To solve this problem, let's start by analyzing the mass-spring system.

The equation of motion for a damped harmonic oscillator subject to an external driving force is:

$m \ddot{x} + k x = F(t)$

Where:

• $m = 0.3 \, \text{kg}$ is the mass,
• $k = 7.1 \, \text{N/m}$ is the spring constant,
• $F(t) = 21 \cos(\omega t) \, \text{N}$ is the driving force with angular frequency $\omega$.

Given this, the equation of motion is:

$0.3 \ddot{x} + 7.1 x = 21 \cos(\omega t)$

This is a second-order linear non-homogeneous differential equation. First, let's find the angular frequency $\omega$ that causes resonance.

1. Resonance Condition

For resonance to occur, the frequency of the external force ($\omega$) must match the natural frequency of the system. The natural angular frequency $\omega_0$ of the mass-spring system is given by:

$\omega_0 = \sqrt{\frac{k}{m}}$

Substitute the values of $k$ and $m$:

$\omega_0 = \sqrt{\frac{7.1}{0.3}} = \sqrt{23.67} \approx 4.86 \, \text{rad/s}$

Thus, the angular frequency that causes resonance is:

$\boxed{\omega_0 = 4.86 \, \text{rad/s}}$

2. Resonance Part of the Solution

At resonance, the system oscillates at its natural frequency with maximum amplitude. The solution to the equation of motion for a driven harmonic oscillator consists of two parts: the complementary solution (which describes the homogeneous case) and the particular solution (which describes the driven response).

For resonance, the particular solution dominates and can be written as:

$x_{\text{resonance}}(t) = \frac{F_0}{m \omega_0^2} \cos(\omega_0 t)$

Where:

• $F_0 = 21 \, \text{N}$ is the amplitude of the driving force,
• $m = 0.3 \, \text{kg}$ is the mass,
• $\omega_0 = 4.86 \, \text{rad/s}$ is the natural angular frequency.

Substitute the known values:

$x_{\text{resonance}}(t) = \frac{21}{0.3 \times (4.86)^2} \cos(4.86 t)$

First, calculate $(4.86)^2$:

$(4.86)^2 = 23.67$

Now, substitute:

$x_{\text{resonance}}(t) = \frac{21}{0.3 \times 23.67} \cos(4.86 t) = \frac{21}{7.101} \cos(4.86 t)$

Simplify:

$x_{\text{resonance}}(t) \approx 2.96 \cos(4.86 t)$

So, the resonance part of the solution is:

$\boxed{x_{\text{resonance}}(t) \approx 2.96 \cos(4.86 t)}$

Summary:

• The positive angular frequency that leads to resonance is $\omega_0 \approx 4.86 \, \text{rad/s}$.
• The resonance part of the solution is $x_{\text{resonance}}(t) \approx 2.96 \cos(4.86 t)$.

Would you like further clarification or details on any part of the solution?

Here are some related questions for further exploration:

1. How do damping effects change the resonance frequency of a system?
2. How would the amplitude of oscillation change if the driving force was $F(t) = 21 \sin(\omega t)$ instead of $\cos(\omega t)$?
3. What is the complementary solution in this case, and how does it relate to the resonance solution?
4. How can you calculate the energy of the system at resonance?
5. How would you derive the general solution for a forced oscillation with no damping?

Tip: The resonance condition occurs when the driving frequency matches the natural frequency, but in real systems, damping can modify this condition.