A bar magnet is oscillating in the Earth's magnetic field with time period T. If its mass is increased four times, then its time period will be

1. Break Down the Problem

We need to understand how the time period $T$ of an oscillating bar magnet is affected by its mass. The time period $T$ of a simple pendulum (which can be likened to the oscillation of a magnet in a magnetic field) is given by the equation: $T = 2\pi \sqrt{\frac{I}{mgl}}$ where:

• $I$ is the moment of inertia,
• $m$ is the mass,
• $g$ is the acceleration due to gravity,
• $l$ is the effective length of the pendulum.

In this case, we notice that increasing the mass also influences the time period.

2. Relevant Concepts

From the formula, we see that the time period $T$ is proportional to the square root of the mass $m$. Therefore, if the mass increases, the time period will also change accordingly.

3. Analysis and Detail

1. Let the original mass of the magnet be $m$.
2. According to the problem, the mass is increased to $4m$.
3. The new time period $T'$ can be expressed as: $T' = 2\pi \sqrt{\frac{I}{4mgl}}$ Since the moment of inertia $I$ and the effective length $l$ remain constant, we can relate $T'$ to the original time period $T$: $T' = 2\pi \sqrt{\frac{I}{mgl}} \cdot \frac{1}{2} = \frac{T}{2}$

4. Verify and Summarize

Thus, the new time period $T'$ becomes half of the original time period $T$. This algebraic manipulation confirms our understanding of the relationship between mass and time period.

Final Answer

The new time period $T'$ will be $\frac{T}{2}$.