To find the amplitude of the particular solution for the undamped system acted on by an external force, we use the provided formula:
[tex]\[
X = \frac{\frac{F}{k}}{1 - \frac{(2 \pi f)^2 m}{k}}
\][/tex]
We are given:
- \( F \) (external force)
- \( k \) (spring constant)
- \( f \) (frequency)
- \( m \) (mass)
For simplicity, let's use the following example values (note that these values are for the sake of explaining the solution):
- \( F = 1 \) (in appropriate units)
- \( k = 1 \) (in appropriate units)
- \( f = 1 \) (in Hz)
- \( m = 1 \) (in kg)
Now, let's substitute these values into the formula and compute step-by-step.
1. Compute \(\frac{F}{k}\):
[tex]\[
\frac{F}{k} = \frac{1}{1} = 1
\][/tex]
2. Compute \((2 \pi f)^2 m\):
- First, compute \(2 \pi f\):
[tex]\[
2 \pi f = 2 \pi \times 1 = 2 \pi
\][/tex]
- Now square it:
[tex]\[
(2 \pi)^2 = (2 \pi)^2 = 4 \pi^2
\][/tex]
- Finally, multiply by \(m\):
[tex]\[
(4 \pi^2) \times 1 = 4 \pi^2
\][/tex]
3. Compute \(\frac{(2 \pi f)^2 m}{k}\):
[tex]\[
\frac{(2 \pi f)^2 m}{k} = \frac{(4 \pi^2) \times 1}{1} = 4 \pi^2
\][/tex]
4. Substitute these results into the denominator:
[tex]\[
1 - \frac{(2 \pi f)^2 m}{k} = 1 - 4 \pi^2
\][/tex]
5. Combine the numerator and the denominator:
[tex]\[
X = \frac{1}{1 - 4 \pi^2}
\][/tex]
Using the values we computed:
[tex]\[
1 - 4 \pi^2 \approx 1 - 39.4784 \approx -38.4784
\][/tex]
So,
[tex]\[
X \approx \frac{1}{-38.4784} \approx -0.025988594704756167
\][/tex]
Therefore, the amplitude of the particular solution \( X \) is approximately:
[tex]\[
X \approx -0.025988594704756167
\][/tex]
