/*! 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 43 The spring-block system above ha... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 43

The spring-block system above has a block of mass \(m\) oscillating in simple harmonic motion. However, instead of one spring, there are two springs connected to each other. The first spring has a spring constant \(k_{1},\) and the second has a spring constant \(k_{2}\) . What would be the effective spring constant of the system above if \(k_{2}=3 k_{1} ?\) (A) 4\(k_{1}\) (B) 3\(k_{1}\) (C) 4\(k_{1} / 3\) (D) 3\(k_{1} / 4\)

Short Answer

Expert verified
The effective spring constant of the system above is 3\(k_{1}\) / 4. Therefore, the correct answer is (D) 3\(k_{1} / 4\).

Step by step solution

01

Understand the concept of springs in series

When springs are connected in series (one after another), the forces in each spring are the same, whereas the total extension or compression is the sum of the extensions or compressions in each spring. Thus, the effective spring constant can be found using the formula for springs in series.
02

Apply the formula for springs in series

The reciprocal of the total spring constant \(k_{\text{total}}\) is equal to the sum of the reciprocals of the individual spring constants. Mathematically, it can be presented as: \(1/k_{\text{total}} = 1/k_{1} + 1/k_{2}\). In the given problem, it is mentioned that \(k_{2}=3 k_{1}\). This can be substituted into the equation.
03

Calculate the total spring constant

Let's substitute \(k_{2}=3 k_{1}\) into the equation from step 2. So the equation becomes \(1/k_{\text{total}} = 1/k_{1} + 1/(3k_{1})\). To solve for \(k_{\text{total}}\), first find a common denominator and simplify the right-hand side. This yields \(1/k_{\text{total}} = 4/(3k_{1})\). To find \(k_{\text{total}}\), take the reciprocal of both sides, which yields \(k_{\text{total}} = 3k_{1} / 4\).

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Key Concepts

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

Springs in Series
Understanding the mechanics of springs in series is crucial when analyzing compound spring systems. When we have multiple springs attached end to end, as in the exercise, we say that they are in series. A key point to remember is that when springs are connected in this fashion, the force exerted by or on each spring is identical because the system is subject to the same tension or compression force.

When dealing with springs in series, an important thing to note is that the overall extension or compression is the sum of that of each spring. This is similar to adding lengths of string together end-to-end; the total length is simply the sum of individual lengths. In the context of springs, if one spring stretches by 2 cm and another by 3 cm, the total stretch would be 5 cm. However, calculating the combined spring constant is less intuitive, as it is not a straightforward sum and requires using the formula for springs in series as shown in the exercise solution.
Effective Spring Constant
The concept of effective spring constant is a vital piece in understanding how different springs behave when combined. The effective spring constant, denoted as \(k_{\text{total}}\), represents how a system of springs will respond to a force when compared to a single spring.With springs in series, finding the effective spring constant can seem counterintuitive because instead of adding the constants directly, we add their reciprocals. The formula \(1/k_{\text{total}} = 1/k_{1} + 1/k_{2}\) allows us to calculate this combined spring constant.

In our exercise, with \(k_{2}=3 k_{1}\), we substitute these values into the formula and find that the system behaves like a single spring with a spring constant of \(3k_{1} / 4\). It's interesting to note that this effective spring constant is always less than the smallest individual spring constant in the series, making the system less stiff when springs are combined in this way.
Oscillatory Motion
Oscillatory motion is a fundamental aspect of simple harmonic motion, which is the type of movement exhibited by the spring-block system in the exercise. This type of motion is characterized by its periodic nature, as objects move back and forth through an equilibrium position. A classic example of such motion can be observed in pendulums and mass-spring systems.

In the context of the spring-block system, once the block is displaced from its equilibrium and let go, it oscillates due to the restoring force of the springs trying to bring it back to the equilibrium position. This motion can be quantified by parameters like amplitude, period, and frequency. The effective spring constant, when paired with the mass of the block, can be used to determine the period of oscillation, using the formula \(T = 2\pi\sqrt{m/k_{\text{total}}}\), making it clear how the properties of the springs directly affect the oscillatory motion of the system.

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Most popular questions from this chapter

If a ball is kicked at an angle of 30 degrees such that it has an initial velocity \(v\) , it will travel some distance, \(d_{1}\) before falling back to the ground. Another ball is kicked at an angle of 45 degrees so that it also has an initial velocity of \(v,\) and it travels a distance, \(d_{2}\) , before falling back to the ground. How much farther will the second ball travel before striking the ground? (A) \(\frac{v^{2}}{10}(2-\sqrt{2})\) (B) \(\frac{v^{2}}{10}(2-\sqrt{3})\) (C) \(\frac{v^{2}}{20}(2-\sqrt{2})\) (D) \(\frac{v^{2}}{20}(2-\sqrt{3})\)

A piano tuner needs to double the frequency of note that a particular string is playing. Should he/she tighten or loosen the string, and by what factor? (A) Tighten, 4 (B) Tighten, 2 (C) Loosen, 2 (D) Loosen, 4

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Directions: Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part. A car of known mass \(m_{1}\) will collide with a second car of known mass \(m_{2}\). The collision will be head on, and both cars will only move linearly both before and after the collision. In a clear, coherent, paragraph-length response, explain a method for determining whether the collision is perfectly elastic, perfectly inelastic, or neither. If the collision is perfectly inelastic, include at least one possible cause of energy loss.

If a hole were dug through the center of a planet and a ball dropped into the hole, which of the following best describes the motion that the ball would undergo? Assume the ball is indestructible and the planet is a perfect sphere. (A) It would continuously gain speed and eventually escape the gravitational pull of the planet. (B) It would fall to the center of the planet and get stuck there because gravity is always pulling things toward the center of the planet. (C) It would fall to the other end of the hole, come to a momentary stop, fall back to the starting location, and then repeat this back-and-forth motion indefinitely. (D) None of the above is correct.
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