/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 136 The nest of two springs is used ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 136

The nest of two springs is used to bring the 0.5 -kg plunger \(A\) to a stop from a speed of \(5 \mathrm{m} / \mathrm{s}\) and reverse its direction of motion. The inner spring increases the deceleration, and the adjustment of its position is used to control the exact point at which the reversal takes place. If this point is to correspond to a maximum deflection \(\delta=200 \mathrm{mm}\) for the outer spring, specify the adjustment of the inner spring by determining the distance \(s .\) The outer spring has a stiffness of \(300 \mathrm{N} / \mathrm{m}\) and the inner one a stiffness of \(150 \mathrm{N} / \mathrm{m}\).

Short Answer

Expert verified
The inner spring must be set to a deflection of approximately 57.7 mm.

Step by step solution

01

Define the Energy Conservation Principle

To solve the problem, we use the principle of energy conservation. The initial kinetic energy of the plunger is converted into potential energy stored within the springs. This can be expressed by the equation \( KE_{initial} = PE_{outer} + PE_{inner} \), where \( KE_{initial} = \frac{1}{2} m v^2 \).
02

Calculate Initial Kinetic Energy

Calculate the initial kinetic energy of the 0.5 kg plunger moving at 5 m/s. The formula is \( KE = \frac{1}{2} m v^2 \). Substitute the given variables: \( KE = \frac{1}{2} \cdot 0.5 \cdot 5^2 = 6.25 \text{ J} \).
03

Calculate Potential Energy in Outer Spring

The potential energy in the outer spring is given by the formula \( PE_{outer} = \frac{1}{2} k_{outer} \delta^2 \). Here \( k_{outer} = 300 \text{ N/m} \) and \( \delta = 0.2 \text{ m} \). Thus, \( PE_{outer} = \frac{1}{2} \cdot 300 \cdot 0.2^2 = 6 \text{ J} \).
04

Calculate Potential Energy in Inner Spring

Since the total energy must equal the sum of the potential energies, use \( PE_{inner} = KE_{initial} - PE_{outer} \). Substitute the known values: \( PE_{inner} = 6.25 - 6 = 0.25 \text{ J} \).
05

Determine Inner Spring Deflection

Use the potential energy equation for the inner spring: \( PE_{inner} = \frac{1}{2} k_{inner} s^2 \). Given \( k_{inner} = 150 \text{ N/m} \), solve for \( s \). Substitute \( PE_{inner} = 0.25 \): \( 0.25 = \frac{1}{2} \cdot 150 \cdot s^2 \). Thus, \( s^2 = \frac{0.25 \times 2}{150} = \frac{0.5}{150} = \frac{1}{300} \).
06

Calculate the Deflection of Inner Spring

Calculate \( s \) by taking the square root of the expression: \( s = \sqrt{\frac{1}{300}} \). This simplifies to \( s = \sqrt{0.00333} \approx 0.0577 \text{ m} \text{ or } 57.7 \text{ mm} \).

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy associated with the motion of an object. In this mechanics problem, we consider the kinetic energy of a moving plunger. The plunger has a mass of 0.5 kg and is moving at a speed of 5 m/s. To find the kinetic energy, we use the formula: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass and \( v \) is the velocity. For the provided values, the calculation will be: \[ KE = \frac{1}{2} \times 0.5 \times 5^2 = 6.25 \text{ J} \] This means the plunger has 6.25 Joules of kinetic energy initially. As the plunger interacts with the springs, this energy converts to potential energy, demonstrating energy conservation principles. It鈥檚 crucial to see how motion energy can be temporarily stored and used later during this energy transformation.
Potential Energy
Potential energy is the stored energy that an object has due to its position. In the context of this mechanics problem, potential energy is stored in the springs when they are compressed or stretched. We have two springs: the outer spring and the inner spring, each with its potential energy formula.The potential energy for a spring is calculated by \[ PE = \frac{1}{2} k x^2 \] where \( k \) is the spring constant and \( x \) is the deflection.For the outer spring with a constant \( k_{outer} = 300 \) N/m and a maximum deflection \( \delta = 0.2 \) m, the potential energy is: \[ PE_{outer} = \frac{1}{2} \times 300 \times 0.2^2 = 6 \text{ J} \]The inner spring contributes to the potential energy with its specific deflection to be determined, showing the balance needed in a mechanics problem to achieve a desired endpoint.
Spring Constant
A spring constant is a measure of a spring's stiffness, indicated by \( k \) in equations. In simpler terms, it tells you how much force is needed to stretch or compress a spring by a unit of distance. The spring constant is crucial in determining the potential energy stored in a spring when it is either compressed or extended.In our example, the spring constant for the outer spring is 300 N/m, and for the inner spring, it is 150 N/m. These values directly influence how much energy each spring can store and thus play a vital role in solving the mechanics problem effectively. Remember, a higher spring constant means a stiffer spring that stores more energy for the same amount of deflection, and this understanding helps adjust the inner spring to achieve the desired adjustment.
Mechanics Problem-Solving
Mechanics problem-solving involves applying principles like energy conservation to find unknown values in physics problems. This problem requires understanding the energy transformations from kinetic to potential forms.The process starts by calculating the kinetic energy and acknowledging that as the object interacts with the springs, energy converts seamlessly into potential energy. Using the given spring constants and desired outcomes, potential energy formulas for the springs allow us to find missing parameters.In this exercise, you work step-by-step:
  • Calculate initial kinetic energy.
  • Determine potential energy in the outer spring.
  • Deduct from total energy to find what's in the inner spring.
  • Utilize these insights, following energy conservation principles, to solve for unknowns (like the inner spring's adjustment \( s \)).
Such systematic approaches are indispensable in dissecting and comprehending intricate mechanics scenarios.

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Most popular questions from this chapter

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