A  m = 2.1 kg object attached to a spring moves without friction (b = 0) and is driven by an external force given by the expression F = 2.66 sin(5.77t) where F  is in newtons and t  is in seconds.  

The force constant of the spring is k = 26.2 N/m.  

5a)  What is the natural angular frequency of the system, 𝜔0
 ? 5b) What is the  angular frequency of the driven system,  𝜔𝐷
 ?5c) What is the amplitude of the driven oscillation?

We are dealing with a spring-mass system driven by an external force, and we need to solve for the natural angular frequency (\(\omega_0\)), the angular frequency of the driven system (\(\omega_D\)), and the amplitude of the driven oscillation.

### Part 5a: Natural Angular Frequency (\(\omega_0\))

The natural angular frequency of a spring-mass system is given by the formula:
\[
\omega_0 = \sqrt{\frac{k}{m}}
\]
where:
- \(k = 26.2 \, \text{N/m}\) is the spring constant,
- \(m = 2.1 \, \text{kg}\) is the mass of the object.

Now, we can calculate \(\omega_0\):

\[
\omega_0 = \sqrt{\frac{26.2}{2.1}}
\]

### Part 5b: Angular Frequency of the Driven System (\(\omega_D\))

The angular frequency of the driven system is simply the angular frequency of the external driving force. Given that the driving force is:
\[
F(t) = 2.66 \sin(5.77t)
\]
The angular frequency of the driving force is \(5.77 \, \text{rad/s}\). Therefore, the angular frequency of the driven system is:
\[
\omega_D = 5.77 \, \text{rad/s}
\]

### Part 5c: Amplitude of the Driven Oscillation

The amplitude of the driven oscillation is given by the formula:
\[
A = \frac{F_0}{m \sqrt{(\omega_0^2 - \omega_D^2)^2}}
\]
where:
- \(F_0 = 2.66 \, \text{N}\) is the amplitude of the driving force,
- \(m = 2.1 \, \text{kg}\) is the mass,
- \(\omega_0\) is the natural angular frequency (calculated in part 5a),
- \(\omega_D = 5.77 \, \text{rad/s}\) is the angular frequency of the driving force.

Let's first calculate \(\omega_0\), and then we'll compute the amplitude of the driven oscillation.

I'll perform these calculations now.### Results:

5a) The natural angular frequency of the system is \(\omega_0 = 3.53 \, \text{rad/s}\).

5b) The angular frequency of the driven system is \(\omega_D = 5.77 \, \text{rad/s}\).

5c) The amplitude of the driven oscillation is \(A = 0.0608 \, \text{m}\).

Would you like further details or explanations? Here are five related questions you might find useful:

1. How would the amplitude change if the driving force increased?
2. What happens to the system if the driving frequency equals the natural frequency?
3. Can damping affect the natural frequency of the system?
4. How would adding friction to the system change the analysis?
5. How can we experimentally determine the spring constant \(k\)?

**Tip**: Resonance occurs when the driving frequency matches the natural frequency, leading to maximum amplitude.

A m = 2.1 kg object attached to a spring moves without friction (b = 0) and is driven by an external force given by the expression F = 2.66 sin(5.77t) where F is in newtons and t is in seconds. The force constant of the spring is k = 26.2 N/m. 5a) What is the natural angular frequency of the system, 𝜔0? 5b) What is the angular frequency of the driven system, 𝜔D? 5c) What is the amplitude of the driven oscillation?

Explore the dynamics of a 2.1 kg mass attached to a spring with a force constant of 26.2 N/m. Learn how to calculate the natural angular frequency (ω₀), the angular frequency of the driven system (ωD), and the amplitude of the oscillation when driven by an external force. Suitable for undergraduate physics students.

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