Problem 34 - Algebra and Trigonometry

Question

The frequency of oscillation of an object suspended on a spring depends on the stiffness \(k\) of the spring (called the spring constant) and the mass \(m\) of the object. If the spring is compressed a distance \(a\) and then allowed to oscillate, its displacement is given by $$f(t)=a \cos \sqrt{k / m} t$$ (a) A 10 -g mass is suspended from a spring with stiffness \(k=3 .\) If the spring is compressed a distance 5 cm and then released, find the equation that describes the oscillation of the spring. (b) Find a general formula for the frequency (in terms of \(k\) and \(m ) .\) (c) How is the frequency affected if the mass is increased? Is the oscillation faster or slower? (d) How is the frequency affected if a stiffer spring is used (larger \(k\) )? Is the oscillation faster or slower?

Step-by-Step Solution

Answer

(a) \( f(t) = 0.05 \cos(10\sqrt{3} t) \); (b) \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \); (c) Slower; (d) Faster.

Step 1: Identify Given Values for Part (a)

We are given a mass \( m = 10 \text{ g} = 0.01 \text{ kg} \), a spring constant \( k = 3 \), and an initial displacement \( a = 5 \text{ cm} = 0.05 \text{ m} \). We need to find the displacement function \( f(t) \).

Step 2: Substitute Values in Displacement Formula

Using the formula \( f(t) = a \cos \left(\sqrt{\frac{k}{m}} t\right) \), substitute \( a = 0.05 \), \( k = 3 \), and \( m = 0.01 \). This gives us:\[ f(t) = 0.05 \cos\left(\sqrt{\frac{3}{0.01}} t\right) = 0.05 \cos(\sqrt{300} t) \]

Step 3: Simplify Expression to Find Part (a)

Since \( \sqrt{300} = 10\sqrt{3} \), the displacement function becomes:\[ f(t) = 0.05 \cos(10\sqrt{3} t) \]This is the equation that describes the oscillation of the spring.

Step 4: Find General Formula for Frequency in Part (b)

The angular frequency \( \omega \) is given by \( \omega = \sqrt{\frac{k}{m}} \). The frequency \( f \) is \( \frac{\omega}{2\pi} \), hence:\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]This is the general formula for frequency in terms of \( k \) and \( m \).

Step 5: Analyze Frequency Change with Increased Mass for Part (c)

If the mass \( m \) increases, the term \( \sqrt{\frac{1}{m}} \) decreases, leading to a decrease in frequency \( f \). This means the oscillation is slower when the mass is increased.

Step 6: Analyze Frequency Change with Stronger Spring for Part (d)

If the spring constant \( k \) increases, the term \( \sqrt{k} \) increases, leading to an increase in frequency \( f \). This means the oscillation is faster when a stiffer spring is used.

Key Concepts

Oscillation Frequency

The frequency of oscillation describes how often an object in a mass-spring system completes a full cycle of movement back and forth during a certain period of time. It is a key characteristic of simple harmonic motion, which mass-spring systems often exhibit. The formula for frequency is crucial in understanding how the system will behave under different conditions. It is given by:\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]- **Frequency decreases with more mass:** As the mass \( m \) increases, the overall frequency \( f \) decreases. This concept reflects that heavier objects tend to oscillate more slowly than lighter ones.- **Frequency increases with a stiffer spring:** Conversely, increasing the spring constant \( k \) boosts the frequency, meaning the system will oscillate faster with a stiffer spring.Understanding these relationships can help in predicting how changes in mass or spring stiffness affect the behavior of oscillating systems.

Spring Constant

The spring constant \( k \) is a measure of a spring's stiffness. It quantifies the force required to compress or extend the spring by a certain amount. This constant plays a crucial role in determining the nature of oscillations in a mass-spring system, influencing the frequency as well as the displacement function.- **Direct Impact on Frequency:** The spring constant directly affects the oscillation frequency. A higher \( k \) results in a faster oscillation, as seen in the frequency formula \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \).- **Characteristics of Stiffness:** Generally, a large spring constant means the spring is quite rigid, requiring more force for displacement, leading to quicker oscillations.The spring constant is fundamental to knowing how different springs will behave under various loads and predicting the speed of oscillations.

Mass-Spring System

A mass-spring system is one of the simplest forms of harmonic oscillators, consisting of a mass attached to a spring. When displaced from its equilibrium position and released, the system exhibits periodic motion due to the restoring force provided by the spring. This force depends on the spring constant \( k \), and the resulting oscillations are influenced by this along with the mass \( m \).- **Behavior in Simple Harmonic Motion:** The system's natural tendency is to return to its equilibrium, setting up an oscillatory motion described mathematically by sinusoidal functions.- **Effect of Mass and Spring Constant:** The dynamics of the mass-spring system are dictated by both the mass itself and the stiffness of the spring, as outlined in the frequency formula.Understanding mass-spring systems helps us comprehend more complex oscillations and is foundational for topics involving wave mechanics and vibrations.

Displacement Function

In a mass-spring system, the displacement function \( f(t) = a \cos(\sqrt{\frac{k}{m}} t) \) describes the position of the mass at any given time \( t \). This function is pivotal in predicting where the mass will be throughout its cycle of motion.- **Initial Displacement and Amplitude:** Here, \( a \) represents the initial displacement and also serves as the maximum distance from the equilibrium point. The amplitude remains constant if no external forces act on the mass-spring system.- **Harmonic Motion Characteristics:** The cosine function models how the displacement changes in a periodic manner, highlighting features such as amplitude, phase, and frequency directly from its expression.The displacement function is critical for analyzing the motion in terms of time, aiding in a deeper understanding of how the components work together in oscillatory systems.