Let’s determine the mass of the truck using the given information: the spring constant ([tex]\(k\)[/tex]) and the frequency ([tex]\(f\)[/tex]) of oscillation.
### Step-by-Step Solution:
1. Understand the Relationship:
The relationship between the spring constant, frequency, and mass in a harmonic oscillator is given by:
[tex]\[
f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}
\][/tex]
where:
- [tex]\( f \)[/tex] is the frequency,
- [tex]\( k \)[/tex] is the spring constant,
- [tex]\( m \)[/tex] is the mass,
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159).
2. Rearrange the Formula to Solve for Mass ([tex]\(m\)[/tex]):
Starting with the equation:
[tex]\[
f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}
\][/tex]
Rearrange it to solve for the mass [tex]\( m \)[/tex] as follows:
[tex]\[
f \times 2 \pi = \sqrt{\frac{k}{m}}
\][/tex]
[tex]\[
(f \times 2 \pi)^2 = \frac{k}{m}
\][/tex]
[tex]\[
m = \frac{k}{(f \times 2 \pi)^2}
\][/tex]
3. Calculate the Intermediate Values:
Calculate the value of [tex]\( 2 \pi f \)[/tex]:
[tex]\[
2 \pi f = 2 \times 3.141592653589793 \times 0.429 \approx 2.6965336943312392
\][/tex]
Next, square this result:
[tex]\[
(2 \pi f)^2 = (2.6965336943312392)^2 \approx 7.2656474543235445
\][/tex]
4. Compute the Mass ([tex]\(m\)[/tex]):
Now, use the spring constant [tex]\( k \)[/tex] and the squared frequency calculated above to find the mass [tex]\( m \)[/tex]:
[tex]\[
m = \frac{k}{(2 \pi f)^2} = \frac{24200}{7.2656474543235445} \approx 3330.742394554168
\][/tex]
### Conclusion:
The mass of the truck is approximately [tex]\( 3330.742 \, \text{kg} \)[/tex].
