A charged cork ball of mass m is suspended on a light string...

A charged cork ball of mass $m$ is suspended on a light string in the presence of a uniform electric field, as shown in Figure P23.53. When $\mathbf{E}=(A \mathbf{i}+B \mathbf{j}) \mathrm{N} / \mathrm{C}$, where $A$ and $B$ are positive numbers, the ball is in equilibrium at the angle $\theta$. Find (a) the charge on the ball and (b) the tension in the string.

Key Concepts

Equilibrium of Forces

This concept involves setting the sum of all forces acting on an object to zero for both horizontal and vertical directions when the object is at rest or moving with constant velocity. It is essential for analyzing systems in static equilibrium by balancing forces such as gravity, tension, and electric forces.

Vector Decomposition

Vector decomposition is the process of resolving forces into their components along chosen axes (typically horizontal and vertical). This allows for the separate analysis of the effects of each force in different directions, using trigonometric functions like sine and cosine to relate the magnitudes to the angles involved.

Electrostatic Force

Charged objects in an electric field experience a force whose magnitude is proportional to the charge and the strength of the electric field. This force, given by the product of the charge and the electric field vector, is central to understanding how electric fields affect charged particles.

Tension Force

Tension force is the pulling force transmitted along a string, rope, or cable. In equilibrium problems, the tension is resolved into components that counterbalance other forces such as the gravitational force and any applied external forces, ensuring that the net force on the suspended object is zero.

Gravitational Force

Gravitational force acts on objects with mass and is directed toward the center of the Earth. It is quantified as the product of the mass and the acceleration due to gravity (mg) and provides a constant downward force that must be balanced by other force components in equilibrium problems.

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