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Chapter 8: Problem 35

Show that if \(c_{1}\) and \(c_{2}\) are any constants, the function $$x=x(t)=c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right)+c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right)$$ is a solution to the differential equation for the vibrating spring. (The corresponding motion of the spring is referred to as simple harmonic motion.)

Short Answer

Expert verified
The function satisfies the vibrating spring differential equation.

Step by step solution

01

Differentiate the function

First, let's find the first derivative of the function with respect to time, \( t \). Here, we apply the chain rule and remember that the derivative of cosine is negative sine and that of sine is cosine:\[\frac{dx}{dt} = \frac{d}{dt}\left(c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right)\right)\]\[= c_{1} \cdot \left(-\sqrt{\frac{k}{m}} \sin \left(\sqrt{\frac{k}{m}} t\right)\right) + c_{2} \cdot \left(\sqrt{\frac{k}{m}} \cos \left(\sqrt{\frac{k}{m}} t\right)\right)\]\[= -c_{1} \sqrt{\frac{k}{m}} \sin \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sqrt{\frac{k}{m}} \cos \left(\sqrt{\frac{k}{m}} t\right)\]
02

Differentiate again for the second derivative

Now, we differentiate the first derivative to find the second derivative \( \frac{d^2x}{dt^2} \). Continue using the chain rule:\[\frac{d^2x}{dt^2} = \frac{d}{dt}\left(-c_{1} \sqrt{\frac{k}{m}} \sin \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sqrt{\frac{k}{m}} \cos \left(\sqrt{\frac{k}{m}} t\right)\right)\]\[= -c_{1} \left(\sqrt{\frac{k}{m}}\right)^2 \cos \left(\sqrt{\frac{k}{m}} t\right) - c_{2} \left(\sqrt{\frac{k}{m}}\right)^2 \sin \left(\sqrt{\frac{k}{m}} t\right)\]\[= -\frac{k}{m}\left( c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right) \right)\]
03

Substitute back into the differential equation

The given differential equation for a spring in simple harmonic motion is:\[m\frac{d^2x}{dt^2} + kx = 0\]Substitute \( \frac{d^2x}{dt^2} \) calculated in Step 2 and the function \( x(t) \).We have:\[m\left(-\frac{k}{m}\left( c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right) \right)\right) + k\left(c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right)\right) = 0\]This simplifies to:\[-k\left( c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right) \right) + k\left( c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right) \right) = 0\]So the equation is satisfied as both terms cancel each other out to zero.
04

Conclusion

The solution \( x(t) = c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right) \) satisfies the differential equation \( m\frac{d^2x}{dt^2} + kx = 0 \), demonstrating that this function is indeed a solution for the simple harmonic motion of a vibrating spring.

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Key Concepts

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

Differential Equation
A differential equation is a mathematical equation that relates a function with its derivatives. In the context of simple harmonic motion, these equations describe how physical quantities, such as displacement, velocity, and acceleration, change over time.

For a vibrating spring system, the classic differential equation is given as \[ m \frac{d^2x}{dt^2} + kx = 0 \].
  • \(m\) represents the mass of the object attached to the spring.
  • \(k\) is the spring constant, a measure of the spring's stiffness.
  • \(\frac{d^2x}{dt^2}\) is the second derivative of displacement, i.e., acceleration.
  • \(x\) denotes the displacement from the equilibrium position.
This equation arises from applying Newton鈥檚 second law of motion. It states that the net force acting on a system is equal to mass times acceleration. Here, the force is due to the spring's restoring force, \(-kx\), opposing the displacement.
Vibrating Spring
A vibrating spring, commonly exemplified in simple harmonic motion, behaves in a predictable, periodic manner.

In simple harmonic motion:
  • The system oscillates around an equilibrium point.
  • The restoring force is proportional to the displacement and acts in the opposite direction.
  • Examples of such systems include the mass-spring system, pendulums, and tuning forks.
The equation \(x(t) = c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right)\) describes the position \(x\) as a function of time \(t\), where \(c_1\) and \(c_2\) are constants determined by initial conditions.

This solution corresponds to the physical observation that a spring's motion repeats in sinusoidal patterns, described mathematically by sine and cosine functions.
Second Derivative
The second derivative, often denoted as \(\frac{d^2x}{dt^2}\), represents the acceleration in context of motion and is a vital component in solving differential equations related to harmonic motion.

The importance of the second derivative in a vibrating spring system includes:
  • It helps determine how the velocity of the object is changing.
  • By finding the second derivative of the position function \(x(t)\), we learn how quickly the spring is accelerating toward or away from the equilibrium position.
  • In the given problem, differentiating the position function twice led to the expression \(-\frac{k}{m} \left( c_{1} \cos \left(\sqrt{\frac{k}{m}} t\right) + c_{2} \sin \left(\sqrt{\frac{k}{m}} t\right) \right)\), crucial for substituting back into the original differential equation.
Therefore, the second derivative not only represents physical acceleration but also confirms the mathematical validity of the spring鈥檚 motion within the differential equation framework.

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Most popular questions from this chapter

Use Euler's Method with the given step size \(\Delta x\) or \(\Delta t\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$ d y / d t=e^{-y}, y(0)=0,0 \leq t \leq 1, \Delta t=0.1 $$

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

True-False Determine whether the statement is true or false. Explain your answer. If \(p(y)\) is a cubic polynomial in \(y\), then the slope field \(d y / d x=p(y)\) has an integral curve that is a horizontal line.

(a) Suppose that a particle moves along an \(s\) -axis in such a way that its velocity \(v(t)\) is always half of \(s(t)\). Find a differential equation whose solution is \(s(t)\). (b) Suppose that an object moves along an \(s\) -axis in such a way that its acceleration \(a(t)\) is always twice the velocity. Find a differential equation whose solution is \(s(t)\).

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?
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