Question

A simple pendulum of length $L$, mass $m$ and electric charge $q$ on its bob is oscillating with a time period $T$ under uniform gravity which is in the $-\hat{z}$ direction. Upon applying a uniform electric field $|E|\hat{n}$ (where $\hat{n}$ is a unit vector in the plane of oscillation), the time period of the pendulum decreases. Which of the following statements is NOT correct?

• A. $q$ is positive and $\hat{n} = \hat{z}$
• B. $q$ is positive and $\hat{n} = -\hat{z}$
• C. $q$ is negative and $\hat{n} = \hat{z}$
• D. $q$ is positive and $\hat{n} \cdot \hat{z} = -\frac{1}{\sqrt{2}}$

Hint:

For a pendulum, any external force with a component acting in the same direction as gravity (downwards) will increase the effective acceleration $g_{eff}$ and thus decrease the time period $T$.
An upward force decreases $g_{eff}$ and increases $T$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The time period of a pendulum is \( T = 2\pi\sqrt{L/g_{\text{eff}}} \). For \( T \) to decrease, the effective acceleration \( g_{\text{eff}} \) must increase.
Key Formula or Approach:
The effective acceleration is the vector sum of gravitational and electric accelerations:
\[ \vec{g}_{\text{eff}} = \vec{g} + \frac{q\vec{E}}{m} \]

Step 2: Detailed Explanation:
1. Gravity: Points in \( -\hat{z} \).
2. Condition: We need \( |\vec{g}_{\text{eff}}| > g \). This happens if the electric force has a downward component (along \( -\hat{z} \)).
3. Analyze Options:
$\bullet$ (B) \( q>0, \hat{n} = -\hat{z} \): Force is downward. \( g_{\text{eff}} = g + qE/m \). Correct (T decreases).
$\bullet$ (C) \( q<0, \hat{n} = \hat{z} \): Force is \( q\vec{E} = (-|q|)(|E|\hat{z}) \), which is downward. Correct (T decreases).
$\bullet$ (D) \( \hat{n} \cdot \hat{z} = -1/\sqrt{2} \): The unit vector has a downward component. For \( q>0 \), force has a downward component. Correct (T decreases).
$\bullet$ (A) \( q>0, \hat{n} = \hat{z} \): Force is upward (along \( +\hat{z} \)). \( g_{\text{eff}} = g - qE/m \), which is less than \( g \). This makes \( T \) increase. Therefore, this statement is incorrect.

Step 3: Final Answer:
Statement (A) is the incorrect one.