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Chapter 8: Problem 10

In the equilibrium position, the 30 -kg cylinder causes a static deflection of \(50 \mathrm{mm}\) in the coiled spring. If the cylinder is depressed an additional \(25 \mathrm{mm}\) and released from rest, calculate the resulting natural frequency \(f_{n}\) of vertical vibration of the cylinder in cycles per second (Hz).

Short Answer

Expert verified
The natural frequency of the system is approximately 2.23 Hz.

Step by step solution

01

Determine the Spring Constant

To find the spring constant, we use the formula for static deflection caused by the cylinder's weight. The weight of the cylinder is given by \( W = mg \), where \( m = 30 \mathrm{kg} \) and \( g = 9.81 \mathrm{m/s}^2 \). Thus, \( W = 30 \times 9.81 = 294.3 \mathrm{N} \). The spring deflects by \(50 \mathrm{mm} = 0.05 \mathrm{m}\), so using \( k = \frac{W}{\Delta} \), we have \( k = \frac{294.3}{0.05} = 5886 \mathrm{N/m} \).
02

Use the Formula for Natural Frequency

The formula for the natural frequency of a mass-spring system is \( f_{n} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \). Using the spring constant \( k = 5886 \mathrm{N/m} \) and mass \( m = 30 \mathrm{kg} \), calculate \( \frac{k}{m} = \frac{5886}{30} = 196.2 \).
03

Calculate the Natural Frequency

Plug \( \frac{k}{m} \) into the natural frequency formula: \( f_{n} = \frac{1}{2\pi} \sqrt{196.2} \). Calculate \( \sqrt{196.2} \approx 14.0 \). Finally, \( f_{n} = \frac{1}{2\pi} \times 14.0 \approx 2.23 \mathrm{Hz} \).

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Key Concepts

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

Mass-Spring System
A mass-spring system is a mechanical model that shows how a mass attached to a spring behaves when a force is applied. This system is foundational in physics because it can represent many physical situations ranging from small-scale molecular vibrations to large-scale seismic activities. Here's how it works:
  • The mass in the system is the moving component; in our exercise, the 30 kg cylinder represents this mass.
  • The spring offers a restoring force that attempts to bring the mass back to its equilibrium position when displaced.
When the cylinder is pushed down and released, it oscillates up and down around its equilibrium position. This oscillation follows a simple harmonic motion, which is crucial to understand natural vibrations. By representing real-world systems in this simple model, physics and engineering can analyze and predict the behavior and response of more complex structures.
Spring Constant
The spring constant, represented as "k," is a measure of a spring's stiffness. It tells us how much force is needed to cause the spring to deflect, or stretch. A higher spring constant means a stiffer spring that requires more force to compress or extend, while a lower spring constant means the spring is more flexible.
  • The spring constant formula is given by: \( k = \frac{F}{\Delta} \)
  • In the exercise, \( k = 5886 \mathrm{N/m} \) was calculated using the weight of the cylinder (294.3 N) and the static deflection (0.05 m).
This concept plays a vital role in determining the natural frequency of the system. A stiffer spring (higher k) results in a higher natural frequency, meaning it oscillates faster. Understanding the spring constant is essential for designing systems that require precise control over oscillation characteristics, such as watches or car suspensions.
Static Deflection
Static deflection is the displacement of a spring when a force, such as the weight of an attached object, is applied to it. In the mass-spring system, this force exerts a downward pull due to gravity, causing the spring to extend or compress to a new equilibrium point.
  • In our example, the 30 kg cylinder causes a static deflection of 50 mm for the spring when at rest.
  • This deflection helps to determine the spring constant by showing how much change the spring undergoes under a specific weight.
Utilizing static deflection, engineers can design the system to ensure it stays within desired operational limits. It helps to guarantee that the natural frequency remains tolerable to prevent resonance, which could lead to mechanical failure. By understanding the relationship between static deflection, spring constant, and mass, engineers can make informed decisions in design and analysis.

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Most popular questions from this chapter

Determine the value of the viscous damping coefficient \(c\) for which the system is critically damped. The cylinder mass is \(m=2 \mathrm{kg}\) and the spring constant is \(k=150 \mathrm{N} / \mathrm{m} .\) Neglect the mass and friction of the pulley.

The instrument shown has a mass of \(43 \mathrm{kg}\) and is spring-mounted to the horizontal base. If the amplitude of vertical vibration of the base is \(0.10 \mathrm{mm}\) calculate the range of frequencies \(f_{n}\) of the base vibration which must be prohibited if the amplitude of vertical vibration of the instrument is not to exceed \(0.15 \mathrm{mm}\). Each of the four identical springs has a stiffness of \(7.2 \mathrm{kN} / \mathrm{m}\)

The seismic instrument shown is attached to a structure which has a horizontal harmonic vibration at \(3 \mathrm{Hz}\). The instrument has a mass \(m=0.5 \mathrm{kg},\) a spring stiffness \(k=20 \mathrm{N} / \mathrm{m},\) and a viscous damping coefficient \(c=3 \mathrm{N} \cdot \mathrm{s} / \mathrm{m} .\) If the maximum recorded value of \(x\) in its steady-state motion is \(X=2 \mathrm{mm}\) determine the amplitude \(b\) of the horizontal movement \(x_{B}\) of the structure.

The semicircular cylindrical shell of radius \(r\) with small but uniform wall thickness is set into small rocking oscillation on the horizontal surface. If no slipping occurs, determine the expression for the period \(\tau\) of each complete oscillation.

During the design of the spring-support system for the 4000 -kg weighing platform, it is decided that the frequency of free vertical vibration in the unloaded condition shall not exceed 3 cycles per second. (a) Determine the maximum acceptable spring constant \(k\) for each of the three identical springs. (b) For this spring constant, what would be the natural frequency \(f_{n}\) of vertical vibration of the platform loaded by the 40 -Mg truck?
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