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Chapter 14: Problem 34

A horizontal spring is attached to a 5 kg block. When the block is pulled 5 cm to the right, the restoring force has a magnitude of 6 N. What is the frequency of the spring? (A) 0.32 Hz (B) 0.56 Hz (C) 0.78 Hz (D) 0.98 Hz

Short Answer

Expert verified
The frequency of the spring is 0.78 Hz. Therefore, the correct answer is (C) 0.78 Hz.

Step by step solution

01

Identify the spring constant using Hooke's Law

According to Hooke's law, force \(F\) is equal to the spring constant \(k\) times the extension \(x\) of the spring. The law is represented by the equation \(F = -kx\), where: - \(F\) is the restoring force (-6N in this case, it's negative because it's a restoring force which operates in the opposite direction). - \(k\) is the spring constant- \(x\) is the distance the spring has been stretched or compressed (5 cm or 0.05 m in this case) Our aim here is to solve for \(k\). By rearranging the equation, we get \(k = -F/x\).
02

Calculate the spring constant

Substituting the known values into the equation: \(k = -(-6N)/0.05m = 120 N/m. Therefore, the spring constant \(k\) is 120 N/m.
03

Calculate the angular frequency

The angular frequency \( \omega \) (in rad/s) of the spring is given by the formula \(\omega = \sqrt {k/m}\) where \(m\) is the mass of the block (5 kg in this case). Substituting the known values into the equation gives \(\omega = \sqrt{ 120 N/m / 5 kg} = 4.9 rad/s\).
04

Calculate the frequency

Frequency \( f \) is related to the angular frequency by the following relationship: \(f = \omega / 2 \pi \). Substituting in the known values, we obtain \(f = 4.9 rad/s / 2 \pi = 0.78 Hz.\)

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Key Concepts

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

Hooke's Law
Understanding Hooke's Law is foundational for analyzing the behavior of springs and other elastic materials. It states that the force exerted by a spring is proportional to its displacement from the equilibrium position. Mathematically, it is expressed as
\[ F = -kx \],
where F represents the restoring force exerted by the spring, \

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Most popular questions from this chapter

Questions 1-3 refer to the following scenario: IMAGE IS NOT AVAILABLE TO COPY An explorer travels 30 \(\mathrm{m}\) east, then 20\(\sqrt{2} \mathrm{m}\) in a direction \(45^{\circ}\) south of east, and then 140 \(\mathrm{m}\) north. The explorer took \(60 \mathrm{s}, 130 \mathrm{s}\) , and 70 \(\mathrm{s}\) to travel the \(30 \mathrm{m}, 2 \mathrm{O} \sqrt{2} \mathrm{m},\) and 140 \(\mathrm{m}\) north distances, respectively. What is the average velocity of the explorer over the total distance traveled? (A) 0.50 \(\mathrm{m} / \mathrm{s}\) (B) 33.3 \(\mathrm{m} / \mathrm{min}\) (C) 0.76 \(\mathrm{m} / \mathrm{s}\) (D) 100 \(\mathrm{m} / \mathrm{min}\)

A student presses a 0.5 \(\mathrm{kg}\) book against the wall. If the \(\mu_{\mathrm{s}}\) between the book and the wall is \(0.2,\) what force must the student apply to hold the book in place? (A) 0 \(\mathrm{N}\) (B) 15 \(\mathrm{N}\) (C) 25 \(\mathrm{N}\) (D) 35 \(\mathrm{N}\)

A rocket is launched into the air. A few moments after liftoff, the rocket is 40 \(\mathrm{m}\) above the ground. After another 5 \(\mathrm{s}\) , the rocket is now 200 \(\mathrm{m}\) off the ground. What is the average velocity of the rocket during the 5 s part of the flight? (A) 16 \(\mathrm{m} / \mathrm{s}\) (B) 32 \(\mathrm{m} / \mathrm{s}\) (C) 48 \(\mathrm{m} / \mathrm{s}\) (D) 64 \(\mathrm{m} / \mathrm{s}\)

As a train approaches the next stop, the conductor blows the horn and applies the breaks on the train. What will a person standing at the train stop hear as the train approaches? (A) A decrease in the frequency and an increase in the intensity of the horn (B) An increase in the frequency and a decrease in the intensity of the horn (C) A decrease in both the frequency and intensity of the horn (D) An increase in both the frequency and intensity of the horn

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