Question

Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position \(y=0\) when the mass hangs at rest. Suppose you push the mass to a position \(y_{0}\) units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $$y=y_{0} \cos (t \sqrt{\frac{k}{m}}),$$ where \(k > 0\) is a constant measuring the stiffness of the spring (the larger the value of \(k,\) the stiffer the spring) and \(y\) is positive in the \(u p-\) ward direction. Use equation (2) to answer the following questions. a. The period \(T\) is the time required by the mass to complete one oscillation. Show that \(T=2 \pi \sqrt{\frac{m}{k}}\). b. Assume \(k\) is constant and calculate \(\frac{d T}{d m}\).. c. Give a physical explanation of why \(\frac{d T}{d m}\) is positive.

Short answer

Question: Show that the period of a spring-mass system with position function \(y(t) = y_0 \cos(t\sqrt{\frac{k}{m}})\) is given by \(T = 2\pi\sqrt{\frac{m}{k}}\), differentiate \(T\) with respect to \(m\), and explain the physical significance of the positive derivative. Answer: The period of the spring-mass system is given by \(T = 2\pi\sqrt{\frac{m}{k}}\). Its derivative with respect to mass is \(\frac{d T}{d m} = \frac{\pi}{\sqrt{mk}}\), which is positive. This means that as the mass of the object increases, the period also increases. Physically, this occurs because a heavier object requires more inertial force to change its velocity, which results in a longer period for heavier masses and a shorter period for lighter masses.

Step-by-step solution

Identify the angular frequency

The angular frequency, denoted by \(\omega\), is the coefficient of \(t\) inside the cosine function in the position function: \(\omega = \sqrt{\frac{k}{m}}\).

Find the period

The period of a function is the time it takes for one complete oscillation, which corresponds to an angular displacement of \(2\pi\). We can write the relationship between the period \(T\) and the angular frequency \(\omega\) as \(T = \frac{2\pi}{\omega}\).

Substitute the angular frequency

Substitute \(\omega = \sqrt{\frac{k}{m}}\) into the period formula: $$ T = \frac{2\pi}{\sqrt{\frac{k}{m}}} = 2\pi\sqrt{\frac{m}{k}}. $$ b. Calculate the derivative of \(T\) with respect to \(m\).

Differentiate the period formula

We want to find \(\frac{d T}{d m}\), so we differentiate the period formula with respect to mass: $$ \frac{d T}{d m} = \frac{d}{d m}(2\pi\sqrt{\frac{m}{k}}). $$

Apply the chain rule

Apply the chain rule to differentiate the product of constants and the square root function: $$ \frac{d T}{d m} = 2\pi \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{mk}} = \frac{\pi}{\sqrt{mk}}. $$ c. Give a physical explanation of why \(\frac{d T}{d m}\) is positive.

Interpret the derivative

The derivative \(\frac{d T}{d m}\) represents the rate of change of the period with respect to the mass. A positive value means that as the mass increases, the period increases as well.

Provide a physical explanation

As the mass of the object increases, more inertial force is required to change its velocity. Therefore, it takes more time for the spring force to accelerate and decelerate the object during each oscillation. This results in a longer period for heavier masses and a shorter period for lighter masses, which is reflected in the positive value of \(\frac{d T}{d m}\).

Key concepts

Harmonic Motion

When we examine the movement of an object attached to a spring, as in the given exercise, we're delving into the realm of harmonic motion. Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement. In this case, the restoring force is provided by the spring, which pushes or pulls the mass back toward its equilibrium position.

Understanding SHM is crucial because it serves as a fundamental model for many physical systems, including pendulums, vibrations in molecules, and even the oscillations of certain electrical circuits. The defining characteristic of SHM is its sine or cosine waveform when we graph it over time - precisely what we see in the exercise equation: \(y=y_{0} \times \text{cos}(t\textwidthk/{m})\).

In SHM, the displacement from the equilibrium position is symmetrical and occurs at a uniform rate, showcasing a beautiful harmony in motion which reflects in the undulating waveform. By studying this simple yet intricate form of motion, students can grasp the basic principles that underpin more complex systems in both classical and quantum physics.

Angular Frequency

Angular frequency, often represented by the Greek letter \(\omega\), is a way to describe the rate of oscillation in harmonic motion by looking at how fast an object goes through its cycle. In our spring-mass example, the angular frequency is the number inside the cosine function, which in this case is \(\sqrt{k/m}\).

It's called 'angular' because it has a direct link to rotational motion - imagine our spring-mass system moving in a circle; the angular frequency would tell us how fast it's rotating. One key thing to remember is that angular frequency isn't measured in 'cycles per time' like regular frequency; it's measured in radians per second, giving us a direct tie to the circular (or angular) aspects of the motion.

When a student understands angular frequency, they're better equipped to analyze oscillations and waves, not just in mechanics but also in optics and electromagnetism. For example, the concepts you learn from angular frequency in mechanical oscillations like this can later be applied to understand light as an electromagnetic wave.

Period of Oscillation

The period of oscillation, represented by the letter \(T\), is the time it takes for one complete cycle of harmonic motion - from the starting point, back to that point again. In the context of our spring-mass system, it's the time for the mass to move up and then down, returning to its initial position.

The formula \(T = 2\text{pi} \times \text{sqrt}(m/k)\) ties back to the angular frequency, highlighting that the period is the inverse of this frequency multiplied by \(2\text{pi}\), which is the full angle of a circle in radians. Understanding the period is essential because it helps us predict how long a system will take to repeat its motion, which is fundamental in systems like clocks, musical instruments, and even in the electrical circuits that time computer processors.

Appreciating the elegance of the relationship between a spring's stiffness, the mass it's attached to, and the cycle time allows students to see the intricate balances in physics that govern not just theoretical exercises but the rhythms of many everyday occurrences.

Differentiation of Physical Concepts

Differentiation is a powerful tool in mathematics, particularly when applied to physical concepts. It's the process by which we can find how changes in one quantity cause changes in another. In the exercise, we differentiate the period of oscillation, \(T\), with respect to the mass, \(m\), to find \(d T/d m\).

This derivative tells us how the period of the oscillation changes as the mass changes - and we find that this relationship is positive, meaning that as you increase the mass, the period also increases. It's a fundamental concept that's at the heart of understanding dynamics because it links an object's properties to its behavior over time.

By mastering differentiation in physics, students learn to predict and model the behavior of systems. It helps us understand why a heavy pendulum swings slower than a light one, or why adding mass to a car changes its acceleration. These kinds of insights are essential for everything from engineering safer vehicles to designing effective medications, making differentiation much more than just a mathematical operation - it's a window into the causes and effects that shape our world.