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Chapter 13: Problem 58

Rank Rank the following mass-spring systems in order of increasing period of oscillation, indicating any ties. System A consists of a mass \(m\) attached to a spring with a spring constant \(k\); system B has a mass \(2 m\) attached to a spring with a spring constant \(k\); system \(\mathrm{C}\) has a mass \(3 m\) attached to a spring with a spring constant \(6 k\); and system \(D\) has a mass \(m\) attached to a spring with a spring constant \(4 k\).

Short Answer

Expert verified
The order is: D, C, A, B.

Step by step solution

01

Understand the Formula for the Period of Oscillation

The period of oscillation for a mass-spring system is given by the formula \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( T \) is the period, \( m \) is the mass, and \( k \) is the spring constant.
02

Calculate the Period for System A

For System A, the mass \( m \) and spring constant \( k \) are given. Plug these values into the formula: \( T_A = 2\pi \sqrt{\frac{m}{k}} \).
03

Calculate the Period for System B

In System B, the mass is \( 2m \) and the spring constant is \( k \). The period is calculated as: \( T_B = 2\pi \sqrt{\frac{2m}{k}} \).
04

Calculate the Period for System C

System C has a mass of \( 3m \) and a spring constant of \( 6k \). The formula becomes: \( T_C = 2\pi \sqrt{\frac{3m}{6k}} = 2\pi \sqrt{\frac{m}{2k}} \).
05

Calculate the Period for System D

In System D, the mass is \( m \) and the spring constant is \( 4k \). Calculate the period: \( T_D = 2\pi \sqrt{\frac{m}{4k}} \).
06

Compare the Periods to Rank the Systems

To rank the periods, consider the expressions: \( T_A = 2\pi \sqrt{\frac{m}{k}} \), \( T_B = 2\pi \sqrt{\frac{2m}{k}} \), \( T_C = 2\pi \sqrt{\frac{m}{2k}} \), and \( T_D = 2\pi \sqrt{\frac{m}{4k}} \). Simplify to find the order: \[T_D = \pi \sqrt{\frac{m}{k}}, \, T_C = \sqrt{2} \pi \sqrt{\frac{m}{k}}, \, T_A = 2\pi \sqrt{\frac{m}{k}}, \, T_B = 2\pi \sqrt{2} \sqrt{\frac{m}{k}}\] So the increasing order is: \( T_D, T_C, T_A, T_B \).

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Key Concepts

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

Mass-Spring System
The mass-spring system is an essential concept in physics, primarily observed in the study of simple harmonic motion. This type of system consists of a mass attached to a spring. When the mass is displaced from its equilibrium position, it experiences a restoring force due to the spring, pulling it back towards equilibrium.
This restoring force is proportional to the displacement, following Hooke's Law:
  • Displacement causes the spring to exert a force in the opposite direction.
  • The force is given by the equation: \( F = - k x \), where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.
When a mass-spring system is set into motion, it oscillates about the equilibrium position, resulting in periodic motion. This oscillation is what we refer to as simple harmonic motion, characterized by its regular and repeatable nature.
Oscillation Period
The oscillation period is an important parameter of a mass-spring system. It represents the duration of time it takes for the system to complete one full cycle of motion, returning to its initial position and state. The formula used to calculate the period of oscillation is: \( T = 2\pi \sqrt{\frac{m}{k}} \).
Here鈥檚 what each part of the formula means:
  • \( T \) is the period: This is what we aim to determine, the time for one complete oscillation.
  • \( m \) denotes the mass attached to the spring. A larger mass results in a longer period, as a greater force is required to change the motion.
  • \( k \) is the spring constant, related to the stiffness of the spring. A stiffer (higher \( k \)) spring decreases the period, meaning the system oscillates faster.
This relationship shows that both the mass and the spring constant significantly impact the behavior and rhythm of the system's oscillations.
Spring Constant
The spring constant, denoted by \( k \), is a crucial parameter in the study of mass-spring systems and simple harmonic motion. It reveals how stiff a spring is. Roughly, the more force you need to extend or compress the spring, the larger the spring constant. This aspect of the system influences how quick the oscillations will occur.
Several key points about the spring constant are:
  • It is measured in units of force per unit length, most commonly Newtons per meter (N/m).
  • A higher spring constant implies a very stiff spring, which oscillates more quickly.
  • A smaller spring constant means the spring is more stretchable, resulting in slower oscillations.
When analyzing a mass-spring system, understanding the spring constant helps in predicting how external forces will alter the system's motion, thus aiding in accurate physics problem solving.
Physics Problem Solving
Effective problem-solving in physics, particularly with systems like mass-spring systems, requires a structured approach. Here's how you can tackle such problems:
  • Begin by identifying all known quantities, such as mass and spring constant, in this case.
  • Write down the physics formulas that apply, like the period formula \( T = 2\pi \sqrt{\frac{m}{k}} \).
  • Substitute known values into the formulas and solve step by step.
  • Pay attention to units and ensure they are consistent throughout the calculation.
  • Cross-check your solution intuitively: Does it make sense that a larger mass leads to a slower oscillation?
By following these steps, physics problems can become more manageable, and solutions more accurate, improving both understanding and confidence.

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Most popular questions from this chapter

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