Section 6.4: Mass-Spring Systems
The mass-spring system is the simplest example of simple harmonic motion (SHM). A mass attached to a spring oscillates back and forth when displaced from its equilibrium position, provided the spring follows Hooke’s Law.
Key Equations:
- Hooke’s Law: \( F = -kx \)
- Equation of Motion: \( m \frac{d^2x}{dt^2} = -kx \)
- Angular frequency: \( \omega = \sqrt{\tfrac{k}{m}} \)
- Period: \( T = 2\pi \sqrt{\tfrac{m}{k}} \)
- Displacement: \( x(t) = A \cos(\omega t + \phi) \)
Example 1
A 0.25 kg mass is attached to a spring with spring constant \( k = 100 \, \text{N/m} \). Find the period of oscillation and the angular frequency.
\( \omega = \sqrt{\tfrac{k}{m}} = \sqrt{\tfrac{100}{0.25}} = \sqrt{400} = 20 \, \text{rad/s} \)
\( T = \tfrac{2\pi}{\omega} = \tfrac{2\pi}{20} = 0.314 \, \text{s} \)
Practice Problems
- A 0.5 kg mass is attached to a spring with \( k = 200 \, \text{N/m} \). Find the period and frequency.
- Find the angular frequency for a spring-mass system with \( m = 0.1 \, \text{kg}, \, k = 50 \, \text{N/m} \).
- A spring-mass system oscillates with a period of 0.5 s. If the mass is doubled, find the new period.
- A 2 kg mass attached to a spring has a frequency of 2 Hz. Find the spring constant \( k \).
- Derive the period equation starting from Newton’s Second Law and Hooke’s Law.