To calculate the speed of the object when its displacement $x = 0$, we use the principles of simple harmonic motion (SHM).

Step 1: Identify relevant equations and given values

The speed of an object in SHM is given by the formula: $v = \omega \sqrt{A^2 - x^2}$

Where:

• $v$ is the speed.
• $\omega$ is the angular frequency.
• $A$ is the amplitude.
• $x$ is the displacement.

Given:

• Mass ($m$) = $260 \, \text{g} = 0.260 \, \text{kg}$ (converted to kilograms)
• Frequency ($f$) = $4.00 \, \text{Hz}$
• Amplitude ($A$) = $24.0 \, \text{cm} = 0.240 \, \text{m}$
• Displacement ($x$) = $0 \, \text{m}$

Step 2: Calculate the angular frequency ($\omega$)

The angular frequency is related to the frequency by: $\omega = 2\pi f$

Substituting the value of $f = 4.00$: $\omega = 2\pi (4.00) = 8\pi \, \text{rad/s}$

Numerically: $\omega \approx 25.13 \, \text{rad/s}$

Step 3: Calculate the speed at $x = 0$

At $x = 0$, the speed equation simplifies to: $v = \omega A$

Substituting $\omega = 25.13 \, \text{rad/s}$ and $A = 0.240 \, \text{m}$: $v = 25.13 \times 0.240 = 6.03 \, \text{m/s}$

Final Answer:

The speed of the object when its displacement is $x = 0$ is: $\boxed{6.03 \, \text{m/s}}$

Would you like to proceed with further details or calculations? 😊

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Related Questions:

1. What is the maximum acceleration of the object in this motion?
2. How much energy is stored in the spring at maximum compression?
3. What is the spring constant ($k$) for this system?
4. How would the speed at $x = A/2$ be calculated?
5. How does the period ($T$) of oscillation relate to the frequency?

Tip:

Always remember that maximum speed occurs at the equilibrium position ($x = 0$) in simple harmonic motion!