Let's solve this step by step, providing detailed information about the motion of the object:

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Known Parameters:

1. Mass of the object, $m$: $m = 55.0 \, \text{g} = 0.055 \, \text{kg}$
   (Converted to kilograms for consistency with SI units.)

2. Spring constant, $k$: $k = 30.0 \, \text{N/m}$

3. Amplitude, $A$: $A = 4.00 \, \text{cm} = 0.040 \, \text{m}$
   (Converted to meters for consistency.)

4. Surface: Frictionless and horizontal.

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Calculations:

(a) Angular frequency, $\omega$:

The formula for angular frequency is: $\omega = \sqrt{\frac{k}{m}}$

Substituting the values: $\omega = \sqrt{\frac{30.0}{0.055}} = \sqrt{545.45} \approx 23.35 \, \text{rad/s}$

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(b) Frequency of oscillation, $f$:

The relationship between angular frequency and frequency is: $f = \frac{\omega}{2\pi}$

Substituting the value of $\omega$: $f = \frac{23.35}{2\pi} \approx 3.71 \, \text{Hz}$

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(c) Maximum speed, $v_{\text{max}}$:

The maximum speed is given by: $v_{\text{max}} = \omega A$

Substituting the values: $v_{\text{max}} = 23.35 \cdot 0.040 \approx 0.934 \, \text{m/s}$

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(d) Maximum acceleration, $a_{\text{max}}$:

The maximum acceleration is: $a_{\text{max}} = \omega^2 A$

Substituting the values: $a_{\text{max}} = (23.35)^2 \cdot 0.040 \approx 21.8 \, \text{m/s}^2$

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Summary of Results:

1. Angular frequency ($\omega$): $23.35 \, \text{rad/s}$
2. Frequency ($f$): $3.71 \, \text{Hz}$
3. Maximum speed ($v_{\text{max}}$): $0.934 \, \text{m/s}$
4. Maximum acceleration ($a_{\text{max}}$): $21.8 \, \text{m/s}^2$

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Would you like a detailed explanation of any step or further exploration of this system?

Related Questions:

1. How would the maximum speed change if the amplitude were doubled?
2. What is the time period of oscillation for this system?
3. How does the angular frequency change if the spring constant is halved?
4. What is the total mechanical energy of the oscillating system?
5. How would adding friction affect the motion of the object?

Tip:

Always keep units consistent (e.g., SI units) when solving physics problems to avoid calculation errors.