Question

A thin magnetic needle oscillates in a horizontal plane with a period \(\mathrm{T}\). It is broken into n equal parts. The time period of each part will be (a) \(\mathrm{T}\) (b) \(\mathrm{n}^{2} \mathrm{~T}\) (c) \((\mathrm{T} / \mathrm{n})\) (d) \(\left(\mathrm{T} / \mathrm{n}^{2}\right)\)

Short answer

The time period of oscillation for each part of the new needle is given by \(T_{n} = \frac{T}{n^{\frac{3}{2}}}\). However, this exact expression is not present among the given options. The closest option to the obtained expression is (d) \(\frac{T}{n^2}\). There could be an error in the question, but based on the available options, the best answer is (d).

Step-by-step solution

Recall the time period formula for a magnetic needle

The time period of oscillation for a magnetic needle is given by the formula \(T = 2\pi \sqrt{\frac{I}{k}}\), where \(T\) is the time period, \(I\) is the moment of inertia of the magnetic needle and k is the torsional constant. Step 2: Express the moment of inertia for the original magnetic needle

Express the moment of inertia for the original magnetic needle

Because the needle is thin, its moment of inertia can be expressed as \(I = m_{orig}\frac{l^{2}}{12}\), where \(m_{orig}\) denotes the mass of the original needle and \(l\) is the length of the original needle. Step 3: Find the moment of inertia for a part of the new needle

Find the moment of inertia for a part of the new needle

We are given that the original needle is broken into n equal parts, so each new part has mass \(m = \frac{m_{orig}}{n}\) and length \(l_{n} = \frac{l}{n}\). The moment of inertia for a part of the new needle can be expressed as \[I_{n} = m\frac{l_{n}^{2}}{12} = \left(\frac{m_{orig}}{n}\right) \frac{\left(\frac{l}{n}\right)^2}{12} = \frac{m_{orig} l^2}{12n^3}\] Step 4: Find the time period for each part of the new needle

Find the time period for each part of the new needle

Plug the moment of inertia of each part, \(I_{n}\), into the time period formula: \[\begin{aligned} T_{n} &= 2\pi \sqrt{\frac{I_{n}}{k}} \\ &= 2\pi \sqrt{\frac{\frac{m_{orig} l^2}{12n^3}}{k}} \\ &= \frac{2\pi \sqrt{\frac{m_{orig} l^2}{12}}}{n^{\frac{3}{2}}}\sqrt{\frac{1}{k}} \end{aligned}\] Since the original time period \(T = 2\pi \sqrt{\frac{I}{k}} = 2\pi \sqrt{\frac{m_{orig}l^2}{12k}}\), we can rewrite the time period for each part as: \[T_{n} = \frac{T}{n^{\frac{3}{2}}}\] Step 5: Match the answer with the options

Match the answer with the options

The time period of each part does not match any of the given options exactly so there seems to be a mistake in the options. However, the closest option is (d) \(T/n^2\), so it is possible there is an error in the question. In this case, we would still choose (d) as the best available answer.

Key concepts

Moment of Inertia

The moment of inertia is like the rotational equivalent of mass; it tells us how hard it is to change the rotational motion of an object. For a thin magnetic needle, we calculate its moment of inertia using the formula:\[ I = m \frac{l^2}{12} \]Here,

• \( m \) is the mass of the needle, and
• \( l \) is its length.

This formula assumes the needle is uniform and thin. When the needle is broken into \( n \) parts, each part has a length \( \frac{l}{n} \) and mass \( \frac{m}{n} \), leading to a different moment of inertia for each part. This concept is crucial because it directly influences the time period of oscillation in torsional systems.

Torsional Oscillation

Torsional oscillations occur when an object twists back and forth around an axis. This is like a magnet oscillating in a horizontal plane. The time period \( T \) of such torsional oscillations depends on both the moment of inertia \( I \) of the object and a torsional constant \( k \) related to stiffness:\[ T = 2\pi \sqrt{\frac{I}{k}} \]Here’s what the terms represent:

• \( T \): time period of one complete oscillation
• \( I \): moment of inertia impacting rotational inertia
• \( k \): torsional constant related to how hard it is to twist the object

Understanding torsional oscillation helps explain why breaking the needle into parts affects the overall oscillatory motion.

Magnetic Needle

The magnetic needle in this scenario is a thin, lightweight object that can rotate freely. When subjected to a magnetic field, it aligns itself accordingly. In the case of oscillation, the needle will twist and then respond to the restoring torque created by the magnetic forces. This creates a back-and-forth motion – a torsional oscillation.When broken into parts:

• Each piece acts independently, with its own moment of inertia and time period \( T_n \).
• The division affects both weight distribution and rotational dynamics.

The magnetic needle’s initial properties impact every subsequent calculation when considering oscillation time.

Physics Problem Solving

When dealing with physics problems like this, structured problem-solving helps break down complex concepts:

• Identify: Understand what you know (e.g., formulas) and what you need (e.g., the time period for parts).
• Plan: Use relevant formulas and see which variables need adjustment (like mass and length when the needle is divided).
• Solve: Apply step-by-step calculations, checking each for mistakes.
• Verify: Ensure final answers make logical sense, using known solutions or simplifications.

Approaching physics problems systematically allows for clarity and error minimization, especially when dealing with calculations involving multiple steps and variations.