To determine the natural vibration frequency of a system consisting of two point masses attached to a spring, we can use the formula for the frequency of a mass-spring system. The formula for the natural frequency \( f \) in hertz (Hz) is given by:
Frequency Formula
The formula is:
f = \frac{1}{2\pi} \sqrt{\frac{k}{m_{eff}}}
Understanding the Components
In this equation:
- f is the natural frequency in hertz.
- k is the spring constant, which in this case is 300 N/m.
- m_{eff} is the effective mass of the system.
Calculating the Effective Mass
For two masses attached to a spring, the effective mass \( m_{eff} \) can be calculated using the formula:
m_{eff} = \frac{m_1 \cdot m_2}{m_1 + m_2}
Here, \( m_1 = 3.0 \, \text{kg} \) and \( m_2 = 1.0 \, \text{kg} \). Plugging in these values:
m_{eff} = \frac{3.0 \cdot 1.0}{3.0 + 1.0} = \frac{3.0}{4.0} = 0.75 \, \text{kg}
Substituting into the Frequency Formula
Now that we have the effective mass, we can substitute \( k \) and \( m_{eff} \) into the frequency formula:
f = \frac{1}{2\pi} \sqrt{\frac{300}{0.75}}
Calculating the value inside the square root:
\frac{300}{0.75} = 400
Now, taking the square root:
\sqrt{400} = 20
Finally, substituting back into the frequency formula:
f = \frac{1}{2\pi} \cdot 20
Calculating this gives:
f \approx \frac{20}{6.2832} \approx 3.18 \, \text{Hz}
Choosing the Closest Answer
From the options provided, the closest value to our calculated frequency of approximately 3.18 Hz is:
Thus, the natural vibration frequency of the system is about 3 Hz. This calculation illustrates how the mass and spring constant interact to determine the frequency of oscillation in a mass-spring system.
