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Chapter 10: Problem 38

Suppose that the block weighs \(w\) pounds and vibrates with a period of 3 s when it is pulled below the equilibrium position and released. Suppose also that if the process is repeated with an additional 4 lb of weight, then the period is 5 s. (a) Find the spring constant. (b) Find \(w\)

Short Answer

Expert verified
Spring constant \( k = \frac{g}{\pi^2} \); Weight \( w = 9 \text{ lb} \).

Step by step solution

01

Understand the Problem

We have a block connected to a spring. The system vibrates with a period that changes when the weight is modified. We need to find the spring constant and the original weight of the block.
02

Use Formula for Period of a Spring System

The period of a spring-mass system is given by the formula: \[ T = 2\pi \sqrt{\frac{m}{k}} \]where \( T \) is the period, \( m \) is the mass, and \( k \) is the spring constant. Convert weights to mass using the relation \( m = \frac{w}{g} \), where \( g \) is the acceleration due to gravity.
03

Apply to First Condition

For the first condition, where the period is 3 seconds and weight is \( w \), we have:\[ 3 = 2\pi \sqrt{\frac{w/g}{k}} \]Simplify this to:\[ \frac{9g}{4\pi^2} = \frac{w}{k} \]
04

Apply to Second Condition

For the second condition, where the period is 5 seconds and weight is \( w + 4 \) lb, we have:\[ 5 = 2\pi \sqrt{\frac{(w+4)/g}{k}} \]Simplify this to:\[ \frac{25g}{4\pi^2} = \frac{w+4}{k} \]
05

Solve Equations for Spring Constant \( k \)

We have two equations:1. \( \frac{9g}{4\pi^2} = \frac{w}{k} \)2. \( \frac{25g}{4\pi^2} = \frac{w+4}{k} \)Subtract the first equation from the second:\[ \frac{16g}{4\pi^2} = \frac{4}{k} \]Solve for \( k \):\[ k = \frac{g}{\pi^2} \]
06

Use Spring Constant to Find Weight \( w \)

Substitute \( k = \frac{g}{\pi^2} \) back into one of the equations. Using the first equation:\[ \frac{9g}{4\pi^2} = \frac{w}{k} \]\[ \frac{9}{4\pi^2} = \frac{w}{g/\pi^2} \]Solve for \( w \):\[ w = 9 \text{ lb} \]

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Key Concepts

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

Spring Constant
In a spring-mass system, the spring constant, represented by the symbol \( k \), plays a crucial role. It measures the stiffness of a spring. A higher spring constant means a stiffer spring, requiring more force to stretch or compress. To find the spring constant in the exercise, we use the equation for the period of a spring-mass system: \[ T = 2\pi \sqrt{\frac{m}{k}} \]Here, \( T \) is the period, \( m \) is the mass, and \( k \) is our spring constant.
  • The relationship is derived from Hooke's Law, which states that the force exerted by a spring is proportional to its displacement.
  • In this specific case, we found \( k \) by solving simultaneous equations based on different periods.
  • The determined spring constant was \( \frac{g}{\pi^2} \), where \( g \) is the acceleration due to gravity.
Weight and Period Relationship
The weight of the block and the period of oscillation are interconnected in a spring-mass system. The period of the spring's vibration changes when the weight, or mass, attached to it varies.
  • Lighter weights result in shorter periods because there is less mass for the spring to move.
  • Heavier weights have longer periods, indicating more mass requires more time to complete an oscillation.
In this exercise, varying the weight affected the period significantly:- With an initial weight \( w \), the period was 3 seconds.- Adding an additional 4 lb increased the period to 5 seconds.This change illustrates how sensitive the period is to weight changes.
Harmonic Motion
The concept of harmonic motion is fundamental to understanding spring-mass systems. Harmonic motion refers to periodic movements where the restoring force is proportional to displacement, following Hooke鈥檚 Law.
  • A classic example is a block attached to a spring, as in this exercise.
  • The system oscillates back and forth around an equilibrium or rest position.
  • This motion is predictable and can be modeled mathematically.
The block in our study moved in simple harmonic motion, described by its constant amplitude and a period affected by changes in weight.
Mass-Spring Equation
The mass-spring equation is a mathematical model capturing the dynamics of spring-mass systems. It relates the spring constant, mass, and period of oscillation, providing insights into the system's behaviors.For this exercise, the relevant equation was:\[ T = 2\pi \sqrt{\frac{m}{k}} \]
  • This equation tells us how the mass \( m \) and spring constant \( k \) affect the period \( T \).
  • The mass must be converted from weight using the relation \( m = \frac{w}{g} \), where \( g \) is gravity.
Understanding this equation allows us to predict how changes in mass or spring stiffness can affect the oscillation period.

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Most popular questions from this chapter

Suppose that a quantity \(y\) has an exponential growth model \(y=y_{0} e^{t t}\) or an exponential decay model \(y=y_{0} e^{-k t},\) and it is known that \(y=y_{1}\) if \(t=t_{1}\). In each case find a formula for \(k\) in terms of \(y_{0} . y_{1},\) and \(t_{1},\) assuming that \(t_{1} \neq 0\)

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of \(x\) \(y^{\prime}=-x y\)

Sketch the direction field for \(y^{\prime}+y=2\) at the gridpoints \((x, y),\) where \(x=0,1, \ldots, 4\) and \(y=0,1, \ldots, 4\)

Radon-222 is a radioactive gas with a half-life of 3.83 days. This gas is a health hazard because it tends to get trapped in the basements of houses, and many health officials suggest that homeowners seal their basements to prevent entry of the gas. Assume that \(5.0 \times 10^{7}\) radon atoms are trapped in a basement at the time it is sealed and that \(y(t)\) is the number of atoms present \(t\) days later. (a) Find an initial-value problem whose solution is \(y(t)\) (b) Find a formula for \(y(t)\) (c) How many atoms will be present after 30 days? (d) How long will it take for \(90 \%\) of the original quantity of gas to decay?

Use Euler's Method with the given step size \(h\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$d y / d t=\sin y, \quad y(0)=1,0 \leq t \leq 2, h=0.5$$
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