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Chapter 20: Problem 42

A bar magnet is oscillating in the Earth's magnetic field with a period \(T\). What happens to its period of motion if its mass is quadrupled? (a) Motion remains SHM with time period \(=T / 2\) (b) Motion remains SHM and period remains nearly constant (c) Motion temains SHM with time period \(=2 T\) (d) Motion remains SHM with time period = \(4 \mathrm{~T}\)

Short Answer

Expert verified
The period of the motion becomes \(2T\), so the correct option is (c).

Step by step solution

01

Understanding the System

A bar magnet oscillating in the Earth's magnetic field can be treated as a physical pendulum or a magnetic dipole oscillating due to the magnetic torque. The period of oscillation for such a system can be analyzed using the principles of simple harmonic motion (SHM).
02

Formula for Period of a Magnetic Oscillator

The period of oscillation for a magnetic dipole in a uniform magnetic field can be given by the formula: \[ T = 2\pi \sqrt{\frac{I}{mB}} \]where \(I\) is the moment of inertia of the bar magnet, \(m\) is the magnetic moment, and \(B\) is the magnetic field strength.
03

Moment of Inertia and Mass Relationship

For a bar magnet, the moment of inertia \(I\) is proportional to its mass \(M\) and the square of its length \(L\): \[ I = \frac{M L^2}{12} \]Since we are quadrupling the mass, the new moment of inertia becomes \(I' = \frac{4M L^2}{12} = \frac{1}{3} \cdot 4ML^2 = 4I\).
04

Analyzing the Effect on Period

Substitute the new moment of inertia \(I' = 4I\) back into the formula for the period:\[ T' = 2\pi \sqrt{\frac{4I}{mB}} = 2\pi \cdot 2\sqrt{\frac{I}{mB}} = 2T \]Therefore, the new period \(T'\) becomes twice the original period \(T\).
05

Conclusion

The period of the bar magnet's motion increases with the square root of the change in mass. Quadrupling the mass results in doubling the period of oscillation. Hence, the motion remains SHM with a new period of \(2T\).

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Key Concepts

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

Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion, where an object oscillates back and forth around an equilibrium position. This kind of motion is common in physics and occurs in many systems such as pendulums, springs, and magnetic oscillators. SHM follows a predictable pattern of movement.
For SHM, two key characteristics are involved:
  • A restoring force that aims to bring the object back to its equilibrium position. This force is proportional to the displacement of the object and acts in the opposite direction, following the formula \( F = -kx \) where \( k \) is the force constant.
  • The motion is sinusoidal in nature, meaning it can be represented by sine or cosine functions, showing smooth and continuous oscillation.
Understanding SHM is crucial as it helps predict the behavior of oscillating systems and calculate key parameters like frequency and period.
Period of Oscillation
The period of oscillation is the time it takes for an oscillating system to complete one full cycle of motion. It is denoted by \( T \) and in the context of SHM, it depends on factors such as the mass of the system and the force constant. Particularly for a bar magnet oscillating in a magnetic field, the period \( T \) is derived from:
  • The moment of inertia \( I \), indicating how the mass is distributed relative to the pivot point.
  • The magnetic moment \( m \), representing the strength and orientation of the magnetic field interaction.
  • The ambient magnetic field strength \( B \).
The formula \( T = 2\pi \sqrt{\frac{I}{mB}} \) reveals how these factors interplay, with an increase or decrease in any element affecting how long it takes for one oscillation to occur. In our problem, quadrupling the mass changes the period to \( 2T \), illustrating the direct relationship between mass distribution and oscillation time.
Magnetic Oscillator
A magnetic oscillator is a system where magnetic properties govern the oscillation behavior. Think of it as a bar magnet swinging in a stable magnetic field, like Earth's. This creates a torque that causes oscillations, similar to how gravity causes a pendulum to swing.
The core components are:
  • The magnetic moment, which plays a crucial role in magnetic torque and oscillation amplitude.
  • The external magnetic field, determining the stability and frequency of the oscillation.
  • The magnetic torque, calculated by \( \tau = m \times B \), which is the force that returns the bar magnet to its equilibrium state.
Understanding a magnetic oscillator helps in applications like measuring magnetic fields or creating stable oscillation systems using magnets. These oscillators exhibit SHM when undamped, illustrating the harmonic nature of magnetic interactions.
Moment of Inertia in Physics
The moment of inertia is a measure of an object's resistance to rotational acceleration around an axis. In the case of a rigid body like a bar magnet, it depends on the object's mass and the distribution of that mass relative to the pivot point.
For a rectangular bar magnet with length \( L \), its moment of inertia \( I \) is given by the formula \( I = \frac{M L^2}{12} \), where \( M \) is the mass.
  • When the mass is increased, the moment of inertia also increases. Specifically, quadrupling the mass results in a moment of inertia of \( 4I \), as calculated in the original exercise.
  • This impacts the period of oscillation since a higher moment of inertia correlates with a longer period, showing how mass distribution affects rotational dynamics.
By grasping moment of inertia, students can understand more complex dynamics such as why heavier or larger objects spin slower around a pivot compared to lighter or smaller objects.

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Most popular questions from this chapter

If a diamagnetic subatance is brought near the north. or the south pole of a bar magnet, it is [Karmatalea CET 2009 ] [a] attracted by both the poles (b) repelled by both the poles (c) repelled by the north pole and attracted by the south pole (d) attracted by the north pole and repelled by the south pole

A magnet freely suspended in a vibration magnetometer makes 40 oscillations per minute at a place \(A\) and 20 oscillations per minute at a place \(B\). If the horizontal component of earth's magnetic field at \(A\) is \(36 \times 10^{-6} \mathrm{~T}\), then its value at \(B\) is (a) \(36 \times 10^{-6} \mathrm{~T}\) (b) \(9 \times 10^{-6} \mathrm{~T}\) (c) \(144 \times 10^{-6} \mathrm{~T}\) (d) \(228 \times 10^{-6} \mathrm{~T}\)

Assertion-Reason type. Each of these contains two Statements: Statement 1 (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion. (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion. (c) If Assertion is true but Reason is false. (d) If Assertion is false but the Reason is true. Assertion Magnetic dipole possesses maximum potential energy when magnetie moment and magnetic field are parallel to each other. Reason Current loop is treated as a magnetic dipole.

Magnetic field of earth is identical to magnetic field of a giant magnet held \(20^{\circ}\) west of geographic \(N-\mathrm{S}\) at the centre of earth. At equator, horizontal component of earth is 0.32 G. Vertical component can be calculated from the relation \(V=H \tan \delta\), where \(\delta\) is angle of dip at the place. The value of \(\delta=0^{\circ}\) at equator and \(8=90^{\circ}\) at poles. What is the order of magnetic declination at a place on earth? (a) 20 - East (b) 10 - East (c) \(20^{\circ}\) West (d) \(10^{-}\)West

A magnet performs 10 oscillations per minute in a horizontal plane at a place where the angle of dip is \(45^{\circ}\) and the total intensity is \(0.707\) CGS units. The number of oscillations per minute at a place where dip angle is \(60^{\circ}\) and total intensity is \(0.5\) CGS units will be (a) 5 (b) 7 (c) 9 (d) 11
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