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Chapter 4: Problem 436

Natural length of a spring is \(60 \mathrm{~cm}\), and its spring constant is \(2000 \mathrm{~N} / \mathrm{m}\). A mass of \(20 \mathrm{~kg}\) is hung from it. The extension produced in the spring is..... $\left(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)$ (A) \(4.9 \mathrm{~cm}\) (B) \(0.49 \mathrm{~cm}\) (C) \(9.8 \mathrm{~cm}\) (D) \(0.98 \mathrm{~cm}\)

Short Answer

Expert verified
The extension produced in the spring is (C) \(9.8 \mathrm{~cm}\).

Step by step solution

01

Convert units

First, it's important to have all units in the same system. Here, we need to convert the natural length of the spring from centimeters to meters. Natural length: \(60 \mathrm{~cm} = 0.6 \mathrm{~m}\)
02

Calculate the force due to gravity on the mass

To find the force acting on the spring due to gravity, we use the formula: Force (\(F\)) = mass (\(m\)) * gravity (\(g\)) Given mass: \(20 \mathrm{~kg}\) and gravity: \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) The force acting on the spring due to mass: \(F = 20 \mathrm{~kg} \times 9.8 \mathrm{~m} / \mathrm{s}^{2} = 196 \mathrm{~N}\)
03

Apply Hooke's Law to find the extension

Hooke's Law states that the force acting on a spring is directly proportional to the extension produced in the spring: \(F = k \times x\) Where: - \(F\) is the force acting on the spring (\(196 \mathrm{~N}\)) - \(k\) is the spring constant (\(2000 \mathrm{~N} / \mathrm{m}\)) - \(x\) is the extension produced in the spring (the quantity we need to find) Now, rearrange the equation to solve for the extension \(x\): \(x = \frac{F}{k}\) Plug in the given values of the force and spring constant: \(x = \frac{196 \mathrm{~N}}{2000 \mathrm{~N} / \mathrm{m}}\)
04

Solve for the extension

Perform the division to find the extension: \(x = 0.098 \mathrm{~m}\) Now, convert the extension back to centimeters: Extension: \(0.098 \mathrm{~m} = 9.8 \mathrm{~cm}\)
05

Match the answer to the given options

The calculated extension is \(9.8 \mathrm{~cm}\), which matches option (C). Therefore, the correct answer is: (C) \(9.8 \mathrm{~cm}\)

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Most popular questions from this chapter

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