Question

A simple pendulum consists of a small sphere of mass \(\mathrm{m}\) suspended by a thread of length \(\ell\). The sphere carries a positive charge q. The pendulum is placed in a uniform electric field of strength \(\mathrm{E}\) directed Vertically upwards. If the electrostatic force acting on the sphere is less than gravitational force the period of pendulum is (A) $\mathrm{T}=2 \pi[\ell /\\{\mathrm{g}-(\mathrm{q} \mathrm{E} / \mathrm{m})\\}]^{(1 / 2)}$ (B) \(\mathrm{T}=2 \pi(\ell / \mathrm{g})^{(1 / 2)}\) \(\left.\left.\left.\mathrm{m}_{\mathrm{}}\right\\}\right\\}\right]^{(1 / 2)}\) (D) \(\mathrm{T}=2 \pi[(\mathrm{m} \ell / \mathrm{qE})]^{(1 / 2)}\) (C) \(\mathrm{T}=2 \pi[\ell /\\{\mathrm{g}+(\mathrm{qE} / \mathrm{t}\)

Short answer

The short answer to the question is: The period of the pendulum is given by (A): \(T = 2\pi\left[\frac{l}{\{g - (\frac{qE}{m})\}}\right]^{\frac{1}{2}}\).

Step-by-step solution

Analyze the forces acting on the pendulum

There are two forces acting on the pendulum sphere: the gravitational force (F_gravity) and the electrostatic force (F_electric). The gravitational force is given by: \[F_{gravity} = m*g\] where m is the mass of the sphere and g is the acceleration due to gravity. The electrostatic force can be determined using Coulomb's Law: \[F_{electric} = q*E\] where q is the charge on the sphere and E is the electric field strength. Given that the electrostatic force is less than the gravitational force, we have: \[F_{electric} < F_{gravity}\]

Determine the effective gravitational force acting on the pendulum

Since both forces are acting vertically, we can determine the effective gravitational force (F_eff) acting on the sphere. \[F_{eff} = F_{gravity} - F_{electric} = m*g - q*E\]

Calculate the period of the pendulum

For a simple pendulum, the period (T) is given by the equation: \[T = 2 \pi \sqrt{\frac{l}{g_{eff}}}\] Substituting the value of the effective gravitational force (F_eff) acting on the pendulum, we get: \[T = 2\pi\sqrt{\frac{l}{\frac{F_{eff}}{m}}}\] Substitute the value of \(F_{eff}\): \[T = 2\pi\sqrt{\frac{l}{\frac{m*g - q*E}{m}}}\] Simplify the equation: \[T = 2\pi\sqrt{\frac{l}{g - \frac{qE}{m}}}\] Comparing this result with the given options, the answer is (A): \[T = 2\pi[\frac{l}{\{g - (\frac{qE}{m})\}}]^{\frac{1}{2}}\]