The question involves the oscillation of two bar magnets, which are placed under different conditions that affect their time periods of oscillation. Let's analyze the problem step-by-step:
- We have two bar magnets with magnetic moments \(m\) and \(2m\). They are placed in such a way that either similar poles are on the same side or one of them is reversed, causing a differential interaction between them.
- The time period of oscillation of a magnet in a magnetic field is given by the formula: \(T = 2\pi \sqrt{\frac{I}{m_{eff} \cdot B}}\) where \(I\) is the moment of inertia of the magnet, \(m_{eff}\) is the effective magnetic moment, and \(B\) is the magnetic field.
- When the similar poles are on the same side (scenario for \(T₁\)), the effective magnetic moment is \(m_{eff} = m + 2m = 3m\).
- When one magnet is reversed (scenario for \(T₂\)), the effective magnetic moment becomes \(m_{eff} = 2m - m = m\) (since they oppose each other).
- The time periods for these scenarios are:
- For \(T₁\): \(T₁ = 2\pi \sqrt{\frac{I}{3m \cdot B}}\)
- For \(T₂\): \(T₂ = 2\pi \sqrt{\frac{I}{m \cdot B}}\)
- From the above expressions, \(T₁\) has a larger denominator due to the larger effective magnetic moment. Thus, \(T₁\) is less than \(T₂\), given that an increase in the denominator for a fixed numerator will decrease the overall expression.
Thus, the correct answer is \(T₁ < T₂\), which makes option \(T₁ < T₂\) the right choice.