An automobile having a mass of 2,000 kg deflects its suspension springs 0.02 m under static conditions. Determine the natural frequency of the automobile in the vertical direction by assuming damping to be negligible.
The correct answer and explanation is :
To determine the natural frequency of the automobile in the vertical direction, we can use the concept of simple harmonic motion (SHM), specifically for a mass-spring system. The natural frequency of oscillation is given by:
[
f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
]
Where:
- (f_n) is the natural frequency (in Hz),
- (k) is the spring constant (in N/m),
- (m) is the mass of the automobile (in kg).
Step 1: Find the spring constant (k)
The spring constant (k) can be determined using Hooke’s law for static deflection, which states:
[
F = k \cdot \Delta x
]
Where:
- (F) is the force (in Newtons),
- (\Delta x) is the static deflection of the spring (in meters).
Since the automobile is in static equilibrium, the force (F) is equal to the weight of the automobile:
[
F = m \cdot g
]
Where:
- (m = 2000 \, \text{kg}) is the mass of the automobile,
- (g = 9.81 \, \text{m/s}^2) is the acceleration due to gravity.
Thus, the weight of the automobile is:
[
F = 2000 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 19,620 \, \text{N}
]
Using Hooke’s law:
[
k = \frac{F}{\Delta x} = \frac{19,620 \, \text{N}}{0.02 \, \text{m}} = 981,000 \, \text{N/m}
]
Step 2: Calculate the natural frequency
Now that we have the spring constant (k = 981,000 \, \text{N/m}), we can use the formula for the natural frequency:
[
f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{2\pi} \sqrt{\frac{981,000}{2000}} \approx \frac{1}{2\pi} \cdot 22.1 \approx 3.52 \, \text{Hz}
]
Explanation:
The natural frequency of a system like a mass-spring system depends on the spring constant and the mass. A higher spring constant (stiffer suspension) or a lower mass leads to a higher natural frequency. In this case, the spring constant (k) is calculated from the static deflection, which gives us an idea of how stiff the automobile’s suspension is. Using this value, we can then calculate the natural frequency, which represents the frequency at which the automobile would oscillate if disturbed, assuming negligible damping. In this case, the natural frequency of the automobile’s suspension system is approximately 3.52 Hz.