A charged sphere of mass 2.8 ×10^-4 kg is suspended from a string. An electric force acts horiz

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A charged sphere of mass 2.8 ×10^-4 kg is suspended from a...

A charged sphere of mass $2.8 \times 10^{-4} \mathrm{~kg}$ is suspended from a string. An electric force acts horizontally on the sphere so that the string makes an angle of $33^{\circ}$ with the vertical when at rest. Find $(a)$ the magnitude of the electric force and $(b)$ the tension in the string.

Key Concepts

Static Equilibrium

This concept states that if an object is at rest, the net force acting on it in any direction must be zero. In problems like this, where multiple forces act on a suspended object, setting up equilibrium equations in both the horizontal and vertical directions is crucial to determine unknown forces.

Force Decomposition

Force decomposition involves breaking a force vector into its orthogonal components using trigonometric functions. For an object hanging by a string, the tension can be resolved into vertical and horizontal components that counteract the gravitational force and any horizontal applied force respectively.

Tension in a String

The tension is the force transmitted through a string when it is pulled tight by forces acting from opposite ends. In equilibrium scenarios, the tension must have components that exactly balance other forces acting on the object, such as the weight and any applied horizontal force.

Gravitational Force

The gravitational force (or weight) of an object is the force due to gravity acting on its mass. It is computed by multiplying the mass of the object by the acceleration due to gravity, and it acts vertically downward.

Trigonometric Functions in Force Analysis

Trigonometric functions like sine and cosine are used to relate the angles in a force diagram to the magnitudes of the force components. These functions help in resolving the tension into horizontal and vertical components according to the angle the string makes with the vertical.

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