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?
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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 $T$.
$q$ is positive and $\hat{n} \cdot \hat{z} = -\frac{1}{\sqrt{2}}$
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The Correct Option isA
Solution and Explanation
Step 1: Understanding the Question:
A simple pendulum with a charged bob oscillates under gravity pointing downwards in the $-\hat{z}$ direction.
An electric field $\mathbf{E} = |E|\hat{n}$ is applied, and the time period of oscillation decreases.
We need to find which of the given options is incorrect. Step 2: Key Formula or Approach:
The time period of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g_{eff}}} \]
For the time period $T$ to decrease, the effective acceleration $g_{eff}$ must increase, i.e., $g_{eff} > g$.
The effective acceleration vector is given by:
\[ \mathbf{g}_{eff} = -g \hat{z} + \frac{q \mathbf{E}}{m} = -g \hat{z} + \frac{q|E|}{m} \hat{n} \]
Step 3: Detailed Explanation:
• Let us analyze each option to see if $g_{eff} > g$ is satisfied:
• Option (A): $q$ is positive and $\hat{n} = \hat{z}$.
The electric force is in the $+\hat{z}$ direction, which is upwards, opposing gravity.
The effective acceleration is:
\[ \mathbf{g}_{eff} = \left( -g + \frac{q|E|}{m} \right) \hat{z} \implies g_{eff} = g - \frac{q|E|}{m} < g \]
Since $g_{eff}$ decreases, the time period $T$ must increase.
This contradicts the statement that the time period decreases, so Option (A) is NOT correct.
• Option (B): $q$ is positive and $\hat{n} = -\hat{z}$.
The electric force is downwards, in the $-\hat{z}$ direction.
The effective acceleration is:
\[ g_{eff} = g + \frac{q|E|}{m} > g \]
This increases $g_{eff}$, so the time period $T$ decreases. This statement is correct.
• Option (C): $q$ is negative and $\hat{n} = \hat{z}$.
Since $q$ is negative, the electric force is $\mathbf{F}_e = q \mathbf{E} = -|q||E|\hat{z}$, which points downwards.
The effective acceleration increases, so $T$ decreases. This statement is correct.
• Option (D): $q$ is positive and $\hat{n} \cdot \hat{z} = -\frac{1}{\sqrt{2}}$.
The unit vector $\hat{n}$ has a downward component, meaning the electric force has a downward component.
This increases the net downward acceleration, making $g_{eff} > g$ and decreasing the period $T$. This statement is correct. Step 4: Final Answer:
The incorrect statement is that "$q$ is positive and $\hat{n} = \hat{z}$".
