/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1988 A thin magnetic needle oscillate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 1988

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

Expert verified
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

01

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
02

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
03

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
04

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
05

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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 鈥� 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鈥檚 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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A small bar magnet of moment \(\mathrm{M}\) is placed in a uniform field of \(\mathrm{H}\). If magnet makes an angle of \(30^{\circ}\) with field, the torque acting on the magnet is (a) \(\mathrm{MH}\) (b) \((\mathrm{MH} / 2)\) (c) \((\mathrm{MH} / 3)\) (d) \((\mathrm{MH} / 4)\)

In the case of bar magnet, lines of magnetic induction (a) Start from the North pole and end at the South pole (b) Run continuously through the bar and outside (c) Emerge in circular paths from the middle of the bar (d) Are produced only at the North pole like rays of light from a bulb.

A long solenoid has 200 turns per \(\mathrm{cm}\) and carries a current of \(2.5 \mathrm{Amp}\). The mag. field at its centre is tesla. (a) \(\pi \times 10^{-2}\) (b) \(2 \pi \times 10^{-2}\) (c) \(3 \pi \times 10^{-2}\) (d) \(4 \pi \times 10^{-2}\)

A long straight wire carrying current of \(30 \mathrm{Amp}\) is placed in an external uniform mag. field of induction \(4 \times 10^{-4}\) tesla. The mag. field is acting parallel to the direction of current. The magnitude of the resultant magnetic induction in tesla at a point \(2 \mathrm{~cm}\) away from the wire is tesla. (a) \(10^{-4}\) (c) \(5 \times 10^{-4}\) (b) \(3 \times 10^{-4}\) (d) \(6 \times 10^{-4}\)

If a magnet of pole strength \(\mathrm{m}\) is divided into four parts such that the length and width of each part is half that of initial one, then the pole strength of each part will be (a) \((\mathrm{m} / 4)\) (b) \((\mathrm{m} / 2)\) (c) \((\mathrm{m} / 8)\) (d) \(4 \mathrm{~m}\)
See all solutions

Recommended explanations on English Textbooks

Discourse

Read Explanation

Multiple Choice Questions

Read Explanation

Single Paragraph Essay

Read Explanation

Syntax

Read Explanation

Lexis and Semantics

Read Explanation

Morphology

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.