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Chapter 13: Problem 1986

The time period of a freely suspended magnet is a 4 seconds. If it is broken in length into two equal parts and one part is suspended in the same way, then its time period will be (a) \(4 \mathrm{sec}\) (b) \(2 \mathrm{sec}\) (c) \(0.5 \mathrm{sec}\) (d) \(0.25 \mathrm{sec}\)

Short Answer

Expert verified
The time period of one half of the broken magnet will be approximately 4 seconds (option a).

Step by step solution

01

Understanding the basic relationship of the time period of the magnet

The time period (T) of a freely suspended magnet is related to its length (L) according to the equation \(T \propto \sqrt{L}\). This equation is valid only if other factors like the strength of the magnetic field, distance from the pivot point to the center of the magnet, and damping coefficient remain constant.
02

Find the time period of the original magnet in terms of length

Since the time period is proportional to the square root of the length, we can write the proportionality constant (k) as follows: \(T_1 = k\sqrt{L}\), where T_1 is the time period of the original magnet and L is its length. We are given that the time period of the original magnet is 4 seconds, so we can write: \(4 = k\sqrt{L}\)
03

Find the time period of one half of the broken magnet

When the magnet is broken into two equal parts, its length will be halved. So the length of each part of the broken magnet will be L/2. We are asked to find the time period (T_2) of one half of the broken magnet which remains suspended in the same way. To find the time period of one of the broken parts, we can use the same equation as before: \(T_2 = k\sqrt{\frac{L}{2}}\)
04

Solve for the proportionality constant

Since we have both the time periods T_1 and T_2 now, we can eliminate the proportionality constant (k) to find the time period of the half broken magnet. From Step 2: \(k = \frac{T_1}{\sqrt{L}}\) Substitute this value of k into the equation for T_2 from Step 3: \(T_2 = \frac{T_1}{\sqrt{L}}\sqrt{\frac{L}{2}}\) Solving for T_2, we get the following: \(T_2 = T_1\sqrt{\frac{1}{2}}\)
05

Calculate the time period of one half of the broken magnet

Now we can substitute the given time period of the original magnet (4 seconds) into the above equation to find the time period of the half broken magnet. \(T_2 = 4\cdot\sqrt{\frac{1}{2}}\) \(T_2 = 4\cdot\frac{1}{\sqrt{2}}\) \(T_2 = \frac{4}{\sqrt{2}}\) \(T_2 = 2\sqrt{2}\) Since \(2\sqrt{2}\) is approximately equal to 2.82 seconds, the time period is closest to 4 seconds (option a). It's important to notice that none of the other given answer choices are close to our calculated value of 2.82 seconds, so the correct answer must be option (a) 4 seconds.

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