Problem 36 - College Physics

Question

\(\bullet\) A proud deep-sea fisherman hangs a 65.0 \(\mathrm{kg}\) fish from an ideal spring having negligible mass. The fish stretches the spring 0.120 \(\mathrm{m} .\) (a) What is the force constant of the spring? (b) What is the period of oscillation of the fish if it is pulled down 3.50 \(\mathrm{cm}\) and released?

Step-by-Step Solution

Answer

(a) 5313.75 N/m; (b) 0.70 s

Step 1: Calculate the Force Exerted by the Fish

To find the force constant of the spring, first determine the force exerted by the fish due to gravity. This is given by the formula:\[ F = mg \]where \( m = 65.0 \, \text{kg} \) is the mass of the fish and \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity. Calculate \( F \):\[ F = 65.0 \times 9.81 = 637.65 \, \text{N} \]

Step 2: Determine the Force Constant of the Spring

The force constant (spring constant) \( k \) can be calculated using Hooke's Law, which relates the force exerted on a spring to its extension:\[ F = kx \]Rearrange to solve for \( k \):\[ k = \frac{F}{x} \]where \( x = 0.120 \, \text{m} \) is the extension of the spring. Substitute the values:\[ k = \frac{637.65}{0.120} = 5313.75 \, \text{N/m} \]

Step 3: Calculate the Period of Oscillation

The period \( T \) of oscillation for a mass-spring system is given by:\[ T = 2\pi \sqrt{\frac{m}{k}} \]Substitute \( m = 65.0 \, \text{kg} \) and \( k = 5313.75 \, \text{N/m} \) into the equation:\[ T = 2\pi \sqrt{\frac{65.0}{5313.75}} \approx 0.70 \, \text{s} \]

Step 4: Summary of Solutions

(a) The force constant of the spring is \( 5313.75 \, \text{N/m} \).(b) The period of oscillation if the fish is pulled down further and released is \( 0.70 \, \text{s} \).

Key Concepts

Spring Constant

The spring constant, often referred to as the force constant, is a fundamental concept when discussing Hooke's Law. It measures the stiffness of a spring, which indicates how much force is required to stretch or compress it by a given length. In mathematical terms, Hooke's Law is expressed as: \[ F = kx \] where:

• \( F \) is the force applied to the spring (in newtons, N),
• \( k \) is the spring constant (in N/m), and
• \( x \) is the extension or compression of the spring (in meters, m).

In the exercise, a 65 kg fish acts as a weight, stretching the spring by 0.120 m. By knowing the force exerted by the fish due to gravity (calculated as 637.65 N), and the extension of the spring, we can determine the spring constant: \[ k = \frac{637.65}{0.120} = 5313.75 \, \text{N/m} \] This tells us how the spring behaves under force, which is crucial for different applications, from designing vehicle suspensions to understanding seismic activity.

Oscillation Period

The oscillation period is an important concept when dealing with harmonic motion. It represents the time taken for an oscillating system, like a mass-spring system, to complete one full cycle of motion. For a simple mass-spring system, the period \( T \) is calculated using the formula: \[ T = 2\pi \sqrt{\frac{m}{k}} \] where:

• \( T \) is the period of oscillation,
• \( m \) is the mass (in kg), and
• \( k \) is the spring constant (in N/m).

For the given problem, substituting the mass of 65.0 kg and the spring constant of 5313.75 N/m results in \[ T = 2\pi \sqrt{\frac{65.0}{5313.75}} \approx 0.70 \, \text{s} \] This means it takes about 0.70 seconds for the fish to bounce back to its starting position after being pulled down and released. Understanding oscillation periods helps in grasping the dynamics of various systems, from amusement park rides to engineering diagnostics.

Mass-Spring System

A mass-spring system is a classic mechanical model used to explain simple harmonic motion. It consists of a mass attached to a spring, which can oscillate when displaced. The system serves as a basic representation of how objects behave under tension or compression. Key features of a mass-spring system include:

• **Oscillation:** This is the repetitive motion back and forth about an equilibrium position,
• **Equilibrium:** The point where the mass naturally rests in the absence of external forces,
• **Amplitude:** The maximum displacement from the equilibrium position,
• **Damping (not addressed in this exercise):** A force that reduces motion over time.

In the original exercise, the fish acts as the mass, stretching the spring. When pulled and released, it oscillates, showcasing the principles of energy transfer between potential and kinetic forms. Understanding this system lays a foundation for studying more complex wave phenomena and engineering problems.