Two small spheres with mass m = 15.0 g are hung by silk...

Two small spheres with mass $m =$ 15.0 g are hung by silk threads of length $L =$ 1.20 m from a common point (Fig. P21.62). When the spheres are given equal quantities of negative charge, so that $q_1 = q_2 = q$, each thread hangs at $\theta = 25.0^\circ$ from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of $q$. (c) Both threads are now shortened to length $L =$ 0.600 m, while the charges $q_1$ and $q_2$ remain unchanged. What new angle will each thread make with the vertical? ($Hint$: This part of the problem can be solved numerically by using trial values for $\theta$ and adjusting the values of $\theta$ until a self-consistent answer is obtained.)

Key Concepts

Coulomb's Law

Coulomb's law describes the force between two point charges. According to this law, the magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This electrostatic force is repulsive if the charges are of the same sign and attractive if they differ, and it is essential for quantifying interactions in charged-particle systems.

Gravitational Force

Gravitational force is the attractive force that acts between any two masses, with the strength proportional to their masses and the universal gravitational constant. In problems involving suspended objects, gravitational force is typically represented as the weight, which is the product of the mass and the acceleration due to gravity, acting vertically downward.

Tension in the Suspension Strings

The tension in a string or thread is the force exerted along its length, balancing other forces acting on a suspended object. In systems where objects are hanging, the tension must counteract the gravitational force and any additional forces such as those arising from electrostatic interactions, with its components determined by the angle the string makes with the vertical.

Static Equilibrium

Static equilibrium occurs when all forces and torques acting on a system are balanced, resulting in no net acceleration. For a suspended charged sphere, this means that the downward gravitational force, the horizontal electrostatic repulsion, and the tension in the string combine such that the net force in both the horizontal and vertical directions is zero. This condition is crucial for setting up the equations that determine quantities like the charge or the equilibrium angle.

Trigonometric Decomposition

Trigonometric decomposition involves breaking vectors into their horizontal and vertical components using sine, cosine, and other trigonometric functions. This is essential in analyzing forces acting at angles, as it allows the separation of forces into components that can be independently balanced using the equilibrium conditions.

Geometric Relations in Pendulum Systems

The geometry in pendulum configurations, such as the length of the strings and the angle they make with the vertical, directly affects the distance between suspended objects and the components of the forces acting on them. Understanding these geometric relationships is vital for relating the physical dimensions of the setup to the forces and angles determined by static equilibrium conditions.

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