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Chapter 9: Problem 15

The periodic time of a mass suspended by a spring (force constant \(k\) ) is \(T\). If the spring is cut in three equal pieces, the force constant of each part and the periodic time if the same mass is suspended from one piece are (A) \(k, T / \sqrt{3}\) (B) \(3 k, T\) (C) \(3 k, \sqrt{3} T\) (D) \(3 k, T / \sqrt{3}\)

Short Answer

Expert verified
The correct answer is (A) \(k, T / \sqrt{3}\), as the new force constant of each part is \(3k\) and the new time period is \(T / \sqrt{3}\) when the same mass is suspended from one piece.

Step by step solution

01

Write down the formula for the time period of a mass-spring system.

The formula for the time period (T) of a mass-spring system is given by: \(T = 2\pi\sqrt{\frac{m}{k}}\) where m is the mass of the object suspended from the spring, and k is the force constant of the spring.
02

Understand the relationship between a spring's force constant and its length.

The force constant k of a spring is inversely proportional to its length. That is, when a spring is cut into smaller pieces, its force constant increases. If a spring's length changes by a factor of n, its force constant changes by the same factor. So, if we cut the spring into three equal parts (n = 3), the force constant of each part will be 3 times the original force constant: \(k_{new} = 3k\)
03

Find the new time period after cutting the spring.

Now let's use the formula for the time period from Step 1, but with the new force constant: \(T_{new} = 2\pi\sqrt{\frac{m}{k_{new}}}\) Since \(k_{new} = 3k\), we can substitute this value into the formula: \(T_{new} = 2\pi\sqrt{\frac{m}{3k}}\)
04

Express the new time period in terms of the original time period.

Since we know that \(T = 2\pi\sqrt{\frac{m}{k}}\), we can rewrite the original time period formula as \(k = \frac{4\pi^2 m}{T^2}\). Now we substitute this value of \(k\) into the formula for the new time period: \(T_{new} = 2\pi\sqrt{\frac{m}{3\frac{4\pi^2 m}{T^2}}}\) Simplifying this expression, we get: \(T_{new} = T / \sqrt{3}\)
05

Identify the correct option.

Now that we have found the new force constant to be \(3k\) and the new time period to be \(T / \sqrt{3}\), we can see that the correct option is: (A) \(k, T / \sqrt{3}\)

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Most popular questions from this chapter

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