Problem 9 - Classical Mechanics - An Undergraduate Text

Question

A heavy uniform rope of mass \(M\) and length \(4 a\) has one end connected to a fixed point. on a smooth horizontal table by light elastic spring of natural length \(a\) and modulus \(\frac{1}{2} M g\), while the other end hangs down over the edge of the table. When the spring has its natural length, the free end of the rope hangs a distance \(a\) vertically below the level of the table top. The system is released from rest in this position. Show that the free end of the rope executes simple harmonic motion, and find its period and amplitude.

Step-by-Step Solution

Answer

The free end of the rope executes Simple Harmonic Motion with period \(T = 2π\sqrt{2a/g}\) and amplitude \(A = a\).

Step 1: Identify the Forces Acting on the System

The forces acting on the rope are gravity, tension from the rope itself, and the restoring force from the spring. Consider an infinitesimal segment of the rope of length dx at a distance x from the top, the forces on this segment are: gravitational force \(dF = \frac{M}{4a}g dx\), tension from the rope \(T\) downward (since the rope is vertical), and restoring force from the spring \(F_{spring} = \frac{1}{2}Mg - T\).

Step 2: Set up the Differential Equation

Applying Newton's second law in the vertical direction and bearing in mind that the acceleration of an infinitesimal segment of the rope is \(d^2x/dt^2\), we can write out force balance as follows: \(\frac{M}{4a}gx = \frac{1}{2}Mg - \frac{d^2x}{dt^2}\). This equation simplifies to \(\frac{d^2x}{dt^2} + \frac{g}{2a}x = 0\). Therefore, this differential equation describes simple harmonic motion (SHM).

Step 3: Derive the Period and Amplitude of the Motion

In SHM, the general solution for \(x(t)\) is \(x(t) = Asin(ωt + φ)\), where \(A\) is the amplitude, \(ω\) is the angular frequency and φ is the phase angle. From the differential equation in Step 2, we see that the angular frequency of the motion \(ω^2 = g/2a\). Therefore \(ω = \sqrt{g/2a}\), and the period of the motion \(T = 2π/ω = 2π\sqrt{2a/g}\). As the initial condition states the rope is released from rest with the spring at its natural length, the maximum extension of the spring is \(a\), which is the amplitude of the motion.

Key Concepts

Differential Equation

A differential equation is a mathematical equation that involves functions and their derivatives. In this context, it helps describe how quantities change over time, which is critical in studying motion. In our exercise, we derived a specific differential equation for the simple harmonic motion of the rope end based on forces acting within the system.
The key equation, which is \[ \frac{d^2x}{dt^2} + \frac{g}{2a}x = 0 \],
represents simple harmonic motion (SHM) because it follows a standard form of \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \). This tells us that the acceleration is proportional to the displacement but in the opposite direction. This opposition is what causes the oscillatory motion.
Understanding differential equations in mechanics allows us to predict the motion of objects, identifying not only motion patterns but also crucial parameters like frequency and amplitude.

Period and Amplitude

In simple harmonic motion, the period and amplitude are two fundamental characteristics describing the motion's behavior. The period \( T \) is the time it takes for one complete oscillation cycle. For our system, the period was determined from the differential equation noted earlier. The angular frequency \( \omega \), from \( \omega^2 = \frac{g}{2a} \), gives us \( \omega = \sqrt{\frac{g}{2a}} \). Thus, the period \( T \) is calculated as \[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{2a}{g}} \].
The amplitude \( A \) represents the maximum displacement of the object from its equilibrium position. In this exercise, with the initial condition being the spring at its natural length, the maximal vertical displacement corresponds directly to the amplitude. Therefore, the amplitude of the motion is \( a \).
Understanding both period and amplitude helps in designing systems and predicting their response over time.

Forces in Mechanics

In studying the forces in mechanics that affect a system, we consider several elements: gravitational force, tension, and any restoring forces like springs. In this exercise, we saw these forces at play, causing the system to execute simple harmonic motion:

• Gravitational Force: Acts downward across the rope, influencing the motion of the free end.
• Tension: Stemming from the rope's weight, tension works against gravity and contributes to the overall balance of forces.
• Spring Force: The restoring force from the spring seeks to return the system to an equilibrium position, an essential feature of simple harmonic motion.

Accurately setting up the differential equation relies on properly accounting for these forces. By understanding the contributing forces, students can predict how changes in system properties might alter motion characteristics, such as period or amplitude.