Problem 48 - University Physics with Modern Physics

Question

A mass of \(0.404 \mathrm{~kg}\) is attached to a spring with a spring constant of \(206.9 \mathrm{~N} / \mathrm{m}\). Its oscillation is damped. with damping constant \(b=14.5 \mathrm{~kg} / \mathrm{s}\). What is the frequency of this damped oscillation?

Step-by-Step Solution

Answer

Answer: The frequency of the damped oscillation is approximately \(3.687 \mathrm{~Hz}\).

Step 1: Identify the given values

As given in the problem, the mass \(m = 0.404 \mathrm{~kg}\), the spring constant \(k = 206.9 \mathrm{~N} / \mathrm{m}\), and the damping constant \(b = 14.5 \mathrm{~kg} / \mathrm{s}\).

Step 2: Calculate intermediate values

Before calculating the damped frequency, we first need to find \((b/2m)^2\). To do this, divide the damping constant by two times the mass: $$ damping\_term = \frac{b}{2m} = \frac{14.5}{2 \times 0.404} = 17.945 \mathrm{~s}^{-1} $$ Now square this value: $$ damping\_term^2 = (17.945)^2 = 322.05 \mathrm{~s}^{-2} $$

Step 3: Calculate the damped frequency

Use the formula for damped frequency to find the solution: $$ f_{damped} = \frac{1}{2 \pi} \sqrt{k/m - (b/2m)^2} $$ Substituting the given values in the equation: $$ f_{damped} = \frac{1}{2 \pi} \sqrt{\frac{206.9}{0.404} - 322.05} $$ Simplify the equation to get the damped frequency: $$ f_{damped} = \frac{1}{2 \pi} \sqrt{512.621 - 322.05} = \frac{1}{2 \pi} \sqrt{190.571} $$ $$ f_{damped} \approx 3.687 \mathrm{~Hz} $$ The frequency of the damped oscillation is approximately \(3.687 \mathrm{~Hz}\).

Key Concepts

Damping Constant

The damping constant, often represented by the symbol \( b \), plays a crucial role in determining how quickly oscillations in a mechanical system will diminish. It represents the resistance exerted to the motion of an oscillating object by damping forces such as friction or air resistance. The unit of the damping constant \( b \) is kilograms per second (\( \mathrm{kg/s} \)). When you have a larger damping constant, oscillations will die out more quickly. Understanding the damping constant is essential for applications like car suspension systems or building structures, where controlling vibrations is necessary for comfort and safety.

• A high damping constant = faster energy loss = faster decay of motion.
• A low damping constant = slower energy loss = slower decay of motion.

In our exercise, the damping constant is 14.5 kg/s, demonstrating a moderate level of damping which balances between rapid suppression of oscillations and allowing some movement.

Spring Constant

The spring constant, denoted as \( k \), is a measure of the stiffness of the spring. It reflects the amount of force required to stretch or compress the spring by a unit length (usually meters). The unit is Newtons per meter (\( \mathrm{N/m} \)). A higher spring constant indicates a stiffer spring. For example, in our exercise, the spring constant is 206.9 \( \mathrm{N/m} \). This tells us that it takes 206.9 Newtons to stretch or compress the spring by one meter.The spring constant is vital when studying harmonic motion, as it influences the system's natural frequency. A stiffer spring (higher \( k \)) will result in a higher natural frequency of oscillation for the mass-spring system.

• High spring constant = stiff spring = higher natural frequency.
• Low spring constant = flexible spring = lower natural frequency.

Oscillating Systems

Oscillating systems are systems that undergo repeated back and forth motion about a central position. Examples of oscillating systems include a pendulum swinging, a mass on a spring, or a guitar string vibrating.These systems are characterized by several key properties:

• Frequency: How fast the oscillations occur, typically measured in Hertz (\( \mathrm{Hz} \)).
• Amplitude: The maximum displacement from the equilibrium position.
• Period: The time taken for one complete cycle of oscillation.

In the context of our problem, the mass-spring system is an oscillating system that's damped, meaning frictional forces are reducing the motion over time. The frequency obtained from the problem is the damped frequency, which accounts for the energy lost due to damping.

Mechanical Vibration

Mechanical vibration involves oscillations of a mechanical component or system. These oscillations can be due to an external force (forced vibration) or internal forces (free vibration). The goal is often to measure, understand, and control these vibrations. Vibrations possess different characteristics:

• Free Vibration: Occurs when a system is set into motion with no external force continuing the motion.
  An example is a tuning fork after being struck.
• Forced Vibration: Occurs when an external time-varying force acts on the system, like a washing machine vibrating during operation.

Our exercise deals with damped mechanical vibration in a spring-mass system, influenced by both damping forces and the elasticity of the spring. The principle behind damping is used extensively in engineering to improve comfort and performance, such as shock absorbers in vehicles, which mitigate vibrations from road bumps.