A ball with a net charge of +450 nC and mass 0.75 g is suspended from a thread of length 12 cm i

step 1 The question is that a ball with a net charge of 450 nal column and mass 0 .75 grams is suspende

A ball with a net charge of +450 nC and mass 0.75 g is...

A ball with a net charge of $+450 \mathrm{nC}$ and mass 0.75 g is suspended from a thread of length $12 \mathrm{cm}$ in a uniform electric field of $8.0 \mathrm{kV} / \mathrm{m}$ downward. Do not ignore gravity. The system acts like a pendulum but with a downward electric force added to the gravitational force. What is the period of oscillation for small amplitudes?

A ball with a net charge of $+450 \mathrm{nC}$ and mass 0.75 g is suspended from a thread of length $12 \mathrm{cm}$ in a uniform electric field of $8.0 \mathrm{kV} / \mathrm{m}$ downward. Do not ignore gravity. The system acts like a pendulum but with a downward electric force added to the gravitational force.
What is the period of oscillation for small amplitudes?

Key Concepts

Simple Pendulum

A simple pendulum is a mass suspended from a fixed point by a string or rod that swings under the influence of gravity. When the amplitudes of the oscillations are small, the system can be treated as undergoing simple harmonic motion, where the restoring force is proportional to the displacement. This concept underpins the derivation of the standard period formula, T = 2??(L/g), for small angular oscillations.

Small-Angle Approximation

For small oscillations, the sine of the angular displacement can be approximated by the angle itself (in radians). This approximation simplifies the nonlinear equations of motion into a linear form, which is essential for obtaining an analytical expression for the period of the pendulum that accurately describes its behavior for small amplitudes.

Effective Gravity in the Presence of Multiple Forces

When additional constant forces are acting on the pendulum, such as an electric force in this case, the total force along the vertical direction becomes the sum of the gravitational force and the additional force. This results in an effective gravitational acceleration (g_eff) that governs the oscillatory dynamics. The period formula is then modified to T = 2??(L/g_eff), where g_eff accounts for all constant forces acting in the vertical direction.

Force Superposition

The principle of superposition states that multiple forces acting simultaneously on a system can be added vectorially to form a net force. In the context of this problem, the gravitational force and the electric force both contribute to the net force acting on the pendulum bob, determining its equilibrium position and the effective restoring force during small oscillations.

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