electrical contact at $a ?$ (See Problem $27.77 . )$

A uniform, 458 -g metal bar 75.0$\mathrm { cm }$ long carries a current $I$ in a uniform,
horizontal $1.25 -$ T magnetic field as shown in
Fig. P27.81. The directions of $I$ and $\vec { B }$ are shown in the figure. The bar is free to rotate
about a frictionless hinge at point $b$ . The other
end of the bar rests on a conducting support at point $a$ but is not attached there. The bar rests
at an angle of $60.0 ^ { \circ }$ above the horizontal. What is the largest value the current $I$ can have without breaking
the electrical contact at $a ?$ (See Problem $27.77 . )$

Key Concepts

Static Equilibrium Conditions

Static equilibrium requires that both the net force and the net torque on a system are zero. This principle governs how multiple forces interact in a statically balanced structure. In the context of the problem, it involves balancing magnetic forces, gravitational forces, and any support reactions to ensure the system remains at rest without unintentional motion or separation at contact points.

Electrical Contact and Constraint Forces

In systems where a component is in physical and electrical contact with a support structure, maintaining that contact involves ensuring that the net normal force at the interface remains sufficient. The concept involves analyzing the forces and torques to determine the limits at which the contact might be lost. Such analysis is crucial when current-induced forces are significant and might alter the effective contact or clamping between components.

Center of Mass and Gravitational Torque

For a uniform body, the gravitational force acts at its center of mass. In problems involving equilibrium, the location of the center of mass influences the magnitude of the torque produced by gravity relative to a pivot point. Understanding how the weight distribution of a body contributes to rotational effects is essential, especially when balancing different torques in a static system.

Magnetic Force on a Current?Carrying Conductor

A conductor carrying an electric current in a magnetic field experiences a force given by the vector equation F = I (L × B). This force, whose magnitude depends on the current, the magnetic field strength, the length of the conductor within the field, and the sine of the angle between the conductor and the field, is central to understanding how electromagnetic devices operate and how forces can be generated and balanced in such systems.

Torque and Moment Equilibrium

When forces act on an object at various points, they create torques (or moments) which can cause the object to rotate about a pivot. In systems with external forces such as magnetic forces and gravitational forces, ensuring that the sum of all torques about a pivot is zero is essential for static rotational equilibrium. This concept is key when analyzing systems where the force may cause a rotational tendency that must be balanced to maintain a desired configuration.

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